Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 14
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They can all be calculated witharbitrary accuracy knowing the discrete components of the type (1.119) in the limit → 0. The functions hn (x) may therefore be used as an approximate basis in thesame way as the previous basis functions f a (x), g b(x), with any desired accuracydepending on the choice of .In general, there are many possible orthonormal basis functions f a (x) in theHilbert space which satisfy the orthonormality relationZ00d3 x f a (x)∗ f a (x) = δ aa ,(1.120)in terms of which we can expandΨ(x, t) =Xaf a (x)Ψa (t),(1.121)with the coefficientsΨa (t) =Zd3 x f a (x)∗ Ψ(x, t).(1.122)Suppose we use other orthonormal basis f˜b (x) with the orthonormality relationZ00d3 x f˜b (x)∗ f˜b (x) = δ bb ,Xbf˜b (x)f˜b (x)∗ = δ (3) (x − x0 ),(1.123)to re-expandΨ(x, t) =f˜b (x)Ψ̃b (t),(1.124)d3 x f˜b (x)∗ Ψ(x, t).(1.125)Xbwith the componentsΨ̃b (t) =ZInserting (1.121) shows that the components are related to each other byΨ̃b (t) =X Zad3 x f˜b (x)∗ f a (x) Ψa (t).(1.126)H.
Kleinert, PATH INTEGRALS211.4 Dirac’s Bra-Ket Formalism1.4.2Bracket NotationIt is useful to write the scalar products between two wave functions occurring in theabove basis transformations in the so-called bracket notation ashb̃|ai ≡Zd3 x f˜b (x)∗ f a (x).(1.127)In this notation, the components of the state vector Ψ(x, t) in (1.122), (1.125) areΨa (t) = ha|Ψ(t)i,(1.128)Ψ̃b (t) = hb̃|Ψ(t)i.The transformation formula (1.126) takes the formhb̃|Ψ(t)i =Xahb̃|aiha|Ψ(t)i.(1.129)The right-hand side of this equation may be formally viewed as a result of insertingthe abstract relationX|aiha| = 1(1.130)abetween hb̃| and |Ψ(t)i on the left-hand side:hb̃|Ψ(t)i = hb̃|1|Ψ(t)i =Xahb̃|aiha|Ψ(t)i.(1.131)Since this expansion is only possible if the functions f b (x) form a complete basis,the relation (1.130) is alternative, abstract way of stating the completeness of thebasis functions.
It may be referred to as completeness relation à la Dirac.Since the scalar products are written in the form of brackets ha|a0 i, Dirac calledthe formal objects ha| and |a0 i, from which the brackets are composed, bra and ket,respectively. In the bracket notation, the orthonormality of the basis f a (x) andg b (x) may be expressed as follows:ha|a0 i =hb̃|b̃0 i =ZZ00d3 x f a (x)∗ f a (x) = δ aa ,00d3 x f˜b (x)∗ f˜b (x) = δ bb .(1.132)In the same spirit we introduce abstract bra and ket vectors associated with thebasis functions hn (x) of Eq. (1.116), denoting them by hxn | and |xn i, respectively,and writing the orthogonality relation (1.117) in bracket notation ashxn |xn0 i ≡Z0d3 x hn (x)∗ hn (x) = δnn0 .(1.133)The components Ψn (t) may be considered as the scalar productsΨn (t) ≡ hxn |Ψ(t)i ≈q3 Ψ(xn , t).(1.134)221 FundamentalsChanges of basis vectors, for instance from |xn i to the states |ai, can be performedaccording to the rules developed above by inserting a completeness relation à laDirac of the type (1.130).
Thus we may expandΨn (t) = hxn |Ψ(t)i =Xahxn |aiha|Ψ(t)i.(1.135)Also the inverse relation is true:ha|Ψ(t)i =Xnha|xn ihxn |Ψ(t)i.(1.136)This is, of course, just an approximation to the integralZd3 x hn (x)∗ hx|Ψ(t)i.(1.137)The completeness of the basis hn (x) may therefore be expressed via the abstractrelationX|xn ihxn | ≈ 1.(1.138)nThe approximate sign turns into an equality sign in the limit of zero mesh size, → 0.1.4.3Continuum LimitIn ordinary calculus, finer and finer sums are eventually replaced by integrals. Thesame thing is done here. We define new continuous scalar products1hx|Ψ(t)i ≈ √ 3 hxn |Ψ(t)i,(1.139)where xn are the lattice points closest to x. With (1.134), the right-hand side isequal to Ψ(xn , t).
