QuantumPhysics.dvi

have a unique quantum Hamiltonian, given by the correspondence principle,

H(Q,P) =

P^2

2 m

+V(Q) (6.32)

In the position representation ofQiandPi, the Hamiltonian becomes a differential operator

acting on wave functions. The time-dependent Schr ̈odinger equation then becomes,

i ̄h

∂ψ(q,t)

∂t

=−

̄h^2

2 m

∆qψ(q.t) +V(q)ψ(q,t) (6.33)

where ∆ = ∆qis the standard Laplace operator on functions of~q, defined by

∆ = ∆q≡

∑N

i=1

∂^2

∂qi∂qi

(6.34)

Specializing to energy eigenvalues and eigenfunctions,

ψ(q,t) =e−iEt/ ̄hψE(q) (6.35)

we recover the standard time-independent Schr ̈odinger equation,

−

̄h^2

2 m

∆ψE(q) +V(q)ψE(q) =EψE(q) (6.36)

6.5 Uniqueness questions of the correspondence principle

But note that this standard realization isnot unique. For example, we could just as well

have chosen themomentum realizationof the canonical commutation relations, in which we

would have instead,Pi=piandQi= +i ̄h∂/∂pi. The Hilbert space is simply the space of

square integrable functions ofpi. Such alternative realizations are sometimes really useful.

Suppose we had to solve the very bizarre looking quantum system for the Hamiltonian,

H=

C 1

|P|

+C 2 X^2 (6.37)

In the momentum realization, the problem is actually just the Coulombproblem !!

For the simplest classical mechanical systems, the corresponding quantum system is

unique, but this need not be so in general. Suppose, for example, that the classical Hamilto-

nian contained not just a potential ofq, but also an interaction of the type 2p^2 U(p) for some

functionU. There are clearly two (and in fact an infinite number) of inequivalentways of

writing down a corresponding (self-adjoint) quantum interaction.For example,

2 pU(q)p 6 = p^2 U(q) +U(q)p^2 (6.38)

From classical mechanics, there is no way to decide. On the quantumside, both produce

self-adjoint Hamiltonians. Thisordering ambiguityalmost never matters much in quantum

mechanics but it does play a crucial role in quantum field theory.