1 Ð@t 2015/01/17 (y) 23:00:19 ID:ctwIRbLQLU CA: 23:10
ɂ́B

^Cǧɂ
_@http://arxiv.org/pdf/quant-ph/0502053v1.pdf
̓̂ق̑ӂ{ɂĂ݂Ă͑̓悤Ƃ݂łB
2 Ð@t 2015/01/17 (y) 23:01:43 ID:ctwIRbLQLU
The role of the rigged Hilbert space in Quantum Mechanics
ʎq͊wRHSʂ

Rafael de la Madrid

Departamento de Fisica Teorica, Facultad de Ciencias, Universidad del Paıs Vasco,
48080 Bilbao, Spain
E-mail: wtbdemor@lg.ehu.es
3 Ð@t 2015/01/17 (y) 23:03:10 ID:ctwIRbLQLU
Abstract. There is compelling evidence that, when continuous spectrum is present,
the natural mathematical setting for Quantum Mechanics is the rigged Hilbert space
rather than just the Hilbert space. In particular, Diracfs bra-ket formalism is fully
implemented by the rigged Hilbert space rather than just by the Hilbert space. In this
paper, we provide a pedestrian introduction to the role the rigged Hilbert space plays
in Quantum Mechanics, by way of a simple, exactly solvable example. The procedure
will be constructive and based on a recent publication. We also provide a thorough
discussion on the physical significance of the rigged Hilbert space.

AuXgNg
AXyNgꍇ̗ʎq͊w̎RȐw@͒PȂqxgԂłȂ
rigged Hilbert space(ȉAuRHSv)ł邱ƂF߂𓾂Ȃ؋B
ɃfBbÑuPbgL@̓qxgԂRHSŏ\Sɉ߂B
̘_ł͒PŌɉƂɂʎq͊wRHSʂ߂Ă̐lɗĂ炤B܂RHS̕IӖڂ_B

4 Ð@t 2015/01/17 (y) 23:06:11 ID:ctwIRbLQLU
1. Introduction
It has been known for several decades that Diracfs bra-ket formalism is mathematically justified not by the Hilbert space alone, but by the rigged Hilbert space (RHS). This is the reason why there is an increasing number of Quantum Mechanics textbooks that
already include the rigged Hilbert space as part of their contents (see, for example,
Refs. [1]-[9]). Despite the importance of the RHS, there is still a lack of simple examples for which the corresponding RHS is constructed in a didactical manner. Even worse, there is no pedagogical discussion on the physical significance of the RHS. In this paper,
we use the one-dimensional (1D) rectangular barrier potential to introduce the RHS at the graduate student level. As well, we discuss the physical significance of each of the
ingredients that form the RHS. The construction of the RHS of such a simple model will unambiguously show that the RHS is needed at the most basic level of Quantum Mechanics.

P@_
fBbÑuPbgL@qxgԂł͐wIɐꂸRHSɂ邱Ƃ͂\NO疾炩ɂȂĂBʎq͊w̋ȏRHŜĂi[1-9nj
RHS͏dvȂ̂RHS㈓Iɍ\Pȗ͂܂񎦂ĂȂB
ƈƂɂ͕włRHS̏dvE炷邱Ƃɂċc_ȂĂȂB
ł1̔^ǃ|eVőwxRHS̓}B܂RHS\vfڍׂɋc_BRHŜ悤ȊȒPȃfō\邱Ƃɂʎq͊w̍łb̃xRHSKvƂƂ킩ɈႢȂB
5 Ð@t 2015/01/17 (y) 23:07:19 ID:ctwIRbLQLU
The present paper is complemented by a previous publication, Ref. [10], to which we
shall refer the reader interested in a detailed mathematical account on the construction
of the RHS of the 1D rectangular barrier. For a general background on the Hilbert
and the rigged Hilbert space methods, the reader may consult Ref. [11] and references
therein

̘_͉ߋoł[10]ɂĂڂ͂QƂꂽBqxgԂRHS̈ʓIwi[11]
6 Ð@t 2015/01/17 (y) 23:09:31 ID:ctwIRbLQLU
Diracfs bra-ket formalism was introduced by Dirac in his classic monograph [12].
Since its inception, Diracfs abstract algebraic model of bras and kets (from the bracket
notation for the inner product) proved to be of great calculational value, although there
were serious difficulties in finding a mathematical justification for the actual

calculations
within the Hilbert space, as Dirac [12] and von Neumann [13] themselves state in their
books [14]. As part of his bra-ket formalism, Dirac introduced the so-called Dirac delta
function, a formal entity without a counterpart in the classical theory of functions. It was L. Schwartz who gave a precise meaning to the Dirac delta function as a functional over a space of test functions [15]. This led to the development of a new branch of functional analysis, the theory of distributions. By combining von Neumannfs Hilbert space with the theory of distributions, I. Gelfand and collaborators introduced the RHS [16, 17]. It was already clear to the creators of the RHS that their formulation was the mathematical
support of Diracfs bra-ket formalism [18]. The RHS made its first appearance in the Physics literature in the 1960s [19, 20, 21], when some physicists also realized that the RHS provides a rigorous mathematical rephrasing of all of the aspects of Diracfs braket
formalism. Nowadays, there is a growing consensus that the RHS, rather than the@Hilbert space alone, is the natural mathematical setting of Quantum Mechanics [22].

fBbN̓uPbgL@𖼒[12]œBfBbN[12]@tHmC}[13]ꂼ꒘ŏqׂ悤ɁAuPbgL@̓qxgԂ̒ŎۂɌvZ鐔wIȐɂ͌ĂAvZɂ͂ւd󂷂̂ł邱Ƃ킩B
uPbgL@̈ꕔƂăfBbN̓֐Ƃ̂𓱓B͓IȊ֐_ɂ΂ȂBVc֐̐mȈӖA܂莎֐ԏ̔Ċ֐ł邱Ƃ𖾂炩ɂBɂĊ֐͂̐VA֐J񂳂ꂽBtHmC}̃qxgԂƒ֐тAQt@gRHS𓱓[16,17]. RHS̓fBbÑuPbgL@x̂ł̂͑nn҂ɂ͖炩[18]BRHSwɂꂽ̂1960N[19,20,21] ꕔ̕w҂RHSuPbgL@ׂ̂Ă̓ɐwIɐ̂ł邱Ƃ𗝉BqxgԂɂ̂łȂRHSʎq͊wɎRɑΉƂRZTXLĂB[22]
7 Ð@t 2015/01/18 () 10:53:20 ID:ctwIRbLQLU CA: 10:56
A note on semantics. The word griggedh in rigged Hilbert space has a nautical
connotation, such as the phrase gfully rigged ship;h it has nothing to do with any
unsavory practice such as gfixingh or predetermining a result. The phrase grigged
Hilbert@spaceh is a direct translation of the phrase gosnashchyonnoe Hilbertovo prostranstvoh
from the original Russian. A more faithful translation would be gequipped Hilbert
space.h Indeed, the rigged Hilbert space is just the Hilbert space equipped with
distribution theory\in Quantum Mechanics, to rig a Hilbert space means simply to
equip that Hilbert space with distribution theory. Thus, the RHS is not a replacement
but an enlargement of the Hilbert space.

pɂẴm[gB@RHSriggedƂ͂Ƃ΁uS~(rigged)DvƂ悤ɊCp

VAosnashchyonnoe Hilbertovo prostranstvo̒ł蒉Ȗ́Aꂽequipped
qxgԁBɁ@RHS̓qxgԂɒ֐𑕔̂̂Aɗʎq͊w
ł́BRHS̓qxgԂ̂ł͂ȂqxgԂĝB

The RHS is neither an extension nor an interpretation of the physical principles
of Quantum Mechanics, but rather the most natural, concise and logic language to
formulate Quantum Mechanics. The RHS is simply a mathematical tool to extract
and process the information contained in observables that have continuous spectrum.
Observables with discrete spectrum and a finite number of eigenvectors (e.g., spin) do
not need the RHS. For such observables, the Hilbert space is sufficient. Actually, as
we shall explain, in general only unbounded observables with continuous spectrum need
the RHS.

RHS͗ʎq͊w̕I̊gł߂łȂAłRŊȌŘ_Iɗʎq͊w킷tB@RHS͘AXyNgIuU[oûЂ鐔wc[ɂȂBUXyNgLȌŗLxNgiƂ΃XsjRHSKvƂȂBqxgԂŏ\BɎ悤ɈʂɋEȂAXyNgIuU[ouRHSKvƂB
8 Ð@t 2015/01/18 () 10:55:00 ID:ctwIRbLQLU
The usefulness of the RHS is not simply restricted to accounting for Diracfs braket
formalism. The RHS has also proved to be a very useful research tool in the
quantum theory of scattering and decay (see Ref. [11] and references therein), and
in the construction of generalized spectral decompositions of chaotic maps [23, 24]. In
fact, it seems that the RHS is the natural language to deal with problems that involve
continuous and resonance spectra.

RHS̗LṕAfBbÑuPbgL@ۏ؂邱Ƃł͂ȂBRHS͎Uƕ[11]AJIX}bv̈ʓIXyNg[23.24]̗ʎq_ɂւ֗ł邱Ƃ킩ĂB@ہARHS͘AXyNg̖鎩RȂƂ΂ł悤ɎvB
9 Ð@t 2015/01/18 () 11:24:03 ID:ctwIRbLQLU
Loosely speaking, a rigged Hilbert space (also called a Gelfand triplet) is a triad of
spaces
 H ~ (1.1)
such that H is a Hilbert space,  is a dense subspace of H [25], and ~ is the space
of antilinear functionals over  [26]. Mathematically,  is the space of test functions,
and ~ is the space of distributions. The space ~ is called the antidual space of .
Associated with the RHS (1.1), there is always another RHS,
 H  , (1.2)
where  is called the dual space of  and contains the linear functionals over  [26].

(1.1)
@H̓qxgԂŁAHfȕ[25]A^X́@̔@Ċ֐̋ԁB@wIɂ΃͎֐̋ԂŃOX͒֐̋ԁBOX́@̔o΋ԂƌĂ΂B
RHS(1.1)ɕtĕʂRHS˂ɂ
(1.2)
f̓̑o΋ԂƂ΂A̐Ċ֐ȂɎ߂Ă
10 hirota 2015/01/18 () 11:45:31 ID:mxZWPl0EEs
̑o΋ԁ  ^Ċ֐̏W
̔o΋ԁ̋o΋ԁ  ^Ċ֐̏W(    ̋𕡑f)
11 Ð@t 2015/01/18 () 14:54:08 ID:ctwIRbLQLU CA: 02/07 (y) 18:04
The basic reason why we need the spaces  and ~ is that the bras and kets
associated with the elements in the continuous spectrum of an observable belong,
respectively, to  and ~ rather than to H. The basic reason reason why we need
the space  is that unbounded operators are not defined on the whole of H but only
on dense subdomains of H that are not invariant under the action of the observables.
Such non-invariance makes expectation values, uncertainties and commutation relations
not well defined on the whole of H. The space  is the largest subspace of the Hilbert
space on which such expectation values, uncertainties and commutation relations are
well defined.

'ԂƃOX KvȗŔAuƃPbgIuU[ouAXyNg̏ꍇɂHłȂꂼꃳ'ƃOX ƌтĂ邩炾B
ԂKvȊ{IȗŔALEȉZq̓IuU[oupĂsςł悤HfȕԂŒĂAĤׂĂŒĂ̂ł͂Ȃ炾B
ĤׂĂɂĒĂ킯łȂҒlAsm萫A֌ẂA̕sϐɂ萬BƂ͊ҒlAsm萫A֌Wƒłő̕ԂB

ӁFLE̒̐referenceɂB
12 Ð@t 2015/01/18 () 15:17:31 ID:ctwIRbLQLU
The original formulation of the RHS [16, 17] does not provide a systematic
procedure to construct the RHS generated by the Hamiltonian of the SchrNodinger
equation, since the space  is assumed to be given beforehand. Such systematic
procedure is important because, after all, claiming that the RHS is the natural setting
for Quantum Mechanics is about the same as claiming that, when the Hamiltonian
has continuous spectrum, the natural setting for the solutions of the SchrNodinger
equation is the RHS rather than just the Hilbert space. The task of developing a
systematic procedure to construct the RHS generated by the SchrNodinger equation was
undertaken in Ref. [11]. The method proposed in Ref. [11], which was partly based
on Refs. [19, 20, 21], has been applied to two simple three-dimensional potentials, see
Refs. [27, 28], to the three-dimensional free Hamiltonian, see Ref. [29], and to the 1D
rectangular barrier potential, see Ref. [10]. In this paper, we present the method of
Ref. [11] in a didactical manner.

ŏɍlĂꂽRHS[16,17]̓Ԃ͂^ĂƂƂ납ô̂ŁAVWK[̃n~gjAŐRHSnė^悤ȏł͂ȂB
VWK[̃n~gjAŐRHSnē邱ƂdvƌƂ́AAXyNg̗ʎq͊wRHSRȐwłƌƂƌǓB[11]͂߂̘_RHSȒPȌnœ邱ƂĂB̘_ł[11]̕@nĐ邱ƂɂB
13 Ð@t 2015/01/18 () 15:20:33 ID:ctwIRbLQLU
The organization of the paper is as follows. In Sec. 2, we outline the major reasons
why the RHS provides the mathematical setting for Quantum Mechanics. In Sec. 3, we
recall the basics of the 1D rectangular potential model. Section 4 provides the RHS of
this model. In Sec. 5, we discuss the physical meaning of each of the ingredients that
form the RHS. In Sec. 6, we discuss the relation of the Hilbert space spectral measures
with the bras and kets, as well as the limitations of our method to construct RHSs.
Finally, Sec. 7 contains the conclusions to the paper.

̘_2͂́ARHSʎq͊wwł邨ȗRqׂB@R͂Ł@Pǃf𕜏KB@S͂ł̌nRHS^B@T͂RHS̍\vf̕IӖc_BU͂ŃqxgԂ̃XyNgxƃuPbg̊֌WƁÂ悤ɂRHS\@̌Ec_BV͂͌_B
14 Ð@t 2015/01/18 () 18:26:23 ID:ctwIRbLQLU
2. Motivating the rigged Hilbert space
The linear superposition principle and the probabilistic interpretation of Quantum
Mechanics are two major guiding principles in our understanding of the microscopic
world. These two principles suggest that the space of states be a linear space (which
accounts for the superposition principle) endowed with a scalar product (which is used
to calculate probability amplitudes). A linear space endowed with a scalar product is
called a Hilbert space and is usually denoted by H [30].

