Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 13
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The equation suggests that the motion of a particle with an arbitrary Hamiltonian H(p, x, t)follows the straightforward generalization of (1.90)Ĥ − ih̄∂t Ψ(x, t) = 0,(1.91)where Ĥ is the differential operatorĤ ≡ H(−ih̄∇, x, t).(1.92)The rule of obtaining Ĥ from the classical Hamiltonian H(p, x, t) by the substitutionp → p̂ = −ih̄∇ will be referred to as the correspondence principle.4 We shall see inSections 1.13–1.15 that this simple correspondence principle holds only in Cartesiancoordinates.The Schrödinger operators (1.89) of momentum and energy satisfy with x and tthe so-called canonical commutation relations[p̂i , xj ] = −ih̄,4[Ê, t] = 0 = ih̄.(1.93)Our formulation of this principle is slightly stronger than the historical one used in the initialphase of quantum mechanics, which gave certain translation rules between classical and quantummechanical relations.
The substitution rule for the momentum runs also under the name Jordanrule.161 FundamentalsIf the Hamiltonian does not depend explicitly on time, the Hilbert space canbe spanned by the energy eigenstates states ΨEn (x, t) = e−iEn t/h̄ ΨEn (x), whereΨEn (x) are time-independent stationary states, which solve the time-independentSchrödinger equationĤ(p̂, x)ΨEn (x) = En ΨEn (x).(1.94)The validity of the Schrödinger theory (1.91) is confirmed by experiment, mostnotably for the Coulomb HamiltonianH(p, x) =p2e2− ,2Mr(1.95)which governs the quantum mechanics of the hydrogen atom in the center-of-masscoordinate system of electron and proton, where M is the reduced mass of the twoparticles.Since the square of the wave function, |Ψ(x, t)|2 , is interpreted as the probabilitydensity of a single particle in a finite volume, the integral over the entire volumemust be normalized to unity:Zd3 x |Ψ(x, t)|2 = 1.(1.96)For a stable particle, this normalization must remain the same at all times.
IfΨ(x, t) is to follow the Schrödinger equation (1.91), this is assured if and only if theHamiltonian operator is Hermitian,5 i.e., if it satisfies for arbitrary wave functionsΨ1 , Ψ2 the equalityZd3 x [ĤΨ2 (x, t)]∗ Ψ1 (x, t) =Zd3 x Ψ∗2 (x, t)ĤΨ1 (x, t).(1.97)The left-hand side defines the Hermitian-adjoint Ĥ † of the operator Ĥ, which satisfies for all wave functions Ψ1 (x, t), Ψ2 (x, t) the identityZd3x Ψ∗2 (x, t)Ĥ † Ψ1 (x, t)≡Zd3 x [ĤΨ2 (x, t)]∗ Ψ1 (x, t).(1.98)An operator Ĥ is Hermitian, if it coincides with its Hermitian-adjoint Ĥ † :Ĥ = Ĥ † .(1.99)5Problems arising from unboundedness or discontinuities of the Hamiltonian and otherquantum-mechanical operators, such as restrictions of the domains of definition, are ignored heresince they are well understood.
Correspondingly we do not distinguish between Hermitian and selfadjoint operators (see J. v. Neumann, Mathematische Grundlagen der Quantenmechanik , Springer,Berlin, 1932). Some quantum-mechanical operator subtleties will manifest themselves in this bookas problems of path integration to be solved in Chapter 12.
The precise relationship between thetwo calls for further detailed investigations.H. Kleinert, PATH INTEGRALS171.3 Quantum MechanicsLetus calculate the time change of the integral over two arbitrary wave functions,Z3d x Ψ∗2 (x, t)Ψ1 (x, t). With the Schrödinger equation (1.91), this time change van-ishes indeed as long as Ĥ is Hermitian:ih̄dd3 x Ψ∗2 (x, t)Ψ1 (x, t)dt ZZ=d3 x Ψ∗2 (x, t)ĤΨ1 (x, t) − d3 x [ĤΨ2 (x, t)]∗ Ψ1 (x, t) = 0.Z(1.100)Thisalso implies the time independence of the normalization integralR 3d x |Ψ(x, t)|2 = 1.Conversely, if Ĥ is not Hermitian, one can always find an eigenstate of Ĥ whosenorm changes with time: any eigenstate of (H − H † )/i has this property.Since p̂ = −ih̄∇ and x are themselves Hermitian operators, Ĥ will automaticallybe a Hermitian operator if it is a sum of a kinetic and a potential energy:H(p, x, t) = T (p, t) + V (x, t).(1.101)This is always the case for nonrelativistic particles in Cartesian coordinates x.
If pand x appear in one and the same term of H, for instance as p2 x2 , the correspondence principle does not lead to a unique quantum-mechanical operator Ĥ. Thenthere seem to be, in principle, several Hermitian operators which, in the above example, can be constructed from the product of two p̂ and two x̂ operators [for instanceαp̂2 x̂2 +β x̂2 p̂2 +γ p̂x̂2 p̂ with α+β+γ = 1]. They all correspond to the same classicalp2 x2 . At first sight it appears as though only a comparison with experiment couldselect the correct operator ordering. This is referred to as the operator-orderingproblem of quantum mechanics which has plagued many researchers in the past.