In the limit → 0, x and xn coincide and we havehx|Ψ(t)i ≡ Ψ(x, t).(1.140)The completeness relation can be used to writeha|Ψ(t)i ≈Xha|xn ihxn |Ψ(t)i≈X3 ha|xihx|Ψ(t)innwhich in the limit → 0 becomesha|Ψ(t)i =Z(1.141)x=xnd3 x ha|xihx|Ψ(t)i.,(1.142)This may be viewed as the result of inserting the formal completeness relation ofthe limiting local bra and ket basis vectors hx| and |xi,Zd3 x |xihx| = 1,(1.143)H. Kleinert, PATH INTEGRALS231.4 Dirac’s Bra-Ket Formalismevaluated between the vectors ha| and |Ψ(t)i.With the limiting local basis, the wave functions can be treated as componentsof the state vectors |Ψ(t)i with respect to the local basis |xi in the same way as anyother set of components in an arbitrary basis |ai.
In fact, the expansionha|Ψ(t)i =Zd3 x ha|xihx|Ψ(t)i(1.144)may be viewed as a re-expansion of a component of |Ψ(t)i in one basis, |ai, intothose of another basis, |xi, just as in (1.129).In order to express all these transformation properties in a most compact notation, it has become customary to deal with an arbitrary physical state vector in abasis-independent way and denote it by a ket vector |Ψ(t)i. This vector may be specified in any convenient basis by multiplying it with the corresponding completenessrelationX|aiha| = 1,(1.145)aresulting in the expansion|Ψ(t)i =Xa|aiha|Ψ(t)i.(1.146)This can be multiplied with any bra vector, say hb|, from the left to obtain theexpansion formula (1.131):hb|Ψ(t)i =Xahb|aiha|Ψ(t)i.(1.147)The continuum version of the completeness relation (1.138) readsZd3 x |xihx| = 1,(1.148)and leads to the expansion|Ψ(t)i =Zd3 x |xihx|Ψ(t)i,(1.149)in which the wave function Ψ(x, t) = hx|Ψ(t)i plays the role of an xth componentof the state vector |Ψ(t)i in the local basis |xi.
This, in turn, is the limit of thediscrete basis vectors |xn i,1(1.150)|xi ≈ √ 3 |xn i ,with xn being the lattice points closest to x.A vector can be described equally well in bra or in ket form. To apply the aboveformalism consistently, we observe that the scalar productsha|b̃i =hb̃|ai =ZZd3 x f a (x)∗ f˜b (x),d3 x f˜b (x)∗ f a (x)(1.151)241 Fundamentalssatisfy the identityhb̃|ai ≡ ha|b̃i∗ .(1.152)Therefore, when expanding a ket vector as|Ψ(t)i =X|aiha|Ψ(t)i,(1.153)hΨ(t)| =XhΨ(t)|aiha|,(1.154)aor a bra vector asaa multiplication of the first equation with the bra hx| and of the second with the ket|xi produces equations which are complex-conjugate to each other.1.4.4Generalized FunctionsDirac’s bra-ket formalism is elegant and easy to handle.
As far as the vectors |xi areconcerned there is, however, one inconsistency with some fundamental postulates ofquantum mechanics: When introducing state vectors, the norm was required to beunity in order to permit a proper probability interpretation of single-particle states.The limiting states |xi introduced above do not satisfy this requirement. In fact,the scalar product between two different states hx| and |x0 i ishx|x0 i ≈113 hxn |xn0 i = 3 δnn0 ,(1.155)where xn and xn0 are the lattice points closest to x and x0 . For x 6= x0 , the states areorthogonal.