Q́@ȂRHSȂ̂
d˂킹̌Ɗm߂ʎq͊w̓wB

œςԂ̓qxgԂƂ΂ʏHƋLB[30]
15 Ð@t 2015/01/18 () 18:35:18 ID:ctwIRbLQLU
In Quantum Mechanics, observable quantities are represented by linear, selfadjoint
operators acting on H. The eigenvalues of an operator represent the possible
values of the measurement of the corresponding observable. These eigenvalues, which
mathematically correspond to the spectrum of the operator, can be discrete (as the
energies of a particle in a box), continuous (as the energies of a free, unconstrained
particle), or a combination of discrete and continuous (as the energies of the Hydrogen
atom).

ʎq͊ẘϑ\ȗʂHł͂炭ŎȋȃIy[^ł킳B
Iy[^̌ŗLlAIuU[oȗłƂ肤l킷B
ŗLl͐wł̓Iy[^̃XyNgɑΉAUi@̂Ȃ̗q̃GlM[jAAi@Rq̃GlM[jAUƘÂ݂킹i@zqƓdq̌ñGlM[jƂ肤B
16 Ð@t 2015/01/18 () 18:54:42 ID:ctwIRbLQLU CA: 02/07 (y) 18:08
When the spectrum of an observable A is discrete and A is bounded [31], then A
is defined on the whole of H and the eigenvectors of A belong to H. In this case, A
can be essentially seen as a matrix. This means that, as far as discrete spectrum is
concerned, there is no need to extend H. However, quantum mechanical observables are
in general unbounded [31] and their spectrum has in general a continuous part. In order
to deal with continuous spectrum, textbooks usually follow Diracfs bra-ket formalism,
which is a heuristic generalization of the linear algebra of Hermitian matrices used for
discrete spectrum. As we shall see, the mathematical methods of the Hilbert space are
not sufficient to make sense of the prescriptions of Diracfs formalism, the reason for
which we shall extend the Hilbert space to the rigged Hilbert space.

IuU[ouUIŋEꍇɂ́A͂gSԂŒǍŗLxNg݂͂Ȃgɂ͂Ă[31]B@@̏ꍇ͍sƂ݂ȂBȂ킿UXyNgȂ΂ggKv͂ȂB
ȂAʎq͊w̃IuU[ou͈ʂɂ͔LE[31]AXyNgɂ͘AȕB
AXyNgɂ͋ȏ͂ĂAUXyNgŎgG~[gs̐㐔̗ސEƂăfBbÑuPbgL@ĂBꂩqxgԂ̐wI@ł̓fBbŇ̕@Ӗɂ͏\łȂqgr֊g邱ƂKvł邱Ƃ݂悤B
17 Ð@t 2015/01/18 () 19:04:27 ID:ctwIRbLQLU
For pedagogical reasons, we recall the essentials of the linear algebra of Hermitian
matrices before proceeding with Diracfs formalism.

IzG~[gs̐㐔𕜏K悤

2.1. Hermitian matrices@@G~[gsi󗪁j
2.2. Diracfs bra-ket formalism@fBbÑuPbgL@
(i)(ii)(iii) @i󗪁j

(iv) Like in the case of two finite-dimensional matrices, all algebraic operations such as the commutator of two observables A and B,
[A,B] = AB | BA, (2.16)
are always well defined.

L̍s̏ꍇƓAӂ̃IuU[ouƂåԂ̌֌WȂǂ̑㐔I֌W͂˂ɂ悭ĂB
18 kafuka 2015/01/18 () 19:32:07 ID:Utjkuz.Osc

ǂ݂Ǝv܂B
19 Ð@t 2015/01/18 () 19:59:41 ID:ctwIRbLQLU
2.3. The need of the rigged Hilbert space
2.3@RHSKvȂ

In Quantum Mechanics, observables are usually given by differential operators. In the
Hilbert space framework, the formal prescription of an observable leads to the definition
of a linear operator as follows: One has to find first the Hilbert space H, then one sees
on what elements of H the action of the observable makes sense, and finally one checks
whether the action of the observable remains in H. For example, the position observable
Q of a 1D particle is given by
Qf(x) = xf(x) . (2.17)
The Hilbert space of a 1D particle is given by the collection of square integrable
functions,
L2 = \{f(x) | Z
|
dx |f(x)|2 < \} , (2.18)

ʎq͊w̃IuU[ou͂ӂZqł邱ƂB
qxgԂ̘gł͐Zq͈̒ȉ̂Ƃ
qxgԂg܂݂Bɂĝǂ̗vf̂ŃIuU[ou̕ԂHӖȂ݂BŌɕԂg̒ɂƂǂ܂Ă邩݂B
Ƃ΂P̗q̍Wp̖
Qf(x) = xf(x) . (2.17)
Pq̃qxgԂ́Aϕ\Ȋ֐ŁA
(2.18)
p̕Ԃ́AIɂׂ͂ĂL^2̗vfŒĂ邯ǂÂ悤ȕԂ̗vfɂĂłȂ΂kOQɋAĂȂB
(2.19)
20 i}Xe 2015/01/18 () 20:24:13 ID:Utjkuz.Osc
the action of the observable
ăIuU[ouZqԃxNgɍp邱Ƃł傤H
21 Ð@t 2015/01/18 () 23:20:10 ID:ctwIRbLQLU
The space D(Q) is the domain of the position operator. Domain (2.19) is not the whole
of L2, since the function g(x) = 1/(x + i) belongs to L2 but not to D(Q); as well,
Q is an unbounded operator, because kQgk = ; as well, QD(Q) is not included in
D(Q), since h(x) = 1/(x2 + 1) belongs to D(Q) but Qh does not belong to D(Q). The
denseness and the non-invariance of the domains of unbounded operators create much
trouble in the Hilbert space framework, because one has always to be careful whether
formal operations are valid. For example, Q2 = QQ is not defined on the whole of L2,
not even on the whole of D(Q), but only on those square integrable functions such that
x2f L2. Also, the expectation value of the measurement of Q in the state ',
(',Q') , (2.20)
is not finite for every ' L2, but only when ' D(Q). Similarly, the uncertainty of
the measurement of Q in ',
'Q = p(',Q2') | (',Q')2 , (2.21)
is not defined on the whole of L2.

D(Q) WZq̗̈ƂȂB
̈ (2.19) ́@L^2Ŝł͂ȂBƂΊ֐ g(x) = 1/(x + i) L^2 ɂ邪A D(Q)ɂ͂܂ȂB
܂||Qg|| = @Ȃ̂Q ͋ÊȂiLEłȂjIy[^łB
QD(Q) ͗̈Ɋ܂܂ȂBƂh(x) = 1/(x^2 + 1)D(Q)ɂ邪AQh D(Q)ɂ܂ȂB
LEłȂIy[^fƉŽʂ̈ɂĂȂƁï̔sϐj̓qxgԂ
lۂ̑傫ȏQɂȂBƂ̂́AZqsρiZʂ̈ɂǂĂ邱Ɓjł邩ǂ
ɐ_oɂȂȂƂȂ炾B
Ƃ Q^2 = QQ L^2Ŝł͒ȂAD(Q)ŜłłȂBł̂
x2f L^2. ł悤ȓϕ֐ɂĂɂȂB

܂p̑̊Ғl
(,Q) , (2.20)
ׂ͂Ẵ L^2ɂėLȂ킯łȂALɂȂ̂̓ D(Q)ɂĂɌB

lɂp̑̕sm肳
, (2.21)
͑S L^2Œł킯ł͂ȂB
22 Ð@t 2015/01/19 () 18:32:28 ID:ctwIRbLQLU
On the other hand, if we denote the momentum observable by
Pf(x) = |i@d/dx@f(x) , (2.22)
then the product of P and Q, PQ, is not defined everywhere in the Hilbert space, but
only on those square integrable functions for which the quantity
PQf(x) = |i@d/dx@xf(x) = |i (f(x) + xf(x)) (2.23)
makes sense and is square integrable. Obviously, PQf makes sense only when f is
differentiable, and PQf remains in L2 only when f, f and xf are also in L2; thus,
PQ is not defined everywhere in L2 but only on those square integrable functions that
satisfy the aforementioned conditions. Similar domain concerns arise in calculating the
commutator of P with Q.

^
Pf(x) = |i@d/dx@f(x) , (2.22)
ɂPQ̐PQ́AqxgԂ̂ǂłĂ킯ł͂ȂA
ϕ\֐ł
PQf(x) = |i@d/dx@xf(x) = |i (f(x) + xf(x)) (2.23)
ɈӖϕ\ł悤ȂɂĂB

PQ́A\łȂΖ炩ɈӖȂȂB
PQ́A f, f,xf݂L^2ɂƂAL^2ɂB
ā@PQL^2̂ƂŒĂ̂łȂ
L̂悤ȏϕ\Ȋ֐ɂĂĂ
Ƃ킩
lPQ̌vZۂɂ̈ɂĂ̌ONB
23 Ð@t 2015/01/19 () 18:38:04 ID:ctwIRbLQLU
As in the case of the position operator, the domain D(A) of an unbounded operator
A does not coincide with the whole of H [32], but is just a dense subspace of H [25];
also, in general D(A) does not remain invariant under the action of A, that is, AD(A)
is not included in D(A). Such non-invariance makes expectation values,
(,A) , (2.24)
uncertainties,

and algebraic operations such as commutation relations not well defined on the whole
of the Hilbert space H [34]. Thus, when the position, momentum and energy operators
Q, P, H are unbounded, it is natural to seek a subspace  of H on which all of these
physical quantities can be calculated and yield meaningful, finite values. Because the
reason why these quantities may not be well defined is that the domains of Q, P and
H are not invariant under the action of these operators, the subspace  must be such
that it remains invariant under the actions of Q, P and H. This is why we take as 
the intersection of the domains of all the powers of Q, P and H [19]:
(2.26)

WZqŐ悤ɁALEłȂZqÄ̗D(A)́AHŜƂ͈vH̕
fȕԂB[32]B

̂悤ɕsςłȂƂҒl
(,A) , (2.24)
sm肳
, (2.25)
֌Ŵ悤ȑ㐔ZH̑SԂł͒łȂ[34].

ā@WA^ʁAGlM[Zq@Q, P, H LEłȂꍇɂ́A̕ʂ
ӖLlɌvZ悤H̕ԂTƂRB

̉Zq͂܂łȂAȂȂQ,P,Ḧ̗悪̉Zq̍pɂsςł͂Ȃ炾B@߂镔ԂP,Q,ĤׂĂׂ̂̍pŕsςɂƂǂ܂悤Ȃ̂łȂ΂ȂȂB [19]:
(2.26)
24 Ð@t 2015/01/19 () 18:42:02 ID:ctwIRbLQLU
ɂ́B
>>20
͂BIuU[ouZqԃxNgɍpĂłxNĝƂƉĂ܂B
󂵂ȂlĂ邽ߖ󂪂Ă܂BmȂ̂͒ʏ̐pƂƑ܂B䒍ӂB
25 Ð@t 2015/01/19 () 18:50:44 ID:ctwIRbLQLU
This space is known as the maximal invariant subspace of the algebra generated by Q, P
and H, because it is the largest subdomain of the Hilbert space that remains invariant
under the action of any power of Q, P or H,
A , A = Q, P,H . (2.27)

̋Ԃ́@Q,P,Hł㐔̍ősϕԁ@ƂĒmĂ̂BȂ킿
̋Ԃ̓qxgԂ̕Ԃ̂Q,P,ĤǂȂׂ̍płsςł悤ȍő̂̂B
A , A = Q, P,H . (2.27)
26 Ð@t 2015/01/19 () 19:22:07 ID:ctwIRbLQLU
On , all physical quantities such as expectation values and uncertainties can be
associated well-defined, finite values, and algebraic operations such as the commutation
relation (2.16) are well defined. In addition, the elements of are represented by
smooth, continuous functions that have a definitive value at each point, in contrast
to the elements of H, which are represented by classes of functions which can vary
arbitrarily on sets of zero Lebesgue measure.

̏ł́AҒlsmƂʂ͂悭ALlŁA֌Wi2.16ĵ悤ȑ㐔Z͂悭łĂB
āA̗vf͊炩ŘAŊe_Ō܂l悤Ȋ֐ŕ\B
Ag̗vf́Ax[Ox[̏Wł͔CӂɂĂ悤Ȋ֐̃NXŕ\B
27 Ggs[ 2015/01/19 () 20:07:27 ID:7BRCjW36HA
Xbh̎|OĐ\ȂłA玸炵܂B

̉p͕@ƂĂ͓Փx͂ǂꂭ炢ł傤H

\Iɂ͓Ȃ̂͂ȂɌ̂ŁiC邾ȂłjA
pƗ͂ł邩ȂƎvA
a󌩂Ȃ̗Kɂ͂ǂȂƎv̂Ŏ₢܂iAɂ͂܂܂EEEj
28 Ð@t 2015/01/19 () 20:46:03 ID:ctwIRbLQLU
Not only there are compelling reasons to shrink the Hilbert space H to , but,
as we are going to explain now, there are also reasons to enlarge H to the spaces
^~ and of Eqs. (1.1) and (1.2).

qxgԂgɏk߂邾łȂAꂩ悤ɂg^Xi1.1jƃfi1.2jɊg債Ȃ΂ȂȂ
ƂɂB
29 Ð@t 2015/01/19 () 20:47:50 ID:ctwIRbLQLU
When the spectrum of A has a continuous part,
prescriptions (2.11b) and (2.10b) associate a bra <a| and a ket |a> to each element a of the
continuous spectrum of A.
Obviously, the bras <a| and kets |a> are not in the Hilbert
space [35], and therefore we need two linear spaces larger than the Hilbert space to
accommodate them.
It turns out that the bras and kets acquire mathematical meaning
as distributions.
More specifically, the bras <a| are linear functionals over the space
, and the kets |a> are antilinear functionals over the space . That is, <a| and
|a> ^~.

̃XyNgɘAȕꍇɂ͘AȒlɂ̓uƃPbgĂ͂߂B

LԂKvɂȂB
uƃPbg͐w̒֐̈ӖƂ킩B܂u͐Ċ֐APbg͔Ċ֐B
<a|
|a> ^~.
30 Ð@t 2015/01/19 () 21:41:43 ID:ctwIRbLQLU
Aside from providing mathematical concepts such as self-adjointness or
unitarity, the Hilbert space plays a very important physical role, namely H selects
the scalar product that is used to calculate probability amplitudes.
The subspace 
contains those square integrable functions that should be considered as physical, because
any expectation value, any uncertainty and any algebraic operation can be calculated
for its elements, whereas this is not possible for the rest of the elements of the Hilbert
space.
The dual space  and the antidual space ^~ contain respectively the bras and
the kets associated with the continuous spectrum of the observables.
These bras and
kets can be used to expand any @  as in Eq. (2.12). Thus, the rigged Hilbert space,
rather than the Hilbert space alone, can accommodate prescriptions (2.10a)-(2.16) of
Diracfs formalism.