Ifthe ordering problem is caused by the geometry of the space in which the particlemoves, there exists a surprisingly simple geometric principle which specifies the ordering in the physically correct way. Before presenting this in Chapter 10 we shallavoid ambiguities by assuming H(p, x, t) to have the standard form (1.101), unlessotherwise stated.1.3.4Particle Current ConservationThe conservation of the total probability (1.96) is a consequence of a more generallocal conservation law linking the current density of the particle probabilityj(x, t) ≡ −i↔h̄ψ(x, t) ∇ ψ(x, t)2m(1.102)with the probability densityρ(x, t) = ψ ∗ (x, t)ψ(x, t)(1.103)∂t ρ(x, t) = −∇ · j(x, t).(1.104)via the relation181 FundamentalsBy integrating this current conservation law over a volume V enclosed by a surfaceS, and using Green’s theorem, one findsZ3Vd x ∂t ρ(x, t) = −ZV3d x ∇ · j(x, t) = −ZSdS · j(x, t),(1.105)where dS are the directed infinitesimal surface elements.
This equation states thatthe probability in a volume decreases by the same amount by which probabilityleaves the surface via the current j(x, t).By extending the integral (1.105) over the entire space and assuming the currentsto vanish at spatial infinity, we recover the conservation of the total probability(1.96).More general dynamical systems with N particles in Euclidean space areparametrized in terms of 3N Cartesian coordinates xν (ν = 1, . . . , N). The Hamiltonian has the formNXp2νH(pν , xν , t) =+ V (xν , t),(1.106)ν=1 2Mνwhere the arguments pν , xν in H and V stand for all pν ’s, xν with ν = 1, 2, 3, . .
. , N.The wave function Ψ(xν , t) satisfies the N-particle Schrödinger equation(1.4−NXν=1"h̄2∂xν 2 + V (xν , t)2Mν#)Ψ(xν , t) = ih̄∂t Ψ(xν , t).(1.107)Dirac’s Bra-Ket FormalismMathematically speaking, the wave function Ψ(x, t) may be considered as a vector inan infinite-dimensional complex vector space called Hilbert space. The configurationspace variable x plays the role of a continuous “index” of these vectors. An obviouscontact with the usual vector notation may be established, in which a D-dimensionalvector v is given in terms of its components vi with a subscript i = 1, . . . D, bywriting the argument x of Ψ(x, t) as a subscript:Ψ(x, t) ≡ Ψx (t).(1.108)The usual norm of a complex vector is defined by|v|2 =Xvi∗ vi .(1.109)Z(1.110)iThe continuous version of this is2|Ψ| =Zd3x Ψ∗x (t)Ψx (t)=d3 x Ψ∗ (x, t)Ψ(x, t).The normalization condition (1.96) requires that the wave functions have the norm|Ψ| = 1, i.e., that they are unit vectors in the Hilbert space.H.
Kleinert, PATH INTEGRALS191.4 Dirac’s Bra-Ket Formalism1.4.1Basis TransformationsIn a vector space, there are many possible choices of orthonormal basis vectors bi alabeled by a = 1, . . . , D, in terms of which6vi =Xbi a va ,(1.111)awith the components va given by the scalar productsva ≡bi a∗ vi .X(1.112)iThe latter equation is a consequence of the orthogonality relation 7X00bi a∗ bi a = δ aa ,(1.113)iwhich in a finite-dimensional vector space implies the completeness relationXabi a∗ bj a = δ ij .(1.114)In the space of wave functions (1.108) there exists a special set of basis functionscalled local basis functions is of particular importance. It may be constructed in thefollowing fashion: Imagine the continuum of space points to be coarse-grained intoa cubic lattice of mesh size , at positionsn1,2,3 = 0, ±1, ±2, . .
. .xn = (n1 , n2 , n3 ),(1.115)Let hn (x) be a function that vanishes everywhere in space, except in a cube of size3 centered around xn , i.e., for each component xi of x,( √|xi − xn i | ≤ /2, i = 1, 2, 3.1/ 3n(1.116)h (x) =0otherwise.These functions are certainly orthonormal:Z00d3 x hn (x)∗ hn (x) = δ nn .(1.117)Consider now the expansionΨ(x, t) =Xhn (x)Ψn (t)(1.118)nP(b)Mathematicians would expand more precisely vi = a bi a va , but physicists prefer to shortenthe notation by distinguishing the different components via different types of subscripts, using forthe initial components i, j, k, .
. . and for the b-transformed components a, b, c, . . . .7An orthogonality relation implies usually a unit norm and is thus really an orthonormalityrelation but this name is rarely used.6201 Fundamentalswith the coefficientsΨn (t) =Z3∗nd x h (x) Ψ(x, t) ≈q3 Ψ(xn , t).(1.119)It provides an excellent approximation to the true wave function Ψ(x, t), as long asthe mesh size is much smaller than the scale over which Ψ(x, t) varies. In fact, ifΨ(x, t) is integrable, the integral over the sum (1.118) will always converge to Ψ(x, t).The same convergence of discrete approximations is found in any scalar product,and thus in any observable probability amplitudes.
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