For x = x0 , on the other hand, the limit → 0 is infinite, approachedin such a way thatX 1(1.156)33 δnn0 = 1.n0 Therefore, the limiting state |xi is not a properly normalizable vector in the Hilbertspace. For the sake of elegance, it is useful to weaken the requirement of normalizability (1.96) by admitting the limiting states |xi to the physical Hilbert space.In fact, one admits all states which can be obtained by a limiting sequence fromproperly normalized state vectors.The scalar product between states hx|x0 i is not a proper function. It is denotedby the symbol δ (3) (x − x0 ) and called Dirac δ-function:hx|x0 i ≡ δ (3) (x − x0 ).(1.157)The right-hand side vanishes everywhere, except in the infinitely small box of width around x ≈ x0 . Thus the δ-function satisfiesδ (3) (x − x0 ) = 0forx 6= x0 .(1.158)H.
Kleinert, PATH INTEGRALS251.4 Dirac’s Bra-Ket FormalismAt x = x0 , it is so large that its volume integral is unity:Zd3 x0 δ (3) (x − x0 ) = 1.(1.159)Obviously, there exists no proper function that can satisfy both requirements, (1.158)and (1.159). Only the finite- approximation in (1.155) to the δ-function are properfunctions. In this respect, the scalar product hx|x0 i behaves just like the states |xithemselves: Both are → 0 -limits of properly defined mathematical objects.Note that the integral Eq. (1.159) implies the following property of the δ function:1 (3)δ (x − x0 ).(1.160)δ (3) (a(x − x0 )) =|a|In one dimension, this leads to the more general relationδ(f (x)) =Xi1δ(x − xi ),|f (xi )|0(1.161)where xi are the simple zeros of f (x).In mathematics, one calls the δ-function a generalized function or a distribution.It defines a linear functional of arbitrary smooth test functions f (x) which yieldsits value at any desired place x:δ[f ; x] ≡Zd3 x δ (3) (x − x0 )f (x0 ) = f (x).(1.162)Test functions are arbitrarily often differentiable functions with a sufficiently fastfalloff at spatial infinity.There exist a rich body of mathematical literature on distributions [3].
Theyform a linear space. This space is restricted in an essential way in comparisonwith ordinary functions: products of δ-functions or any other distributions remainundefined. In Section 10.8.1 we shall find, however, that physics forces us to gobeyond these rules. An important requirement of quantum mechanics is coordinateinvariance. If we want to achieve this for the path integral formulation of quantummechanics, we must set up a definite extension of the existing theory of distributions,which specifies uniquely integrals over products of distributions.In quantum mechanics, the role of the test functions is played by the wave packetsΨ(x, t). By admitting the generalized states |xi to the Hilbert space, we also admitthe scalar products hx|x0 i to the space of wave functions, and thus all distributions,although they are not normalizable.1.4.5Schrödinger Equation in Dirac NotationIn terms of the bra-ket notation, the Schrödinger equation can be expressed in abasis-independent way as an operator equationĤ|Ψ(t)i ≡ H(p̂, x̂, t)|Ψ(t)i = ih̄∂t |Ψ(t)i,(1.163)261 Fundamentalsto be supplemented by the following specifications of the canonical operators:hx|p̂ ≡ −ih̄∇hx|,hx|x̂ ≡ xhx|.(1.164)(1.165)Any matrix element can be obtained from these equations by multiplication fromthe right with an arbitrary ket vector; for instance with the local basis vector |x0 i:hx|p̂|x0 i = −ih̄∇hx|x0 i = −ih̄∇δ (3) (x − x0 ),(1.166)hx|x̂|x0 i = xhx|x0 i = xδ (3) (x − x0 ).(1.167)The original differential form of the Schrödinger equation (1.91) follows by multiplying the basis-independent Schrödinger equation (1.163) with the bra vector hx|from the left:hx|H(p̂, x̂, t)|Ψ(t)i = H(−ih̄∇, x, t)hx|Ψ(t)i= ih̄∂t hx|Ψ(t)i.(1.168)Obviously, p̂ and x̂ are Hermitian matrices in any basis,ha|p̂|a0 i = ha0 |p̂|ai∗ ,(1.169)ha|x̂|a0 i = ha0 |x̂|ai∗ ,(1.170)ha|Ĥ|a0 i = ha0 |Ĥ|ai∗ ,(1.171)Ô(t) ≡ O(p̂, x̂, t).(1.172)and so is the Hamiltonianas long as it has the form (1.101).The most general basis-independent operator that can be constructed in thegeneralized Hilbert space spanned by the states |xi is some function of p̂, x̂, t,In general, such an operator is called Hermitian if all its matrix elements havethis property.
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