ȋƂj^[̂悤ȐwITO͂܂ʂɂ킦Ƃ
qxgԂ́AmUvZ邽߂̓ς^AƂdvȕIڂʂB
ҒlAsm肳A㐔ŽvZvfׂ̂ĂɂĂł邱ƂA̓ϕ֐܂
Ԃ͕Iƍl邱ƂłBvZ̓qxgԂ̂ق̗vfł͕s\B
o΋ԃfƔo΋ԃ^X͂ꂼAŗLl̃uAPbg܂łB
uAPbǵi2.12jł̂悤ɔCӂ̃ @ WĴɎgB
ăqxgԂł͂߂RHS̓fBbN̋L@ɂ
31 Ð@t 2015/01/20 () 09:39:13 ID:ctwIRbLQLU
It should be clear that the rigged Hilbert space is just a combination of the Hilbert
space with distribution theory. This combination enables us to deal with singular objects
such as bras, kets, or Diracfs delta function, something that is impossible if we only use
the Hilbert space.
Even though it is apparent that the rigged Hilbert space should be an essential
part of the mathematical methods for Quantum Mechanics, one may still wonder if the
rigged Hilbert space is a helpful tool in teaching Quantum Mechanics, or rather is a
technical nuance. Because basic quantum mechanical operators such as P and Q are
in general unbounded operators with continuous spectrum [36], and because this kind
of operators necessitates the rigged Hilbert space, it seems pertinent to introduce the
rigged Hilbert space in graduate courses on Quantum Mechanics.
From a pedagogical standpoint, however, this sectionfs introduction to the rigged
Hilbert space is not sufficient. In the classroom, new concepts are better introduced
by way of a simple, exactly solvable example. This is why we shall construct the RHS
of the 1D rectangular barrier system. We note that this system does not have bound
states, and therefore in what follows we shall not deal with discrete spectrum.

RHSqxgԂƒ֐_ɂ̂ɂȂƂ͖炩B
qxgԂł͂łȂuAPbgAfBbÑf^֐Ƃ
ɂȂ邱Ƃłł悤ɂȂB
RHSʎq͊w̐wI@̗vł邱Ƃ炩ƂĂARHSʎq͊w
ۂɎgʂŖ𗧂ɂ͋^₪낤BRHS͑w@ŋ̂KɎvB
IȌńȀ͂̐͏\łȂBuł͐VTO͗eՂɌɉ
ŐB̂߂P̒@̕ǂ̌nRHS\Ă݂悤B
̌nł͑Ԃ͂Ȃ܂藣UIȃXyNg͂ȂB
32 Ð@t 2015/01/20 () 17:07:07 ID:ctwIRbLQLU
ɂ́B
>>27
>a󌩂Ȃ̗Kɂ͂ǂȂƎv̂Ŏ₢܂

ӂƂȗɑłĂ܂B
p̎ɂ͂悭ȂeAƐSz܂B

Ȃł

EDirac "The Principles of Quantum Mechanics" 񂾂ƂełȂpꂪł΂炵Ƃ܂B

EuZpŉȊw׋Γ񒹂łˁBvƖ^w̐w̐搶ɘb
uw̖{ŉpmߎSȂƂɂȂBvƂ͂ꂽ̂vo܂B
33 Ð@t 2015/01/20 () 22:46:33 ID:ctwIRbLQLU
2.4. Representations
2.4 \

In working out specific examples, the prescriptions of Diracfs formalism have to be
written in a particular representation. Thus, before constructing the RHS of the 1D
rectangular barrier, it is convenient to recall some of the basics of representations.
In Quantum Mechanics, the most common of all representations is the position
representation, sometimes called the x-representation. In the x-representation, the
position operator Q acts as multiplication by x. Since the spectrum of Q is (|,),
the x-representation of the Hilbert space H is given by the space L2. In this paper, we
shall mainly work in the position representation.
-------------------------
1ǂ̗ɂ͂܂ɕ\̊b𕜏K悤B
߃fBbNL@̕\ǂꂩɂ߂KvB
Ƃ悭͍̂W\i-\jB
W\ł͍WZqQ͂邱ƂƂēB
Q̃XyNg (|,)@qxgHx-\́@L^2ԂƂė^B
̘_ł͍W\gĂB
34 Ð@t 2015/01/20 () 22:48:49 ID:ctwIRbLQLU
In general, given an observable B, the b-representation is that in which the operator
B acts as multiplication by b, where the bfs denote the eigenvalues of B. If we denote
the spectrum of B by Sp(B), then the b-representation of the Hilbert space H is given
by the space L2(Sp(B), db), which is the space of square integrable functions f(b) with
b running over Sp(B). In the b-representation, the restrictions to purely continuous
spectrum of prescriptions (2.10a)-(2.13) become
<b|A|a> = a<b|a> , (2.28a)
<a|A|b> = a<a|b> , (2.28b)
<b|> = Z da <b|a><a|> , (2.28c)
(b | b) = <b|b> =@ da <b|a><a|b>. (2.28d)
The gscalar producth <b|a> is obtained from Eq. (2.28a) as the solution of a differential
eigenequation in the b-representation. The <b|a> can also be seen as transition elements
from the a- to the b-representation. Mathematically, the <b|a> are to be treated as
distributions, and therefore they often appear as kernels of integrals. In this paper, we
shall encounter a few of these gscalar productsh such as <x|p>, <x|x> and <x|E}>l,r.

--------------------
ʂɁAIuU[ouBƁA|\Ƃ́@B̍pŗLl|΂悤ȕ\B
B̃XyNgSpiB)ƂƁ@qxgHb-\͋ԁ@L^2iSp(B),db)łB
œϕ\֐iĵSp(B)̏𑖂BAȃXyNg̏ꍇ@|\ł
(2.10a)-(2.13) ͂̂悤ɂȂB

<b|A|a> = a<b|a> , (2.28a)
<a|A|b> = a<a|b> , (2.28b)
<b|> = da <b|a><a|> , (2.28c)
(b | b) = <b|b> =@ da <b|a><a|b>. (2.28d)

uρv<b|a>(2.28)̂悤ɁAb-\̔ŗLl̉ƂēB
<b|a>́Aa-\b-\ւ̕ϊWƂ݂ȂB
wł<b|a> ͒֐ƂĈAϕ̊jɂ悭B
̘_ł<x|p>, <x|x> <x|E}>l,r ̂悤ȁuρvłĂB
35 Ð@t 2015/01/20 () 22:52:18 ID:ctwIRbLQLU
3. Example: The one-dimensional rectangular barrier potential
3.@@1ǃ|eV

The example we consider in this paper is supposed to represent a spinless particle moving
in one dimension and impinging on a rectangular barrier. The observables relevant to
this system are the position Q, the momentum P, and the Hamiltonian H. In the
position representation, Q and P are respectively realized by the differential operators
(2.17) and (2.22), whereas H is realized by
(3.1)
where V(x)=
0 | < x < a
V0 a < x < b
0 b < x < (3.2)
is the 1D rectangular barrier potential.

---------------
XŝȂqǂ̂1^Ƃ肠悤B
IuU[ou͍WQ@^P@n~gjAHŁAW\B
Q́i2.17jṔi2.22jŁ@H
(3.1)
V(x)=
@0 | < x < a
@V0 a < x < b
@0 b < x < (3.2)
łB
36 Ð@t 2015/01/21 () 19:08:24 ID:ctwIRbLQLU CA: 02/07 (y) 20:35
Formally, these observables satisfy the following
commutation relations:
, (3.3a)
, (3.3b)
. (3.3c)
Since our particle can move in the full real line, the Hilbert space on which the
differential operators (2.17), (2.22) and (3.1) should act is L2 of Eq. (2.18). The
corresponding scalar product is
, (3.4)
where    denotes the complex conjugate of f(x).
The differential operators (2.17), (2.22) and (3.1) induce three linear operators on
the Hilbert space L^2. These operators are unbounded [10], and therefore they cannot
be defined on the whole of L^2, but only on the following subdomains of L2 [10]:
D(Q) ={f L^2 | xf L^2}, (3.5a)
D(P) ={f L^2 | f AC, Pf L^2}, (3.5b)
D(H) ={f L^2 | f AC^2, Hf L^2}, (3.5c)
where, essentially, AC is the space of functions whose derivative exists, and AC2 is the
space of functions whose second derivative exists (see Ref. [10] for more details). On
these domains, the operators Q, P and H are self-adjoint [10].
---------------
̃IuU[oǔ͎֌W𖞂B
, (3.3a)
, (3.3b)
. (3.3c)
q͐̂ǂɂĂZq(2.17), (2.22) (3.1) ͂炭qxg
(2.18)L^2B

Ής
, (3.4)
f(x)̕f킵ĂB

Zq (2.17), (2.22) (3.1) ̓qxgL^2݂̂̐ZqB
̉Zq͗LEłȂ [10],đS L^2ł͒ꂸɎ̂悤ȕԂŒ [10]:
D(Q) ={f L^2 | xf L^2}, (3.5a)
D(P) ={f L^2 | f AC, Pf L^2}, (3.5b)
D(H) ={f L^2 | f AC^2, Hf L^2}, (3.5c)
AC ͓֐݂悤Ȋ֐̋ԂB AC^2 ͓K֐݂悤Ȋ֐̋ԂB (ڂ [10]QƁj. ̗̈łQ, P H ͎ȋ [10]B
37 Ð@t 2015/01/21 () 19:10:40 ID:ctwIRbLQLU
In our example, the eigenvalues (i.e., the spectrum) and the eigenfunctions of the
observables are provided by the Sturm-Liouville theory. Mathematically, the eigenvalues
and eigenfunctions of operators extend the notions of eigenvalues and eigenvectors of a
matrix to the infinite-dimensional case. The Sturm-Liouville theory tells us that these
operators have the following spectra [10]:
Sp(Q) = (|,) , (3.6a)
Sp(P) = (|,) , (3.6b)
Sp(H) = [0,) . (3.6c)
-----------------
̗ł,ŗLl (܂XyNg)ƃIuU[oǔŗL֐ Sturm-Liouville _
߂BwIɂ̓Iy[^̌ŗLlƌŗL֐͖sŋLB
Sturm-Liouville ̕@Iy[^͎̃XyNg [10]:
Sp(Q) = (|,) , (3.6a)
Sp(P) = (|,) , (3.6b)
Sp(H) = [0,) . (3.6c)
38 Ð@t 2015/01/21 () 19:14:04 ID:ctwIRbLQLU CA: 01/22 () 12:30
These spectra coincide with those we would expect on physical grounds. We expect the
possible measurements of Q to be the full real line, because the particle can in principle
reach any point of the real line. We also expect the possible measurements of P to be
the full real line, since the momentum of the particle is not restricted in magnitude or
direction. The possible measurements of H have the same range as that of the kinetic
energy, because the potential does not have any wells of negative energy, and therefore
we expect the spectrum of H to be the positive real line.
----------------
͕ƂvBW Q ׂ͎̑Ăɂ킽邾낤AȂȂ痱q͌Ƃ
̂ǂ̓_ɂ邾낤B^ P ׂ͎̑Ăɂ킽邾낤AȂȂΎRq
^ʂ̂ƕɐ͂ȂBGlM[H͉̑^GlM[̑ƓXyNg
낤AȂȂ}CiX̃GlM[˂݂͑ĂȂ̂ŁBGlM[ P ׂ͎̑̐̂Ăɂ킽邾낤B
39 Ð@t 2015/01/21 () 19:17:57 ID:ctwIRbLQLU
To obtain the eigenfunction corresponding to each eigenvalue, we have to solve the
eigenvalue equation (2.10b) for each observable. Since we are working in the position
representation, we have to write Eq. (2.10b) in the position representation for each
observable:
<x|Q|x> = x<x|x> , (3.7a)
<x|P|p> = p<x|p> , (3.7b)
<x|H|E> = E<x|E> . (3.7c)
By recalling Eqs. (2.17), (2.22) and (3.1), we can write Eqs. (3.7a)-(3.7c) as
x<x|x> = x<x|x>, (3.8a)
, (3.8b)
(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V (x))<x|E> = E<x|E>. (3.8c)
For each position x, Eq. (3.8a) yields the corresponding eigenfunction of Q as a delta
function,
<x|x>=(x | x) . (3.9)
For each momentum p, Eq. (3.8b) yields the corresponding eigenfunction of P as a plane
wave,
<x|p> = \frac{e^{\frac{ipx}{\hbar}}{\sqrt{2\pi\hbar}. (3.10)
For each energy E, Eq. (3.8c) yields the following two linearly independent
eigenfunctions [10]:
-----------------------

ŗLlƑΉŗLxNgꂼꋁ߂邽߂ɂ
ŗLl (2.10b) ꂼ̃IuU[ouɂĂƂ˂ΐȂ
͍W\Ă邩 Eq. (2.10b)W\ŏȂƂȂ:
<x|Q|x> = x<x|x> , (3.7a)
<x|P|p> = p<x|p> , (3.7b)
<x|H|E> = E<x|E> . (3.7c)
(2.17), (2.22) (3.1)ɒӂāA (3.7a)-(3.7c)
x<x|x> = x<x|x>, (3.8a)
, (3.8b)
. (3.8c)
ʒux'ƂɎ (3.8a) Q̌ŗL֐֐A
<x|x>=(x | x) . (3.9)
ƂB
^ pƂ, (3.8b) P ̑ΉŗL֐𕽖ʔg,
. (3.10)
ƂB
GlM[ EƂ, (3.8c) ͎̂ӂ̐ƗȌŗL֐ [10]B
40 Ð@t 2015/01/21 () 19:19:41 ID:ctwIRbLQLU
To obtain the eigenfunction corresponding to each eigenvalue, we have to solve the
eigenvalue equation (2.10b) for each observable. Since we are working in the position
representation, we have to write Eq. (2.10b) in the position representation for each
observable:
<x|Q|x> = x<x|x> , (3.7a)
<x|P|p> = p<x|p> , (3.7b)
<x|H|E> = E<x|E> . (3.7c)
By recalling Eqs. (2.17), (2.22) and (3.1), we can write Eqs. (3.7a)-(3.7c) as
x<x|x> = x<x|x>, (3.8a)
, (3.8b)
(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V (x))<x|E> = E<x|E>. (3.8c)
For each position x, Eq. (3.8a) yields the corresponding eigenfunction of Q as a delta
function,
<x|x>=(x | x) . (3.9)
For each momentum p, Eq. (3.8b) yields the corresponding eigenfunction of P as a plane
wave,
.
For each energy E, Eq. (3.8c) yields the following two linearly independent
eigenfunctions [10]:
-----------------------

