Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 20
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(1.101). The correspondence principle requires replacing the momenta pµ by the momentum operators p̂µ ,but it does not specify the position of these operators with respect to the coordinates q µ contained in the inverse metric g µν (q). An important constraint is providedby the required Hermiticity of the Hamiltonian operator, but this is not sufficientfor a unique specification.
We may, for instance, define the canonical Hamiltonianoperator as1 µĤcan ≡p̂ gµν (q)p̂ν + V (q),(1.392)2Min which the momentum operators have been arranged symmetrically around theinverse metric to achieve Hermiticity. This operator, however, is not equal to the561 Fundamentalscorrect Schrödinger operator in (1.380).
The kinetic term contains what we maycall the canonical Laplacian∆can = (∂µ + 12 Γµ ) g µν (q) (∂ν + 21 Γν ).(1.393)It differs from the Laplace-Beltrami operator (1.378) in (1.380) by∆ − ∆can = − 21 ∂µ (g µν Γν ) − 14 g µν Γν Γµ .(1.394)The correct Hamiltonian operator could be obtained by suitably distributing pairs ofdummy factors of g 1/4 and g −1/4 symmetrically between the canonical operators [5]:Ĥ =1 −1/4gp̂µ g 1/4 g µν (q)g 1/4 p̂ν g −1/4 + V (q).2M(1.395)This operator has the same classical limit (1.391) as (1.392).
Unfortunately, thecorrespondence principle does not specify how the classical factors have to be orderedbefore being replaced by operators.The simplest system exhibiting the breakdown of the canonical quantization rulesis a free particle in a plane described by radial coordinates q 1 = r, q 2 = ϕ:x1 = r cos ϕ, x2 = r sin ϕ.(1.396)Since the infinitesimal square distance is ds2 = dr 2 + r 2 dϕ2 , the metric reads1 00 r2gµν =!.(1.397)µνIt has a determinantg = r2and an inverseµν1 00 r −2(1.398)!µν.(1.399)11∆ = ∂r r∂r + 2 ∂ϕ 2 .rr(1.400)g=The Laplace-Beltrami operator becomesThe canonical Laplacian, on the other hand, reads1 2∂r2 ϕ111= ∂r 2 + ∂r − 2 + 2 ∂ϕ 2 .r4rr∆can = (∂r + 1/2r)2 +(1.401)The discrepancy (1.394) is therefore∆can − ∆ = −1.4r 2(1.402)H. Kleinert, PATH INTEGRALS571.14 Particle on the Surface of a SphereNote that this discrepancy arises even though there is no apparent ordering problemin the naively quantized canonical expression p̂µ gµν (q) p̂ν in (1.401).
Only the need tointroduce dummy g 1/4 - and g −1/4 -factors creates such problems, and a specificationof the order is required to obtain the correct result.If the Lagrangian coordinates qi do not merely reparametrize a Euclidean spacebut specify the points of a general geometry, we cannot proceed as above and derive the Laplace-Beltrami operator by a coordinate transformation of a CartesianLaplacian. With the canonical quantization rules being unreliable in curvilinearcoordinates there are, at first sight, severe difficulties in quantizing such a system.This is why the literature contains many proposals for handling this problem [6].Fortunately, a large class of non-Cartesian systems allows for a unique quantummechanical description on completely different grounds. These systems have thecommon property that their Hamiltonian can be expressed in terms of the generators of a group of motion in the general coordinate frame.
For symmetry reasons,the correspondence principle must then be imposed not on the Poisson brackets ofthe canonical variables p and q, but on those of the group generators and the coordinates. The brackets containing two group generators specify the structure of thegroup, those containing a generator and a coordinate specify the defining representation of the group in configuration space. The replacement of these brackets bycommutation rules constitutes the proper generalization of the canonical quantization from Cartesian to non-Cartesian coordinates.
It is called group quantization.The replacement rule will be referred to as group correspondence principle. Thecanonical commutation rules in Euclidean space may be viewed as a special caseof the commutation rules between group generators, i.e., of the Lie algebra of thegroup. In a Cartesian coordinate frame, the group of motion is the Euclidean groupcontaining translations and rotations. The generators of translations and rotationsare the momenta and the angular momenta, respectively. According to the groupcorrespondence principle, the Poisson brackets between the generators and the coordinates are to be replaced by commutation rules. Thus, in a Euclidean space, thecommutation rules between group generators and coordinates lead to the canonical quantization rules, and this appears to be the deeper reason why the canonicalrules are correct.
