QuantumPhysics.dvi

formulation. A second is via thefunctional or path integralwhich directly produces prob-

ability amplitudes from a classical mechanics system in the Lagrangianformulation; this

method will be discussed in the next section.

Let the classical mechanics system be given by a HamiltonianH(q,p) on a phase space

qi,pj, withi,j= 1,···,N,

{qi,pj}=δi,j a ̇(q,p) ={a,H} (6.26)

The correspondence principle states that, to this classical system, there corresponds a quan-

tum system with the following observables,

classical variable Quantum Observable

qi → Qi

pi → Pi

a(p,q) → A(P,Q)

{a,b} → −

i

̄h

[A,B] (6.27)

In particular, the classical Hamiltonian has a quantum counterpartH(P,Q), and the Poisson

bracket relations betweenqiandpjbecome the canonical commutation relations,

[Qi,Pj] =i ̄hδi,j [Qi,Qj] = [Pi,Pj] = 0 (6.28)

The correspondence principle maps the Hamilton time-evolution equations into the Schr ̈odinger

equation for observables, i.e. given in the Heisenberg formulation,

i ̄h

d

dt

A= [A,H] +i ̄h

∂

∂t

A (6.29)

The evolution operator may be used to translate the Heisenberg formulation into the Schr ̈odinger

formulation.

6.4 Schr ̈odinger equation with a scalar potential

In the usualposition realizationof the canonical commutation relations, given by

Qi=qi Pi=−i ̄h

∂

∂qi

(6.30)

the Hilbert space is that of of square integrable functions ofqi. The simplest non-relativistic

classical mechanical systems, given in terms of a potentialV(q), with Hamiltonian,

H(q,p) =

p^2

2 m

+V(q) (6.31)