ŗLlƑΉŗLxNgꂼꋁ߂邽߂ɂ
ŗLl (2.10b) ꂼ̃IuU[ouɂĂƂ˂ΐȂ
͍W\Ă邩 Eq. (2.10b)W\ŏȂƂȂ:
<x|Q|x> = x<x|x> , (3.7a)
<x|P|p> = p<x|p> , (3.7b)
<x|H|E> = E<x|E> . (3.7c)
(2.17), (2.22) (3.1)ɒӂāA (3.7a)-(3.7c)
x<x|x> = x<x|x>, (3.8a)
, (3.8b)
. (3.8c)
ʒux'ƂɎ (3.8a) Q̌ŗL֐֐A
<x|x>=(x | x) . (3.9)
ƂB
^ pƂ, (3.8b) P ̑ΉŗL֐𕽖ʔg,
. (3.10)
ƂB
GlM[ EƂ, (3.8c) ͎̂ӂ̐ƗȌŗL֐ [10]B
41 Ð@t 2015/01/21 () 19:21:03 ID:ctwIRbLQLU CA: 01/22 () 13:35
GlM[ EƂ, (3.8c) ͎̂ӂ̐ƗȌŗL֐ [10]B
(3.11a)
(3.11b)
where
,
, (3.12)

and where the coefficients that appear in Eqs. (3.11a)-(3.11b) can be easily found by
the standard matching conditions at the discontinuities of the potential [10]. Thus, in
contrast to the spectra of Q and P, the spectrum of H is doubly degenerate.
-------------------
(3.11a)-(3.11b) ̌W̓|eV̕sAȂƂł̕Ŵ킹ɂ
ȒPɋ߂B Q A P̃XyNgƂׂ H ̃XyNg͓dɏkdĂB
42 Ð@t 2015/01/21 () 22:05:31 ID:ctwIRbLQLU CA: 01/22 () 13:33
Physically, the eigenfunction <x|E+>_r represents a particle of energy E that impinges
on the barrier from the right (hence the subscript r) and gets reflected to the right with
probability amplitude Rr(k) and transmitted to the left with probability amplitude T(k),
see Fig. 1a. The eigenfunction hx|E+>_l represents a particle of energy E that impinges
on the barrier from the left (hence the subscript l) and gets reflected to the left with
probability amplitude Rl(k) and transmitted to the right with probability amplitude
T(k), see Fig. 1b.
Note that, instead of (3.11a)-(3.11b), we could choose another pair of linearly
independent solutions of Eq. (3.8c) as follows [10]:
---------------------
Iɂ͌ŗL֐ <x|E+>_r ̓GlM[ E ŉEiYjāAقɔ˂mURr(k)Ɠ߂mU T(k)킷B}1aQƁB
ŗL֐<x|E+>_l ̓GlM[EōiYjāAقɔ˂mUR_l(k)Ɠ߂mU T(k)킷B}1bQƁB
(3.11a)-(3.11b)̂ɕ(3.8c)̕ʂ̐ƗȉƂ邱Ƃł [10]:Ȃ킿

,(3.13a)
,(3.13b)
where the coefficients of these eigenfunctions can also be calculating by means of the
standard matching conditions at x = a, b [10].
-------------------
ŁAŗL֐̌Wx=a,bł̒ʏ̐ڑ@ɂvZłB
43 Ð@t 2015/01/21 () 22:07:39 ID:ctwIRbLQLU
The eigenfunction <x|E|>_r represents
two plane waves\one impinging on the barrier from the left with probability amplitude
T∗(k) and another impinging on the barrier from the right with probability amplitude
R∗r (k)\that combine in such a way as to produce an outgoing plane wave to the right, see
Fig. 2a. The eigenfunction hx|E|il represents two other planes waves\one impinging
on the barrier from left with probability amplitude R∗l (k) and another impinging on the
barrier from the right with probability amplitude T∗(k)\that combine in such a way
as to produce an outgoing wave to the left, see Fig. 2b. The eigensolutions hx|E|ir,l
correspond to the final condition of an outgoing plane wave propagating away from the
barrier respectively to the right and to the left, as opposed to hx|E+ir,l, which correspond
to the initial condition of a plane wave that propagates towards the barrier respectively
from the right and from the left.
The eigenfunctions (3.9), (3.10), (3.11a)-(3.11b) and (3.13a)-(3.13b) are not square
integrable, that is, they do not belong to L2. Mathematically speaking, this is the reason
why they are to be dealt with as distributions (note that all of them except for the delta
function are also proper functions). Physically speaking, they are to be interpreted in
analogy to electromagnetic plane waves, as we shall see in Section 5.
------------------------

ŗL֐ <x|E|>_r ͓̕ʔgiǂɂԂmUT∗(k)ƉEǂɂԂmU R∗_r (k)j킳E֏očsʔgl킵ĂB} 2a.Q
ŗL֐ <x|E|>_l ͓̕ʔgiǂɂԂmUR*_l(k)ƉEǂɂԂmU T∗ (k)j킳荶֏očsʔgl킵ĂB} 2b.Q

ŗL <hx|E|>r,l͂ꂼȈԂEւgɂȂ邱ƂɑΉĂB
A<x|E+>r,l͎n߂̏ԂEƍɂgB
ŗL֐ (3.9), (3.10), (3.11a)-(3.11b) (3.13a)-(3.13b) ͂ǂ݂ȓϕ\łȂB
L^2ɑȂB@琔wIɂ͒֐ƂĈ˂΂ȂȂB
i֐قׂ̂Ă͌ŗL֐ł邱Ƃɒځj
Iɂ5͂ł݂悤ɓdg̕ʔgƂ̃AiW[Ă邱ƂłB
44 Ð@t 2015/01/21 () 22:11:19 ID:ctwIRbLQLU
4. Construction of the rigged Hilbert space
4.@RHS̍\z

In the previous section, we saw that the observables of our system are implemented by
unbounded operators with continuous spectrum. We also saw that the eigenfunctions
of the observables do not belong to L2. Thus, as we explained in Sec. 2, we need to
construct the rigged Hilbert spaces of Eqs. (1.1) and (1.2) [see Eqs. (4.8) and (4.21)
below]. We start by constructing .
----------------
O͂ŁAIuU[ouLEłȂAAXyNgAŗL֐L^2 ɑȂƂ݂B2͂ł݂悤Ɂi1.1j

i1.2j[ȉ̎ (4.8) (4.21)lRHSKvɂȂBɍ\zĂB
45 Ð@t 2015/01/22 () 21:04:48 ID:ctwIRbLQLU
4.1. Construction of S(R {a, b})
------------
4.1@ S(R {a, b})邱Ɓ@

The subspace  is given by Eq. (2.26). In view of expressions (2.17), (2.22) and (3.1),
the elements of  must fulfill the following conditions:
• they are infinitely differentiable, so the differentiation operation can be applied as
many times as wished,
• they vanish at x = a and x = b, so differentiation is meaningful at the discontinuities
of the potential [37],
• the action of all powers of Q, P and H remains square integrable.
--------------
i2.26jł܂镔ԃӂ̗vf́A(2.17), (2.22) (3.1)玟̏𖞂B
EK\A܂Z͂Ȃ{Ƃł邱ƁB
E x = a x = bŃ[ł邱ƁA܂̓|eV|eV̕sAȂƂłӖ [37],
EQ,P,ĤǂȂׂpĂϕ\ł邱
46 Ð@t 2015/01/22 () 21:07:11 ID:ctwIRbLQLU
Hence,
, (4.1)

where C^(R) is the collection of infinite differentiable functions, and ^(n) denotes the
nth derivative of '. From the last condition in Eq. (4.1), we deduce that the elements
of  satisfy the following estimates:
-----------------
C^(R) ͖K\Ȋ֐̏WA^(n) nK̃ӂ̓֐.

From the last condition in Eq. (4.1), we deduce that the elements
of  satisfy the following estimates:
----------------
(4.1j̍Ō̏A̗vf͎̕]𖞂B

(4.2)
47 Ð@t 2015/01/22 () 21:08:54 ID:ctwIRbLQLU CA: 02/07 (y) 20:58
These estimates mean that the action of any combination of any power of the observables
remains square integrable. For this to happen, the functions '(x) must be infinitely
differentiable and must fall off at infinity faster than any polynomial. The estimates (4.2)
induce a topology on , that is, they induce a meaning of convergence of sequences, in
the following way. A sequence {_} -converges to when {_} converges to with
respect to all the estimates (4.2),
-------------------
̕]́A܂IuU[oûǂȂׂ̑gݍ킹pĂϕ\ƂƂB
Ȃ邽߂ɂ͊֐@(x) ͖K\ŁAɂނĂǂȑ邱ƂKvB

] (4.2)@Ɉʑtopology сA܂邱ƂłB
{_} ӂɎƂ́A_]i4.2ĵׂĂɊւăӂɎ邱Ƃ@܂

if (4.3)
48 Ð@t 2015/01/22 () 21:10:16 ID:ctwIRbLQLU CA: 21:53
Intuitively, a sequence _ converges to if whenever we follow the terms of the sequence,
we get closer and closer to the limit point with respect to a certain sense of closeness.
In our system, the notion of closeness is determined by the estimates || ||n,m,l, which
originate from the physical requirements that led us to construct .
-----------------
łǂ΁A邫܂Ӗ́u߂vɂĂłɌ_ӂɔȂΗ _ Ӂ@Ɏ
Ƃ̂ϓIȐB
ł݂Ănł́Au߂vƂ Iȗvł|| ||n,m,l@݂邱ƂB
49 Ð@t 2015/01/23 () 23:27:59 ID:ctwIRbLQLU CA: 02/07 (y) 21:52
From Eqs. (4.1) and (4.2), we can see that is very similar to the Schwartz space
S(R), the major differences being that the derivatives of the elements of vanish at
x = a, b and that is not only invariant under P and Q but also under H. This is why
we shall write
-----------------
(4.1)(4.2)A̓VcS(R)ɂ߂ċ߂Ƃ킩B
ȈႢ̗̓vfׂ̂Ă̎̓֐=a,bŃ[ł邱Ƃ
P,płȂgɂsςł邠邱ƁABĈȉ̂悤ɏB

 S(R- {a, b}) . (4.4)

It is always a good, though lengthy exercise to check that S(R-{a, b}) is indeed
invariant under the action of the observables,
---------
S(R-{a, b})ہAIuU[ou̍pŕsςł邱ƂmF͖̂ʓ|ǂ悢KɂȂB

AS(R {a, b}) S(R {a, b}) , A = P,Q,H . (4.5)

This invariance guarantees that the expectation values
̕sϐҒl

(,AOn) , S(R {a, b}) , A = P,Q,H, n = 0, 1, . . . (4.6)
are finite, and that the commutation relations (3.3a)-(3.3c) are well defined [38]. It
can also be checked that P, Q and H, which are not continuous with respect the
topology of the Hilbert space L2, are now continuous with respect to the topology 
of S(R-{a, b}) [10, 11].
---------
͗LŁA֌W(3.3a)-(3.3c) 悭ł [38]B
P, Q, H@̓qxgL^2ɂĂ͘AłȂA S(R-{a, b})̈ʑtopology ɂĂ͘Ał邱ƂmFł [10, 11]B
50 Ð@t 2015/01/24 (y) 08:53:13 ID:ctwIRbLQLU
4.2. Construction of ^~ SO~(R- {a, b}). The Dirac kets
4.2 fBbNPbg@^~ SO~(R- {a, b})̍\

The space ^~ is simply the collection of у-continuous antilinear functionals over@  [26].@By combining the spaces , H and ^~, we obtain the RHS of our system,
 H O~ , (4.7)
which we denote in the position representation by
S(R- {a, b}) L^2 S~(R- {a, b}) . (4.8)
------------
^~ Ƃ̏̃_-AȔĊ֐̏W [26].@
O̋ , H ^~ɂāAlĂn RHS łB
 H O~ , (4.7)
W\ł͈͂ȉ̂悤ɋLB
S(R- {a, b}) L^2 SO~(R- {a, b}) . (4.8)
51 Ð@t 2015/01/24 (y) 10:50:19 ID:ctwIRbLQLU
The space S^~(R- {a, b}) is meant to accommodate the eigenkets |p>, |x> and |E}>_il,r of
P, Q and H. In the remainder of this subsection, we construct these eigenkets explicitly
and see that they indeed belong to S^~(R- {a, b}). We shall also see that |p>i, |x> and
|E}>_l,r are indeed eigenvectors of the observables.
The definition of a ket is borrowed from the theory of distributions as follows [16].
Given a function f(x) and a space of test functions, the antilinear functional F that
corresponds to the function f(x) is an integral operator whose kernel is precisely f(x):
---------------
S^~(R- {a, b}) ɂ́@P, Q ,ȞŗLPbg |p>, |x> , |E}>_il,r ܂B
̏͂̎cł͂ƃPbg\Ă S^~(R- {a, b})ɂ邱Ƃ悤B
܂|p>, |x>,|E}>_l,r ɃIuU[oǔŗLxNgł邱Ƃ悤B

Pbg̒ɂ͈ȉqׂ悤ɒ֐̗_؂Ă邱Ƃł [16]B
֐f(x) Ǝ֐ԃ,f(x)ɑΉ锽Ċ֐ F Ƃf(x)jƂϕZq

(4.9a)

which in Diracfs notation becomes
fBbN̋L@ł

(4.9b)
52 Ð@t 2015/01/24 (y) 11:03:01 ID:ctwIRbLQLU
It is important to keep in mind that, though related, the function f(x) and the functional
F are two different things, the relation between them being that f(x) is the kernel of F
when we write F as an integral operator. In the physics literature, the term distribution
is usually reserved for f(x).
----------
֐f(x)ƔĊ֐éAeϕZqłf(x)ϕjɂȂƂ֌W͂ĂAقȂӂ̂̂ł邱ƂL悤B
ł͐ϕjf(x)ɗpu֐v𓖂ĂB
53 Ð@t 2015/01/24 (y) 12:19:44 ID:ctwIRbLQLU
Definition (4.9a) provides the link between the quantum mechanical formalism
and the theory of distributions. In practical applications, what one obtains from the
quantum mechanical formalism is the distribution f(x) (in this paper, the plane waves
, the delta function    and the eigenfunctions <x|E}>_l,r). Once f(x) is
given, one can use definition (4.9a) to generate the functional |F>. Then, the theory of
distributions can be used to obtain the properties of the functional |F>, which in turn
yield the properties of the distribution f(x).
---------
 (4.9a) ͗ʎq͊w̋L@ƒ֐̗_̌т킷̂B
ʎq͊w̋L@ł͂ij͒֐B (̘_ł͕ʔg
, f^֐    ,GlM[ŗL֐ <x|E}>_l,r)ɂj.
f(x) ^΁@ (4.9a) ŔĊ֐l |F>BĒ֐̗_ŔĊ֐ |F>, ̐m邱ƂłA͒֐ f(x)̐łB
54 Ð@t 2015/01/24 (y) 13:16:13 ID:ctwIRbLQLU
By using prescription (4.9a), we can define for each eigenvalue p the eigenket |p>
associated with the eigenfunction (3.10):
--------------
(4.9a)̂肩ŁAŗLlŌŗL֐i3.10ǰŗLPbg|p>邱ƂłB

, (4.10a)
which, using Diracfs notation for the integrand, becomes
ϕ֐ɂăfBbN̋L@

. (4.10b)
Similarly, for each x, we can define the ket |x> associated with the eigenfunction (3.9)
of the position operator as
, (4.11a)
which, using Diracfs notation for the integrand, becomes
ϕ֐ɂăfBbN̋L@
. (4.11b)

The definition of the kets |E}il,r that correspond to the Hamiltonianfs eigenfunctions
(3.11a)-(3.11b) and (3.13a)-(3.13b) follows the same prescription:
------------
Pbg |E}>l,r ̓n~gjǍŗL֐(3.11a)-(3.11b) (3.13a)-(3.13b)
ŒB

, (4.12a)
that is, ܂
. (4.12b)

(Note that this equation defines four different kets.)
(̎͂S̈ႤPbg邱ƂɒӁj

One can now show that the
definition of the kets |p>, |x> and |E}>_l,r makes sense, and that these kets indeed belong
to the space of distributions S~(R {a, b}) [10].