In systems whose energy depends on generators of the group ofmotion other than those of translations, for instance on the angular momenta, thecommutators between the generators have to be used for quantization rather thanthe canonical commutators between positions and momenta.The prime examples for such systems are a particle on the surface of a sphere ora spinning top whose quantization will now be discussed.1.14Particle on the Surface of a SphereFor a particle moving on the surface of a sphere of radius r with coordinatesx1 = r sin θ cos ϕ, x2 = r sin θ sin ϕ, x3 = r cos θ,(1.403)581 Fundamentalsthe Lagrangian readsL=Mr 2 2(θ̇ + sin2 θ ϕ̇2 ).2(1.404)The canonical momenta arepθ = Mr 2 θ̇,pϕ = Mr 2 sin2 θ ϕ̇,(1.405)and the classical Hamiltonian is given by11 22H=p .2 pθ +2Mrsin2 θ ϕ(1.406)According to the canonical quantization rules, the momenta should become operators1(1.407)p̂θ = −ih̄ 1/2 ∂θ sin1/2 θ, p̂ϕ = −ih̄∂ϕ .sin θBut as explained in the previous section, these momentum operators are not expected to give the correct Hamiltonian operator when inserted into the Hamiltonian(1.406).
Moreover, there exists no proper coordinate transformation from the surface of the sphere to Cartesian coordinates17 such that a particle on a sphere cannotbe treated via the safe Cartesian quantization rules (1.269):[p̂i , x̂j ] = −ih̄δi j ,[x̂i , x̂j ] = 0,[p̂i , p̂j ] = 0.(1.408)The only help comes from the group properties of the motion on the surface of thesphere. The angular momentumL=x×p(1.409)can be quantized uniquely in Cartesian coordinates and becomes an operatorL̂ = x̂ × p̂(1.410)whose components satisfy the commutation rules of the Lie algebra of the rotationgroup[L̂i , L̂j ] = ih̄L̂k(i, j, k cyclic).(1.411)Note that there is no factor-ordering problem since the x̂i ’s and the p̂i ’s appearwith different indices in each Lˆk . An important property of the angular momentum17There exist, however, certain infinitesimal nonholonomic coordinate transformations which aremultivalued and can be used to transform infinitesimal distances in a curved space into those in aflat one.
They are introduced and applied in Sections 10.2 and Appendix 10A, leading once moreto the same quantum mechanics as the one described here.H. Kleinert, PATH INTEGRALS591.14 Particle on the Surface of a Sphereoperator is its homogeneity in x.
It has the consequence that when going fromCartesian to spherical coordinatesx1 = r sin θ cos ϕ, x2 = r sin θ sin ϕ, x3 = r cos θ,(1.412)the radial coordinate cancels making the angular momentum a differential operatorinvolving only the angles θ, ϕ:L̂1 =ih̄ sin ϕ ∂θ + cot θ cos ϕ ∂ϕ ,L̂2 = −ih̄ cos ϕ ∂θ − cot θ sin ϕ ∂ϕ ,L̂3 = −ih̄∂ϕ .(1.413)There is then a natural way of quantizing the system which makes use of theseoperators L̂i .
We re-express the classical Hamiltonian (1.406) in terms of the classicalangular momentaL1 = Mr 2 − sin ϕ θ̇ − sin θ cos θ cos ϕ ϕ̇ ,L2 = Mr 2 cos ϕ θ̇ − sin θ cos θ sin ϕ ϕ̇ ,22L3 = Mr sin θ ϕ̇(1.414)as1L2 ,(1.415)2Mr 2and replace the angular momenta by the operators (1.413). The result is the Hamiltonian operator:H=11h̄212∂θ (sin θ ∂θ ) +∂2 .L̂=−Ĥ =222Mr2Mr sin θsin2 θ ϕ(1.416)The eigenfunctions diagonalizing the rotation-invariant operator L̂2 are well known.They can be chosen to diagonalize simultaneously one component of L̂i , for instancethe third one, L̂3 , in which case they are equal to the spherical harmonicsYlm (θ, ϕ) = (−1)m"2l + 1 (l − m)!4π (l + m)!#1/2Plm (cos θ)eimϕ ,(1.417)with Plm (z) being the associated Legendre polynomialsPlm (z)l+m12 m/2 d2l= l (1 − z )l+m (z − 1) .2 l!dx(1.418)The spherical harmonics are orthonormal with respect to the rotation-invariantscalar productZ0πdθ sin θZ02π∗dϕ Ylm(θ, ϕ)Yl0m0 (θ, ϕ) = δll0 δmm0 .(1.419)601 FundamentalsTwo important lessons can be learned from this group quantization.
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