ŃPbg |p>, |x> A|E}>_l,r ̒Ӗ̃Pbg֐̋  ɂ邱Ƃ킩 [10]B
55 Ð@t 2015/01/24 (y) 14:18:25 ID:ctwIRbLQLU
As in the general case of Eqs. (4.9a)-(4.9b), it is important to keep in mind the
difference between eigenfunctions and kets. For instance, hx|pi is an eigenfunction of a
differential equation, Eq. (3.8b), whereas |pi is a functional, the relation between them
being given by Eq. (4.10b). A similar relation holds between hx|xi and |xi, and between
hx|E}il,r and |E}il,r. It is also important to keep in mind that gscalar productsh like
hx|pi, hx|xi or hx|E}il,r do not represent an actual scalar product of two functionals;
these gscalar productsh are simply solutions to differential equations.
-----------------
(4.9a)-(4.9b)ʓIɌŗL֐ƃPbg̈Ⴂ̂ɖĂ̂dvB
Ƃ΁x|p͔(3.8b)̌ŗL֐BA, |p͔Ċ֐B
҂̊֌W͎(4.10b)̂ƂB
x|x |x, x|E}l,r |E}l,rɂĂႢ͓lB

܂x|p, x|x x|E}l,r̂悤ȔĊ֐ǂ́uρv ͖{ɂӂ̔Ċ֐̓ςȂ̂łȂ
ƂɂӁB@́uρv͒Pɔ̉AƂB
56 Ð@t 2015/01/24 (y) 14:59:58 ID:ctwIRbLQLU
We now turn to the question of whether the kets |pi, |xi and |E}il,r are eigenvectors
of the corresponding observable [see Eqs. (4.17)-(4.19) below]. Since the observables act
in principle only on their Hilbert space domains, and since the kets lie outside the Hilbert
space, we need to extend the definition of the observables from  into O~, in order to
specify how the observables act on the kets. The theory of distributions provides us
with a precise prescription of how an observable acts on O~, and therefore of how it
acts on the kets, as follows [16].
------------
Pbg |p>, |x>, |E}>l,r AΉIuU[oǔŗLxNgł̂ǂ݂Ă݂悤 [ȉ̎ Eqs. (4.17)-(4.19)].@

IuU[ou͌qxgԂ̗̈ɂȂB
ăPbg̓qxgԂ̊Oɂ邩,Pbgɍpł悤IuU[o̒烳OwɂЂ낰ȂƍsȂB֐̗_ɂ萸ɂ̒̊g傪ł[16].
57 Ð@t 2015/01/24 (y) 16:01:29 ID:ctwIRbLQLU
The action of a self-adjoint operator A on a functional
|F>    is defined as
, for all    . (4.13)
Note that this definition extends the Hilbert space definition of a self-adjoint operator,
, (4.14)
which is valid only when f and g belong to the domain of A.
In turn, Eq. (4.13) can
be used to define the notion of eigenket of an observable: A functional |a> in ^~ is an
eigenket of A with eigenvalue a if
------------

ȋZqĊ֐|F>   ɍp邵
@ׂẮ@   ɂā@(4.13)
ƒB ͈̒ȉ̎ȋZq̒̊gł邱ƂɒӁB
, (4.14)
̎ f g ̈ AɑƂɓKp̂B
tɁA (4.13) ̓IuU[oǔŗLPbg̋L@̂ƂĂgB
: O~ ̗vf̔Ċ֐|a> @A ̌ŗLl ǎŗLPbgłƂ͈ȉ̎Ȃ肽ƂBA

, for all Ӂ@in@
̗vfӂׂ̂Ăɂ. (4.15)
58 Ð@t 2015/01/24 (y) 18:20:55 ID:ctwIRbLQLU
When the gleft sandwichingh of this equation with the elements of is understood and
therefore omitted, we shall simply write
, (4.16)
which is just Diracfs eigenket equation (2.10b). Thus, Diracfs eigenket equation acquires
a precise meaning through Eq. (4.15), in the sense that it has to be understood as gleft
sandwichedh with the wave functions of .
By using definition (4.15), one can show that |p>, |x> and |E}>l,r are indeed
eigenvectors of P, Q and H, respectively [10]:
-----------
̎ƗvfŁg͂ށhƂ񑩂̉ł̗vf̂ȗΊȒP
, (4.16)
ƏB̓fBbŇŗLPbg(2.10b)B

̂悤ɃfBbŇŗLPbg̕ɂ͐mȈӖ́A
̔g֐ӂŁg͂ށhƂƂɂāA(4.15)ŗ^B

(4.15)̗̒̍pɂ |p>, |x> |E}>l,r ͂ɂꂼ P, Q ȞŗLxNg[10]B

, (4.17)
, (4.18)
. (4.19)
59 Ð@t 2015/01/24 (y) 20:27:43 ID:ctwIRbLQLU
4.3. Construction of  S(R- {a, b}). The Dirac bras
4.3.  S(R- {a, b})̍\. fBbNu

In complete analogy with the construction of the Dirac kets, we construct in this
subsection the Dirac bras <p|, <x| and l,r_<}E| of P, Q and H. Mathematically, the
Dirac bras are distributions that belong to the space , which is the space of linear
functionals over [26]. The corresponding RHS is
fBbNPbg̂ƑSlɂāȀ͂łP, Q AH. fBbNu<p|, <x| A l,r_<}E| 낤B
fBbNúA̐Ċ֐̋ԃfɑ钴֐BΉRHS
, (4.20)
which we denote in the position representation by@W\ŋL
. (4.21)
60 Ð@t 2015/01/24 (y) 20:31:49 ID:ctwIRbLQLU
Likewise the definition of a ket, the definition of a bra is borrowed from the theory
of distributions [16]. Given a function f(x) and a space of test functions , the linear
functional    generated by the function f(x) is an integral operator whose kernel is the
complex conjugate of f(x):

PbĝƂƓu̒͒֐؂Ă[16]B
֐f(x)Ǝ֐ԃ^΁A֐f(x)琶Ċ֐  f(x)̕fjƂϕB
, (4.22a)
which in Diracfs notation becomes@@fBbN̋L@
. (4.22b)
61 Ð@t 2015/01/24 (y) 20:44:55 ID:ctwIRbLQLU
Note that this definition is very similar to that of a linear functional, Eq. (4.9a), except
that the complex conjugation affects f(x) rather than (x), which makes    linear rather than antilinear. Likewise the antilinear case (4.9a), it is important to keep in mind that,
though related, the function f(x) and the functional    are two different objects, the
relation between them being that    is the kernel of    when we write ˜    as an integral operator.

̒͐Ċ֐i4.9a)ƓAfƂ̂(x)łȂf(x)̂قɂȂĂ̂
ӁBɂ  ͔łȂɂȂB
lɁ@i4.9a)̂ƂƓlA֐ f(x) ƔĊ֐       ϕZqƂ    ̊jłƂ֌W͂̂Aӂ̈قȂ̂ł邱Ƃɒӂ悤B

By using prescription (4.22a), we can now define for each eigenvalue p the eigenbra
<p| associated with the eigenfunction (3.10):
i4.22ajł̂肩ŌŗL֐(3.10)ƂނтŗLu<p|́A

, (4.23a)
which, using Diracfs notation for the integrand, becomes@ϕ֐ɃfBbN̋L@

. (4.23b)

Comparison with Eq. (4.10a) shows that the action of <p| is the complex conjugate of
the action of |p>,
(4.10a)Ƃ̔r@<p|ւ̍pƂ|p>ւ̍p̕fB

, (4.24)
and that
. (4.25)
62 Ð@t 2015/01/24 (y) 21:27:15 ID:ctwIRbLQLU
The bra <x| is defined as u<x|̒
, (4.26a)
which, using Diracfs notation for the integrand, becomes@fBbN̋L@ϕ֐ɂ
. (4.26b)
Comparison with Eq. (4.11a) shows that the action of <x| is complex conjugated to the
action of |x>,
(4.18a)Ɣׂ<x|ւ̍pƂ|x>ւ̍p̕fƂ킩B
, (4.27)
and that@܂
. (4.28)
Analogously, the eigenbras of the Hamiltonian are defined as
ƓlɃn~gjǍŗLu̒
, (4.29a)
that is,@Ȃ킿
, (4.29b)
where
. (4.30)
(Note that in Eq. (4.29a) we have defined four different bras.)
((4.29a)͂S̈قȂuĂ邱ƂɒӁj

Comparison of Eq. (4.29a)with Eq. (4.12a) shows that the actions of the bras l,r<}E| are the complex conjugates of the actions of the kets |E}i>l,r:
(4.29a)(4.12ajׂƁAul,r<}E|ւ̍ṕAPbg |E}i>l,rւ̍p̕f
Ƃ킩B
. (4.31)

Now, by using the RHS mathematics, one can show that the definitions of <p|, <x| and
l,r<}E| make sense and that <p|, <x| and l,r<}E| belong to S(R-{a, b}) [10].

RHS̐wKp <p|, <x| ,r<}E| ӖȂƁA <p|, <x| and l,r<}E| S(R-{a, b})ɂ邱Ɓ@킩B
63 Ð@t 2015/01/25 () 10:49:49 ID:ctwIRbLQLU
Our next task is to see that the bras we just defined are left eigenvectors of the
corresponding observable [see Eqs. (4.35)-(4.37) below]. For this purpose, we need to
specify how the observables act on the bras, that is, how they act on the dual space
S(R-{a, b}). We shall do so in analogy to the definition of their action on the kets, by
means of the theory of distributions [16]. The action to the left of a self-adjoint operator
A on a linear functional <F|  is defined as
--------
͂ĒuΉIuU[oǔŗLxNgł邱Ƃ悤B[ȉ (4.35)-(4.37)]
̂߂ɂ͂܂IuU[ouuɍp邩Aǂ̂悤ɑo΋S(R-{a, b})ɍp邩悤B
Pbĝ΂ɂȂĒ֐_Ē悤 [16].
Ċ֐<F| ȋZqɍpƂ

, for all    in    . (4.32)
64 Ð@t 2015/01/25 () 17:21:34 ID:ctwIRbLQLU
Likewise definition (4.13), this definition generalizes Eq. (4.14). In turn, Eq. (4.32) can
be used to define the notion of eigenbra of an observable: A functional <a| in is an
eigenbra of A with eigenvalue a if
----
(4.13)Ɠ悤,̒(4.14)ʉ̂BtɎ (4.32)
IuU[oǔŗLűL@̒ɎgB
̂Ȃ̔Ċ֐ <a| ŗLlǎŗLułƂׂ͂Ă   ̂Ȃ    ɂ

for    in    , (4.33)

When the gright sandwichingh of this equation with the elements of    is understood
and therefore omitted, we shall simply write
------
̗̎vfŉE͂ނƂOŁ@ƂĂȂĂ

, (4.34)

which is just Diracfs eigenbra equation (2.11b).
Thus,Diracfs eigenbra equation acquires
a precise meaning through Eq. (4.33), in the sense that it has to be understood as gright
sandwichedh with the wave functions    of   .
By using definition (4.33), one can show that <p|, <x| and l,r<}E| are indeed left
eigenvectors of P, Q and H, respectively [10]:
-------
͂Ȃ̂Ƃ͂ȂfBbŇŗLu̕(2.11b)
fBbŇŗLu(4.33)ʂāAɂg֐ӂŁuE͂ށvƂӖŗB
(4.33)A<p|, <x|Al,r<}E| P, QAH́uvŗLxNgƎƂo[10]BȂ킿

, (4.35)
, (4.36)
. (4.37)

It is worthwhile noting that, in accordance with Diracfs formalism, there is a oneto-
one correspondence between bras and kets [39]; that is, given an observable A, to
each element a in the spectrum of A there correspond a bra <a| that is a left eigenvector
of A and also a ket |a that is a right eigenvector of A. The bra a| belongs to ,
whereas the ket |a belongs to ^~.
------
ӂׂƂƂ fBbNɏ],uƃPbg͂P΂PΉ[39];
܂, IuU[ou A΁AA ̃XyNg̊evf́@A̍ŗLxNgłu<a|
ƉEŗLxNgłPbg |a>@ΉB
u a| ,Pbg |a ^~@ɑB
65 Ð@t 2015/01/25 () 18:20:42 ID:ctwIRbLQLU CA: 01/26 () 20:14
4.4. The Dirac basis expansions
----------------
4.4. fBbN̊WJ̕@

A crucial ingredient of Diracfs formalism is that the bras and kets of an observable form
a complete basis system, see Eqs. (2.12) and (2.13). When applied to P, Q and H,
Eq. (2.13) yields
-------------
fBbNL@̗v̓uƃPbgIuU[oůSȂƂƂBi2.12ji2.13jQƁB
P,Q,H^
, (4.38)

, (4.39)

, (4.40)
66 Ð@t 2015/01/26 () 20:18:44 ID:ctwIRbLQLU
In the present subsection, we derive various Dirac basis expansions for the algebra of
the 1D rectangular barrier potential. We will do so by formally sandwiching Eqs. (4.38)-
(4.40) in between different vectors.@If we sandwich Eqs. (4.38)-(4.40) in between <x| and bӁ, we obtain

̐߂ł́A낢ȃfBbNWJPǃ|eVł݂Ă݂悤B
낢ȃxNgi4.38)-(4.40jŃThCbĂ݂悤B<x| |>
ł͂߂

, (4.41)

, (4.42)

. (4.43)
67 Ð@t 2015/01/27 () 22:55:32 ID:ctwIRbLQLU
Equations (4.41)-(4.43) can be rigorously proved by way of the RHS [10]. In proving
these equations, we give meaning to Eqs. (4.38)-(4.40), which are just formal equations:
Equations (4.38)-(4.40) have always to be understood as part of a gsandwich.h Note
that Eqs. (4.41)-(4.43) are not valid for every element of the Hilbert space but only
for those that belong to S(R-{a, b}), because the action of the bras and kets is well
defined only on S(R {a, b}) [40]. Thus, the RHS, rather than just the Hilbert space,
fully justifies the Dirac basis expansions. Physically, the Dirac basis expansions provide
the means to visualize wave packet formation out of a continuous linear superposition
of bras and kets.

(4.41)-(4.43) RHS̕@ŌɏؖłB[10]
ؖ邽߂ (4.38)-(4.40)ɉ߂B܂莮 (4.38)-(4.40) ͂u͂ށv
Ӗŉ߂Ȃ΂ȂȂB

(4.41)-(4.43) ̓qxgԂׂ̂Ă̗vfɂĐ藧킯ł͂Ȃ
S(R-{a, b})ɑӂɂĐ藧̂BƂ̂́AuƃPbg̍p
S(R- {a, b})̏ゾŒł邩炾B [40].
ăqxgԂłȂRHSɂā@fBbNWJB
Iɂ̓fBbNWJ͔guƃPbg̐d˂킹ƂĂ킷̂B
68 Ð@t 2015/01/28 () 17:05:30 ID:ctwIRbLQLU
We can obtain similar expansions to Eqs. (4.41)-(4.43) by sandwiching Eqs. (4.38)-
(4.40) in between other vectors. For example, sandwiching Eq. (4.39) in between hp|
and ' yields [10]
---
(4.41)-(4.43) ƓlȎ (4.38)-(4.40)𑼂̃xNgł͂ނƂœ邱ƂłB
Ƃ Eq. (4.39) <p|Ɓb>ł͂ނ [10]

, (4.44)

and sandwiching Eq. (4.39) in between l,rh}E| and ' yields [10]
---
(4.39) l,r<}E| Ɓb>ł͂߂ [10]

. (4.45)

It is worthwhile noting the parallel between the Dirac basis expansions and the Fourier
expansions (4.41) and (4.44) [10].
This parallel will be used in Sec. 5 to physically interpret the Dirac bras and kets.
We can also sandwich Eqs. (4.38)-(4.40) in between two elements and of@S(R-{a, b}), and obtain [10]
---
fBbNWJƃt[GϊA (4.41) (4.44)ȂΉ݂Ă̂؂B [10].
̑Ή̓fBbNuƃPbg߂̂ɑT͂ŎgB
(4.38)-(4.40) S(R-{a, b})̗vf̃Ղƃӂł͂߂

, (4.46)
, (4.47)
(4.48)
69 Ð@t 2015/01/29 () 22:33:26 ID:ctwIRbLQLU
Equations (4.46)-(4.48) allow us to calculate the overlap of two wave functions ' and
by way of the action of the bras and kets on those wave functions.
The last aspect of Diracfs formalism we need to implement is prescription (2.14),
which expresses the action of an observable A in terms of the action of its bras and kets.
When applied to P, Q and H, prescription (2.14) yields
---

(4.46)-(4.48) ɂӂ̔g֐ӂƃՂ̓ςAg֐ւ̃uAPbg̍pɂ
킷ƂłB
fBbN̋L@łƂ݂Ȃ΂ȂȂƂ (2.14)̂肩A܂IuU[ouA̍puƃPbgւ
pɂĂ킷ƂB
A P, QA HɂƂ, (2.14)

, (4.49)
, (4.50)
, (4.52)
, (4.53)
,

(4.54)

and sandwiching them in between two elements and of S(R-{a, b}) yields [10]
----
S(R-{a, b})ɑӂƃՂł͂߂

, (4.55)
, (4.56)
. (4.57)
70 Ð@t 2015/01/30 () 17:20:09 ID:ctwIRbLQLU
Note that, in particular, the operational definition of an observable—according to which
an observable is simply an operator whose eigenvectors form a complete basis such that
Eqs. (2.12), (2.13) and (2.14) hold, see for example Ref. [41]—acquires meaning within
the RHS.
--------
ɋLׂƂ́@ŗLxNg̑IȒAȂ킿
Zq̌ŗLxNg (2.12), (2.13) , (2.14)悤ȊSnȂi[41]j
݂͂Ȃqgrl̂ȂňӖƂƂB

The sandwiches we have made so far always involved at least a wave function of
S(R-{a, b}). When the sandwiches do not involve elements of S(R-{a, b}) at all, we
obtain expressions that are simply formal. These formal expressions are often useful
though, because they help us understand the meaning of concepts such as the delta
normalization or the gmatrix elementsh of an operator. Let us start with the meaning
of the delta normalization. When we sandwich Eq. (4.39) in between hp| and |pi, we
get
---------
܂ł񂪂ĂThCbɂ͂ȂƂЂƂ́AS(R-{a, b})ɑg֐ӂ
ĂB
ThCbS(R-{a, b})̗vfЂƂӂ܂ȂȂ炻͌̂̂B
܂ɗ̂ł͂邪AƂ΁uZq̍s\ṽ֐Kî悤ɁB@
̃֐KiƋᖡĂ݂悤Bi4.39j<p'|,|p>ł͂߂΁A

. (4.58)

This equation is a formal expression that is to be understood in a distributional sense,
that is, both sides must appear smeared out by a smooth function (p) = <p|Ӂ in an
integral over p:
---
̌IȎ͒֐̈ӖŗȂƂȂBȂ킿ӂ͊炩Ȋ֐(p) = <p|Ӂ
Őϕ̂ƁB

. (4.59)

The left-hand side of Eq. (4.59) can be written as
-----
i4.59j̍ӂ͈ȉ̂悤ɕόłB

(4.60)
71 Ð@t 2015/01/31 (y) 10:48:30 ID:ctwIRbLQLU
Plugging Eq. (4.60) into Eq. (4.59) leads to
--
(4.60) (4.59)Ȃ

. (4.61)

By recalling the definition of the delta function, we see that Eq. (4.61) leads to
------
f^֐̒ƁAi4.61j

, (4.62)

and to@܂

. (4.63)

By using Eq. (4.25), we can write Eq. (4.63) in a well-known form:
----
(4.25) (4.63)͂悭Ă鎟̌ɂB

. (4.64)

This formal equation is interpreted by saying that the bras and kets of the momentum
operator are delta normalized. That the energy bras and kets are also delta normalized
can be seen in a similar, though slightly more involved way [28]:

----
̌́A܂ŉ^ʉZq̃uƃPbgKiꂽ̂悤ɂ݂B
GlM[̃uƃPbglɁuKivł [28]:

, (4.65a)

,

(4.65b)
where , stand for the labels l, r that respectively denote left and right incidence.
----
, ̓xs l, r܂EˁA˂킷B

The derivation of expressions involving the Dirac delta function such as Eqs. (4.62),
(4.64) or (4.65a)-(4.65b) shows that these formal expressions must be understood in a
distributional sense, that is, as kernels of integrals that include the wave functions ' of
S(R-{a, b}), like in Eq. (4.59).
--------
f^֐̊܂܂ (4.62),(4.64j@(4.65a)-(4.65b) ̓o݂Ă킩悤
I\́A֐̈Ӗŉ˂΂ȂȂB܂S(R-{a, b})ɑӂ܂ސϕ̊jƂāAB
72 Ð@t 2015/01/31 (y) 12:08:47 ID:ctwIRbLQLU
Equations (4.66)-(4.68) can be obtained by formally inserting Eq. (4.39) into respectively
Eq. (2.17), (2.22) and (3.1).
-----
(4.66)-(4.68) (4.39) ꂼ (2.17), (2.22) (3.1)ɑ邱ƂłB

It is illuminating to realize that the expressions (4.66)-(4.68) generalize the matrix
representation of an observable A in a finite-dimensional Hilbert space. If a1, . . . , aN
are the eigenvalues of A, then, in the basis {|a1i, . . . , |aNi}, A is represented as
----
\ (4.66)-(4.68)LqxgԂ̃IuU[ouA̍s\ʉ̂ł̂

, (4.69)

which in Diracfs notation reads as
---
ŃfBbN̋L@̈Ӗ

. (4.70)

Clearly, expressions (4.66)-(4.68) are the infinite-dimensional extension of expression (4.70).
---

73 Ð@t 2015/02/01 () 17:57:50 ID:ctwIRbLQLU
5. Physical meaning of the Dirac bras and kets
---
5. fBbÑuƃPbg̕IӖ

The bras and kets associated with eigenvalues in the continuous spectrum are not
normalizable. Hence, the standard probabilistic interpretation does not apply to
them straightforwardly. In this section, we are going to generalize the probabilistic
interpretation of normalizable states to the non-normalizable bras and kets. As well, in
order to gain further insight into the physical meaning of bras and kets, we shall present
the analogy between classical plane waves and the bras and kets.
---
AXyNǧŗLl̃uƃPbg͋KiłȂB
WIȊm̉߂͂̂܂܂ł͂Ă͂߂ȂB
̏͂ł͋KiꂽԂ̕WIȊm߂KiłȂuƃPbgɂg
Ă͂߂邱Ƃ݂Ă݂悤B
܂uƃPbg̕IӖɂB̂߂ɃuAPbgƌÓTIȕʔg̗ގɒڂB

In Quantum Mechanics, the scalar product of the Hilbert space is employed to
calculate probability amplitudes. In our example, the Hilbert space is L2, and the
corresponding scalar product is given by Eq. (3.4). That an eigenvalue of an observable
A lies in the discrete or in the continuous part of the spectrum is determined by this
scalar product. An eigenvalue an belongs to the discrete part of the spectrum when its
corresponding eigenfunction fn(x) hx|ani is square normalizable:
---
ʎq͊wł̓qxgԂ̓ςmUvẐɂB
qxgԂ L^2,ς(3.4)B
IuU[ouǍŗLl͗U܂͘ÃXyNgłꂼ̃XyNgœς
ĂBUŗLl ̃XyNg͋Ki\ȌŗL֐ fn(x) x|a_n ƑΉB

. (5.1)

An eigenvalue a belongs to the continuous part of the spectrum when its corresponding
eigenfunction fa(x) <x|a is not square normalizable:
---
AŗLlΉŗL֐fa(x) <x|a͋Kis\

. (5.2)

In the latter case, one has to use the theory of distributions to gnormalizeh these states,
e.g., delta function normalization:
---
̂Ƃ̂قɂĂ͏Ԃ𒴊֐̈ӖŁuKivKvBA

. (5.2)

This Dirac delta normalization generalizes the Kronecker delta normalization of
gdiscreteh states:
---
̃fBbNf^Ki͗UԂ̃NlbJf^ʉ̂B

. (5.3)

Because they are square integrable, the gdiscreteh eigenvectors fn(x) <x|a_n> can be
interpreted in the usual way as probability amplitudes. But because they are not square
integrable, the gcontinuoush eigenvectors fa(x) <x|a> must be interpreted as gkernelsh
of probability amplitudes, in the sense that when we multiply <x|a> by <|x> and then
integrate, we obtain the density of probability amplitude <|a>:
---
UŗLxNg͓ϕ\Ȃ̂ fn(x) <x|a_n> ʏǂmUƉ߂
AŗLxNg͓ϕs\Ȃ̂ fa(x) <x|a>@͊mŮjƉ߂A܂
<x|a> <|x>@|̂ϕƊmUx <|a>@ƂƂB:

. (5.5)

Thus, in particular, <x|p>, <x|x> and    represent gkernelsh of probability
amplitudes.
---
̂悤Ɂ@<x|p>, <x|x>    ͊mx̐ϕjƂȂB
74 Ð@t 2015/02/01 () 20:26:48 ID:ctwIRbLQLU
Another way to interpret the bras and kets is in analogy to the plane waves
of classical optics and classical electromagnetism. Plane waves eikx represent
monochromatic light pulses of wave number k and frequency (in vacuum) w = kc.
Monochromatic light pulses are impossible to prepare experimentally; all that can be
prepared are light pulses '(k) that have some wave-number spread. The corresponding
pulse in the position representation, '(x), can be gFourier decomposedh in terms of the
monochromatic plane waves as
---
uƃPbg߂ЂƂ̕@́AÓTIȌwƌÓTIȓdCwƂ̃AiW[B
ʔg@   ͔g@Ui^󒆂Łjw = kc@̒PFU킷B
ێŒPFÛ͕s\B; ł̂̓(k)ŐF̕dgB
̃t[GϊőΉ(x) gt[Gh A܂PFʔg̏d˂킹

, (5.6a)
which in Diracfs notation becomes
---
fBbN̋L@

. (5.6b)

Thus, physically preparable pulses can be expanded in a Fourier integral by the
unpreparable plane waves, the weights of the expansion being   . When@  (k) is highly
peaked around a particular wave number k0, then the pulse can in general be represented
for all practical purposes by a monochromatic plane wave e^ik0x. Also, in finding out how
a light pulse behaves under given conditions (e.g., reflection and refraction at a plane
interface between two different media), we only have to find out how plane waves behave
and, after that, by means of the Fourier expansion (5.6a), we know how the light pulse
(x) behaves. Because obtaining the behavior of plane waves is somewhat easy, it is
advantageous to use them to obtain the behavior of the whole pulse [42].
---
܂AIɏłǵ@邱Ƃ͂łȂPFʔg̏d˂킹ł킷Ƃł
̏d݂   @.@@  (k) g@k_0̂܂ɍs[NƁAg
ۓIɂ͒PFʔg e^ik0x@ł킷ƂłB
܂A^Čǂӂ܂i@قȂ}̂ɂ锽˂܁j݂ɂ
ʔgǂӂ܂܂݂āAŃt[GWJ (5.6a)@gǂӂ܂킩B
ʔĝӂ܂͔rIȒP炻Sĝӂ܂m̂ɂ̂֗B[42].

The quantum mechanical bras and kets can be interpreted in analogy to the classical
plane waves. The eigenfunction    represents a particle of sharp
momentum p; the eigenfunction <x|x> = (x|x) represents a particle sharply localized
at x; the monoenergetic eigenfunction hx|E}il,r represents a particle with well-defined
energy E (and with additional boundary conditions determined by the labels } and l, r).
---
ʎq͊w̃uƃPbg͌ÓTIʔgƂ̃AiW[ŗłB
ŗL֐    ͉s^ʂ̃s[N
q킷B; ŗL֐ <x|x> = (x|x) ͂fɋǍ݂闱q킷B;
PGlM[ŗL֐ <x|E}>_l,r ͂͂ƃGlM[܂Ă ( } l, r@ł܂
E܂߂ājq킷B
75 Ð@t 2015/02/01 () 21:19:56 ID:ctwIRbLQLU
In complete analogy to the Fourier expansion of a light pulse by classical plane waves,
Eq. (5.6a), the eigenfunctions hx|pi, <x|x> and <x|E}>_l,r expand a wave function , see
Eqs. (4.41)-(4.43). When the wave packet '(p) is highly peaked around a particular
momentum p0, then in general the approximation (x) ∼    holds for all
practical purposes; when the wave packet (x) is highly peaked around a particular
position x0, then in general the approximation (x) ∼ (x | x0) holds for all practical
purposes; and when (E) is highly peaked around a particular energy E0, then in general
the approximation (x) ∼    holds for all practical purposes (up to the boundary
conditions determined by the labels } and l, r). Thus, although in principle <x|p>, <x|x>
and <x|E}>_l,r are impossible to prepare, in many practical situations they can give good
approximations when the wave packet is well peaked around some particular values p0,
x0, E0 of the momentum, position and energy. Also, in finding out how a wave function
behaves under given conditions (e.g., reflection and transmission off a potential barrier),
all we have to find out is how the bras and kets behave and, after that, by means of
the Dirac basis expansions, we know how the wave function (x) behaves. Because
obtaining the behavior of the bras and kets is somewhat easy, it is advantageous to use
them to obtain the behavior of the whole wave function [43].
---
̔ǧÓTIʔgł̃t[GWJƂ̊SȃAiW[ (5.6a),ŗL֐ <x|p>, <x|x> <x|E}>_l,r ͔g֐ ӂWJB (4.41)-(4.43)QƁB
g(p) ^ʂ̒l p0s[N̂܂ɂ, ߎI (x) ∼    Ƃ͎pエĂ܂Ȃ; g (x) W x0s[N̂܂
ɂ (x) ∼ (x | x0)ƎpߎĂ܂ȂB; (E) GlM[̒l E0̍s[N̂܂
ɂ΁Aߎ (x) ∼    p͂łB ( } and l, rł܂鋫E܂߂邱). āAIɂ <x|p>, <x|x>@ <x|E}>_l,r ͗pӂ邱ƂłȂǁA̎pʂł͍s[Ngi^ʂȂ p0́AWȂx0́AGlM[ȂE0́jłւ悭ߎł
܂A^ꂽŔg֐ǂӂ܂ (Ƃ |eVǂɂ锽˂ⓧ߁j݂ɂ
܂ǂuƃPbgӂ܂݂΂悢BɃfBbN̊WJɂg֐ (x)̂ӂ܂BuƃPbgǂӂ܂݂̂͂₷̂ŁAg֐̂ӂ܂킩 [43].

From the above discussion, it should be clear that there is a close analogy between
classical Fourier methods and Diracfs formalism. In fact, one can say that Diracfs
formalism is the extension of Fourier methods to Quantum Mechanics: Classical
monochromatic plane waves correspond to the Dirac bras and kets; the light pulses
correspond to the wave functions '; the classical Fourier expansion corresponds to the
Dirac basis expansions; the classical Fourier expansion provides the means to form light
pulses out of a continuous linear superposition of monochromatic plane waves, and the
Dirac basis expansions provide the means to form wave functions out of a continuous
linear superposition of bras and kets; the classical uncertainty principle of Fourier
Optics corresponds to the quantum uncertainty generated by the non-commutativity
of two observables [44]. However, although this analogy is very close from a formal
point of view, there is a crucial difference from a conceptual point of view. To wit,
whereas in the classical domain the solutions of the wave equations represent a physical
wave, in Quantum Mechanics the solutions of the equations do not represent a physical
object, but rather a probability amplitude\In Quantum Mechanics what is gwavingh
is probability.
---
ȏ̋c_ÓTIt[GϊƃfBbN̋L@ɋގ邱Ƃ͖炩
ہAfBbN̋L@́Aʎq͊wւ̃t[Gϊ̊gƂ邾낤B:
ÓTIPFʔg̓fBbÑuƃPbgɑΉB
; ̔g͔g֐ӂɑΉB
; ÓTIt[GWJ̓fBbNWJɑΉB
; ÓTIt[GWJ͒PFʔg̘A̐dˍ킹ŘA̔g@AfBbNWJ̓uƃPbg̘A̐d˂킹Ŕg֐@B
; ÓTIt[Gw̕sm萫͂ӂ̃IuU[ou̔ɂʎqIsm萫ɑΉB[44].

̃AiW[͌̏ł͂ƂĂ߂̂́ATOƂĂ͑傫ȈႢB
ÓTIɔg͕IgAʎqIɂ͕͕̉Î킷̂ł͂ȂmU
킷Bʎq͊wŐUĂ̂͊mȂ̂B

76 Ð@t 2015/02/04 () 16:59:14 ID:ctwIRbLQLU
6. Further considerations
---
6. i񂾍l@

In Quantum Mechanics, the main objective is to obtain the probability of measuring
an observable A in a state . Within the Hilbert space setting, such probability can be
obtained by means of the spectral measures Ea of A (see, for example, Ref. [8]). These
spectral measures satisfy
----
ԃӂɂăIuU[ouA𑪒肵Ăl𓾂m邱Ƃʎq͊w̎ႾB
qxgԂ̘gg݂łÃXyNgxEa@iƂ[8]QƁj炦B
XyNgx̊֌W

(6.1)

and
(6.2)

Comparison of these equations with Eqs. (2.13) and (2.14) yields
--------
̎ (2.13) (2.14) Ɣr

. (6.3)

Thus, the RHS is able to gfactor outh the Hilbert space spectral measures in terms of
the bras and kets [45]. For the position, momentum and energy observables, Eq. (6.3)
---
Ȃ킿ARHSŃuƃPbgăqxgԂ̃XyNgxuǂovƂłB
WA^ʁAGlM[̃IuU[ouɂāi6.3j͈ȉ̂悤ɓǂݒB

, (6.4)
, (6.5)
. (6.6)
77 Ð@t 2015/02/04 () 17:35:29 ID:ctwIRbLQLU
Although the spectral measures dEa associated with a given self-adjoint operator A are
unique, the factorization in terms of bras and kets is not. For example, as we can see
from Eq. (6.6), the spectral measures of our Hamiltonian can be written in terms of the
basis {|E+il,r} or the basis {|E−>l,r}. From a physical point of view, those two basis are
very different. As we saw in Sec. 3, the basis {|E+>l,r} represents the initial condition
of an incoming particle, whereas the basis {|E−>l,r} represents the final condition of an
outgoing particle. However, the spectral measures of the Hilbert space are insensitive to
such difference, in contrast to the RHS, which can differentiate both cases. Therefore,
when computing probability amplitudes, the RHS gives more precise information on
how those probabilities are physically produced than the Hilbert space.
---
ȋZqA΁@̃XyNgx dEa ͂ЂƂ܂邪, uƃPbg̐ς͂ЂƂƂłȂB
ƂΎ (6.6)݂, n~gjÃXyNgx͊ {|E+>il,r} ł {|E−>l,r}ł킹B
Iɂ݂΂ӂ͈̊قȂBR͂ł݂悤Ɋ {|E+>l,r} ͓˂̏@ {|E−>l,r} ͎ˏȍI
킷B
qxgԂ̑x͂Ⴂɂ͖ڒBRHSł͗҂ʂł̂ɁB
mxvZȂ RHS ̂قqxgԂ蕨IʂɑvẐɓKĂB

In this paper, we have restricted our discussion to the simple, straightforward
algebra of the 1D rectangular barrier. But, what about more complicated potentials?
In general, the situation is not as easy. First, the theory of rigged Hilbert spaces as
constructed by Gelfand and collaborators is based on the assumption that the space 
has a property called nuclearity [16, 17]. However, it is not clear that one can always
find a nuclear space  that remains invariant under the action of the observables.
Nevertheless, Roberts has shown that such  exists when the potential is infinitely
often differentiable except for a closed set of zero Lebesgue measure [19]. Second, the
problem of constructing the RHS becomes more involved when the observable A is
not cyclic [16]. And third, solving the eigenvalue equation of an arbitrary self-adjoint
operator is rarely as easy as in our example.
---
̘_ł͊ȒPł킩₷Pǂł̐wɌĐBƕGȃ|eVȂǂ낤H
ʓIɂȒPɂ͂ȂB

ЂƂFQt@gRHSɂԂunuclearityvƂ
O𗧂ĂB [16, 17]. AIuU[ou̍pŕsςɕۂ悤ȁ@nuclear space
ł݂邱Ƃł̂͂͂肵ĂȂB
Ƃ͂ĂRoberts |eVK\ix[OxO̕W͏ājł悤Ȋ֐ɂ
Ԃ݂邱ƂB[19].

ӂFIuU[ouAIłȂƂɂRHS̂͂Ƃ₱iHjƂ͂킩ĂB[16].

݂FCӂ̎ȋZq̌ŗLlƂ̗͂̂悤ɂ͊ȒPɂȂB
78 Ð@t 2015/02/05 () 17:32:40 ID:ctwIRbLQLU CA: 02/07 (y) 23:05
7. Summary and conclusions
---
7. ܂Ƃ߂ƌ_

We have used the 1D rectangular barrier model to see that, when the spectra of the
observables have a continuous part, the natural setting for Quantum Mechanics is the
rigged Hilbert space rather than just the Hilbert space. In particular, Diracfs bra-ket
formalism is fully implemented by the rigged Hilbert space rather than just by the
Hilbert space.
---
1ǃfăIuU[ouAXyNgꍇ̗ʎq͊w
RȘgg݂̓qxgԂRHSł邱Ƃ݂ĂB
ɃfBbÑuPbgL@́AqxgԒPƂłȂRHSŏ\Sɂ邱Ƃ킩B

We have explained the physical and mathematical meanings of each of the
ingredients that form the rigged Hilbert space. Physically,
the space@ S(R@- {a, b})
is interpreted as the space of wave functions, since its elements can be associated
well-defined, finite physical quantities, and algebraic operations such as commutation
relations are well defined on . Mathematically,  is the space of test functions. The
---
RHS\邻ꂼɂĂ̕IAwIȈӖB

Iɂ͋ԃ S(R@- {a, b})͔g֐̋ԂƉ߂Bƕʂ͗LɂƂǂ܂
֌Ŵ悤ȑ㐔ZƒłBwIɂ͎֐̋ԂB

The spaces S(R- {a, b}) and ^~ SO~(R- {a, b}) contain respectively the bras and
kets associated with the eigenvalues that lie in the continuous spectrum. Physically, the
bras and kets are interpreted as gkernelsh of probability amplitudes. Mathematically,
the bras and kets are distributions. The following table summarizes the meanings of
each space:
---
S(R- {a, b}) Ƌ ^~ SO~(R- {a, b}) ͂ꂼAŗLlAł
uAPbg܂łB@Iɂ̓uƃPbg͊mÚ@ϕjgkernelsh B
wIɂ̓uƃPbg͒֐@distributions@B.

The following table summarizes the meanings of each space:
---
ȉ̃e[û悤ɐB

Space Physical Meaning Mathematical Meaning
@Space of wave functions @ Space of test functions
H Probability amplitudes Hilbert space
Space of kets |a> Antidual space
Space of bras <a| Dual space
---
ԁ@@IӖ@wIӖ
@g֐̋ Ӂ@ ֐̋
H mU qxg
Pbg̋ kets |a> o΋
ű <a| o΋

We have seen that, from a physical point of view, the rigged Hilbert space does
not entail an extension of Quantum Mechanics, whereas, from a mathematical point of
view, the rigged Hilbert space is an extension of the Hilbert space. Mathematically, the
rigged Hilbert space arises when we equip the Hilbert space with distribution theory.
Such equipment enables us to cope with singular objects such as bras and kets.
We have also seen that formal expressions involving bras and kets must be
understood as gsandwichedh by wave functions '. Such gsandwichingh by 'fs is
what controls the singular behavior of bras and kets. This is why mathematically the
sandwiching by 'fs is so important and must always be implicitly assumed. In practice,
we can freely apply the formal manipulations of Diracfs formalism with confidence, since
such formal manipulations are justified by the rigged Hilbert space.
We hope that this paper can serve as a pedagogical, enticing introduction to the
rigged Hilbert space.

---
Iɂ݂RHS͂Ȃʎq͊ẘg𔺂̂ł͂ȂAwIɂ̓qxgԂ̊gB
wIɂRHS̓qxgԂɒ֐𑕔̂B
̑ŃuAPbgƂ̂ƂłB
uƃPbg܂ޕ\́Ag֐ӂł͂܂邱ƂOɗ˂΂ȂȂB
ӂł̃ThCb@ŃuƃPbg̓قȂӂ܂łB

Ƃ͂ۂ̂Ƃ̓fBbN̋L@RɂĂ܂ȂBRHS𗠕tĂ邩炾B̘_RHS̓ƂĊw̖ɗƂFOB

Acknowledgments
Research supported by the Basque Government through reintegration fellowship
No. BCI03.96, and by the University of the Basque Country through research project
No. 9/UPV00039.310-15968/2004.
---
ӎ
͈̌ȉ̔ԍ̃oXN{̃tF[VbvƃvWFNgŎxꂽB
79 Ð@t 2015/02/07 (y) 09:02:32 ID:ctwIRbLQLU
References
---

[1]-[13]
[14] In Ref. [12], page 40, Dirac states that gthe bra and ket vectors that we now use form a more
general space than a Hilbert space.h
In Ref. [13], page viii, von Neumann states that gDirac has given a representation of quantum
mechanics which is scarcely to be surpassed in brevity and elegance, [...].h On pages viii-ix, von
Neumann says that gThe method of Dirac, mentioned above, (and this is overlooked today in a
great part of quantum mechanical literature, because of the clarity and elegance of the theory) in
no way satisfies the requirements of mathematical rigor – not even if these are reduced in a
natural and proper fashion to the extent common elsewhere in theoretical physics.h On page ix, von
Neumann says that g[...],this requires the introduction of eimproperf functions with self-
contradictory properties. The insertion of such mathematical efictionf is frequently necessary in
Diracfs approach,[...].h Thus, essentially, although von Neumann recognizes the clarity and
beauty of Diracfs formalism, he states very clearly that such formalism cannot be
implemented within the framework of the Hilbert space.
---
[14] fBbNA[12]P. 40, ggĂuƃPbg̓qxgԂƈʓIȋԂh
tHmC}. [13]@P. viii, gfBbN̍lʎq͊w̕\͂ƂȌŃGKgBh
P. viii-ix, g̃fBbN̕@ (ĂƉؗ킳ő̗ʎq͊w̕ɂ邪) wI
͖ĂȂB_̂ƂӂɎRɂƂĂABh P. ix, g[...],
ꂩ@ȂɖuqłȂv֐ KvɂȂBwI\ꂴȂ̂
fBbN̕@B[...].h

̂悤ɁAtHmC}̓fBbN̕@̖, qxgԂ̘gɂ
܂Ȃ̂Ƃ͂qׂĂB

[15]- [17].
[18] In Ref. [17], page 7, Maurin states that gIt seems to us that this is the formulation which
was anticipated by Dirac in his classic monograph.h
---
[18] [A[17], 7y[W g̋L@fBbbN͏̌ÓTI_łłɗ\Ă悤.h

[19]-[21]
[22] The following quotation, extracted from Ref. [3], page 19, gives a clear idea of the status
the RHS is achieving: g...rigged Hilbert space seems to be a more natural mathematical setting for
quantum mechanics than Hilbert space.h
---
[22] Ref. [3], 19y[W, RHS ̖ӎB: g...RHS̓qxgԂ莩RȐwIgg݂Bh
80 ********** 2015/02/07 (y) 13:13:58 ID:********** 폜: 21:16
ie҂ɂč폜܂j
81 Ð@t 2015/02/07 (y) 13:41:44 ID:ctwIRbLQLU
[23] -[24]
[25] A subspace S of H is dense in H if we can approximate any element of H by an element of S as
well as we wish. Thus, for any f of H and for any small ǫ > 0, we can find a ϕ in S such that
If | ϕI < ǫ. In physical terms, this inequality means that we can replace f by ϕ within an
accuracy ǫ.
[26] A function F : C is called a linear [respectively antilinear] functional over if for any
complex numbers , and for any ϕ, , it holds that F (ϕ + ) = F (ϕ) + F ()
[respectively F (ϕ + ) = ∗F (ϕ) + ∗F ()].
---
[25] H̕W S H fA܂H̔Cӂ̗vfłł߂Âł邱ƁAB
܂@Cӂ f H ƂłƂĂ ǫ > 0@ɂ, ŜȂɁ@If | ϕI < ǫ@𖞂ӂ݂BIȂł́A Cӂ̐x ǫ@f ϕ@ɒu@ƂB.
[26] ֐ F : C ́@Cӂ̕f , ƔCӂ ϕɂ, , F (ϕ + ) = F (ϕ) + F () [ F (ϕ + ) = ∗F (ϕ) + ∗F ()]𖞂ꍇ.
̐Ċ֐ [Ċ֐] Ƃ΂B

[27] -[29]
[30] Strictly speaking, a Hilbert space possesses additional properties (e.g., it must be complete
with respect to the topology induced by the scalar product). For a more technical definition of the
Hilbert space, see for example Ref. [11].
[31] An operator A is bounded if there is some finite K such that IAf I < KIf I for all f H,
where I I denotes the Hilbert space norm. When such K does not exist, A is said to be unbounded.
For a detailed account of the properties of bounded and unbounded operators, see for example Ref.
[11].
[32] The mathematical reason why quantum mechanical unbounded operators cannot be defined on all
the vectors of the Hilbert space can be found, for example, in Ref. [33], page 84.
---
[30] ɂ,qxgԂ͂Ƃق̑ (Ƃ΁@ς̂炷ʑɂĊ
łȂ΂ȂȂ).@肭킵qxgԂ̒͂ƂRef. [11]Q.
[31] Iy[^ A LEłƂ́@LȐ K ׂĂ f HɂābAfb < Kbfb@,
Łbb̓qxgԂ̃m̂
̂悤K݂ȂꍇAA ͔LEłƂB
ڂ[11]Q.
[32] LẼIy[^ȂqxgԂ̑SxNgĂł̂̐wIɂĂ Ref. [33], page 84@QƁB

[33] .
[34] If we nevertheless insisted in for example calculating the expectation value (2.24) for
elements of H
that are not in D(A), we would obtain an unphysical infinity value. For instance, if A represents
an unbounded Hamiltonian H, then the expectation value (2.24) would be infinite for those ϕ
of H that lie outside of D(H). Because they have infinite energy, those states do not represent
physically preparable wave packets.
[35] If they were in the Hilbert space, |aj and ia| would be square integrable, and a would belong
to the discrete spectrum.
[36] It is well known that Heisenbergfs commutation relation necessarily implies that either P or Q
is unbounded. See, for example, Ref. [33], page 274.
[37] The reason why the derivatives of ϕ(x) must vanish at x = a, b is that we want to be able to
apply the Hamiltonian H as many times as we wish. Since repeated applications of H to ϕ(x) involve
the derivatives of V (x)ϕ(x), and since V (x) is discontinuous at x = a, b, the function V (x)ϕ(x)
is infinitely differentiable at x = a, b only when the derivatives of ϕ(x) vanish at x = a, b. For
more details, see Ref. [19]. The vanishing of the derivatives of ϕ(x) at x = a, b must be viewed as
a mathematical consequence of the unphysical sharpness of the discontinuities of the potential,
rather than as a physical consequence of Quantum Mechanics. Note also that in standard numerical
simulations, for example, Gaussian wave packets impinging on a rectangular barrier, one never sees
that the wave packet vanishes at x = a, b. This is due to the fact that on a Gaussian wave packet,
the Hamiltonian (3.1) can only be applied once.
[38] We note that, when acting on elements ϕ of S(R {a, b}), the commutator [H, P ] = i V
reduces to [H, P ] = 0, due to the vanishing of the derivatives of ϕ at x = a, b.
[39] We recall that some authors have erroneously claimed that gthere are more kets than brash
[19], and that therefore such one-to-one correspondence between bras and kets does not hold.
[40] We can nevertheless extend Eqs. (4.41) and (4.43) to the whole Hilbert space L2 by a limiting
procedure, although the resulting expansions do not involve the Dirac bras and kets any more, but
simply the eigenfunctions of the differential operators.
---
[34] ɊҒl (2.24) D(A)ɂȂH̗vfɂČvZ悤Ƃ, 񕨗IȖ𓾂B
Ƃ A 񑩔Ԃ̃n~gjAHƁA D(H)̊Oɂ ϕHɂĊҒl(2.24)͖ɂȂ.
̃GlM[ȏԂ킷悤Ȕg͂ȂB
[35] ꂪqxgԂƂ, |aƁa|͓ϕ\ƂȂ藣UXyNg킷
Ƃ낤B
[36] nC[xŇ֌W P QLEł邱Ƃ𓱂Ƃ͂悭mĂBƂ
Ref. [33], page 274.
[37] ϕ(x)̓֐ x = a, b łOłȂ΂ȂȂŔAn~gjAHȂǂłp

, V (x)ϕ(x)́Aϕ(x) ̓֐ x = a, błOłꍇɌȂǂł\ɂȂB

sAłƂIłȂ肩̐wIAłėʎq͊w̕IAł͂ȂƂɒӁB
ӂ悭ڂɂKEXgł̐lV~[Vł x = a, bŔgOɂȂĂȂ̂̓KEXg
n~gjAi3.PjxpĂȂ炾ƂƂɂӁB

[41]
[42]@This is one of the major reasons why plane waves are so useful in practical calculations.
[43] This@is one of the major reasons why bras and kets are so useful in practical calculations.
[44] There are many other links between the classical and the quantum worlds, such as for example
the de Broglie relation p =   k, which entails a formal identity between the classical eikx and the
quantum eipx/   plane waves.
[45] We recall that the direct integral decomposition of the Hilbert space falls short of such
factorization, see Ref. [20].
---
[41]
[42]@ꂪp㕽ʔgƂĂ֗ȂȗR̂ЂƂB
[43]@ꂪpuƃPbgƂĂ֗ȂȗR̂ЂƂB
[44] ÓTƗʎqȂق̊֌W񂠂AƂ΃huC̊֌Wp =   kł͌ÓTI e^ikx ʎqIg e^ipx/  @Ȃ.

[45] qxgԂ́uڐϕvHł͏łȂƂɒ, Ref. [20]Q.

=Ð@t=
82 Ð@t 2015/02/07 (y) 18:02:10 ID:ctwIRbLQLU
zƍl@

>>9

͒疾炩A

ĤׂĂ̗vfɍp悤ȁ@[]Ċ֐ԁ@H*͓Rɂp邩

HH*Ƃɂ͊ĂȂB

H̗vf̓xNg
H*̗vf͊֐lɑΉĊ֐
܂ŗvf̎ނ܂łނ̂悤ɂ݂B

ԂƂā@xNgɁAf̉w̎̂悤ɁAς̎肪̂C[WĂB
ӂ̗vfƎRɎςԂ悤Ȃ̂̃C[WB

܂@H̗vf̃xNgɂ͓ς̎AH*̗vf̔Ċ֐Ƃ͗vɂ邫܂xNg
ςƂ邱ƂƂ݂ȂāAȂƂHAH*̂ӂ𓯂̂ɂ݂B

>>10 肪Ƃ܂B@ref. >>81 B

=Ð@t=
83 Ð@t 2015/02/07 (y) 19:51:56 ID:ctwIRbLQLU
>>22
@\ƂƂ̗̒ɎMȂBƂΉEAł͂߂B
@
̓I1˃|eVɂŌqB

>>23 >>25
@Q,P,Hł㐔̍ősϕ
(2.26)
ׂĂׂ̂̐ς̕Ԃ̋Ԃ̋ʕ

ȂɂĂ̊OɏoȂ
A , A = Q, P,H . (2.27)@

@Ƃ͕̂IɎێłĂԁ@ƑΉ̂ƂĂ͂킩₷

@ȂQ,P,HőĂ邩͖ȂB
@
>>29

<a| @u
|a> ^~.@Pbg

Ƃ̂́AĂC[WƂ͐EtBςƂ鑊iԁj̐ƎvȂĔ[Ă݂B
=Ð@t=
84 Ð@t 2015/02/07 (y) 20:54:32 ID:ctwIRbLQLU
@1ǌn

>>35@

V(a),V(b)͒ĂȂB
@VWK[i3.1jȂ肽ɂf(a)=f(b)=0łȂ΂ȂȂBŏڏqB

>>36 AC ͓֐݂悤Ȋ֐̋ԂB AC^2 ͓K֐݂悤Ȋ֐̋ԂB

Ƃ\L͓̂˂A܂\L̂Ȃ̂낤B

>>46@@@ősϕ
, (4.1)

ƂŖ\Ȃ̂͂Ƃā@ȂC[WɂB
ق̂Ƃł͂낢ȒlĂĂx=a,bł͔g֐̓֐ׂ̂Ă[Ƃ̂́@x=a,b̂łׂ肵Ă銴Be[[WJ̌W݂ȃ[Ȃ킯B

=Ð@t=
85 Ð@t 2015/02/07 (y) 21:43:09 ID:ctwIRbLQLU
>>49

 S(R- {a, b}) . (4.4)

Vc֐A}֐ƓƂ̓C[W₷AR- {a, b}̈Ӗɂ

Wiki Vc
---
VcԁA邢 Rn ̋}̋ԂƂ́A̔Ԃ̂ƂB

A ͑dwłAC(Rn) Rn C ւ̊炩Ȕ̏WłB
---
dŵƂ͂悭킩Ȃ̂łĂ

@R- {a, b}@͊JWȂ̂ŔƂłāA֐a̍ƉEŒlĂ͂ȂB

@x=a,bł͂ǂȋł悭Ał̒ĺi4.1j
@ƌ߂̂ƗB

^Fx={a,b}Ŋ֐lO^̂͂܂悢ƂāBǗ_Ȃ̂ɔAKAłĂlOƂ͓̂s悷ȂB
O̔̂Ȃ̐SzpB
86 Ð@t 2015/02/11 () 18:29:46 ID:ctwIRbLQLU
>>76
IuU[ou̘AȊϑlɂ
qxgԁFŗLxNg͖BɃXyNgxB
RHSF[]Ċ֐ԃfOXŁuŗLxNgvlBXyNgxlp͂ȂȂB

Uϑl̎ˉeZq
.
̃AiW[

. (6.3)
ӁFXyNgx@EӁFuPbgŕ\悤ɂȂB

.

ӂ̂悤ɕ\LĂǂ͊młȂB

Ql@XyNg藝
Theorem: Let A be a Hermitian operator. Then there exists a spectral measure E such
that for all v,w in H,

87 Ð@t 2015/02/14 (y) 15:20:02 ID:ctwIRbLQLU
RHŜƂƉĂu^2009NPDFB

http://www2.math.kyushu-u.ac.jp/~hara/lectures/09/QM_structure2.pdf
w҂̂߂̗ʎq͊w∗
@
Bww@@w@

3.1.1 Dirac ̋L@ƎOg

u3.9 (Gelfand̎Og)
ŌɁCDirac̋L@ʂ߂鎎݂ɂĊȒPɐGĂD
Dirac̋L@̍ő́iB́Hj_͌X݂Ȃ̘AXyNgɑ΂uŗLԁv|a〉Ƃ̋b〈a|iƂɏԋԁqxgԂ̌Ƃāj݂邩̂悤ȂӂƂɂD]āC̍瓦ɂ|·〉̋ԂH菬C〈·|̋Ԃ͂̋bԂƂH傫ƂCHƕĂ̂R̊Kw\lĂ΂悢D͐wIɂGelfand̎Og݂ƂĎ\łiႦS(Rn)H=L2(Rn)S(Rn)C[6]ȂǂQƁjv

Ȃ݂Ɍ搶Dirac̋ȏ@ref. 32>> ɂ
û[5]͐wIɓŝƂSĉ肵Đił܂ɋ낵{ł邪Cȕ͖{͂Ƃ񂾂ȁĈƂ͂ȉ肪񂾂ȁvƌƂ𑼂̖{CႦ[9, 7]CŕĂ΁C͑ϖɓǂ߁C҂̋CĂ閼łDv
ƂĂ̂́A䂪ӂ𓾂B