QuantumPhysics.dvi

Since the number of dynamical variables on phase space has been doubled up, Hamilton’s

equations are now first order.

There is a somewhat formal structure on phase space, thePoisson bracketwhich is very

useful for quantum mechanics. For any two functionsa(q,p) andb(q,p), it is defined by

{a,b}≡

∑N

i=1

(

∂a

∂qi

∂b

∂pi

−

∂a

∂pi

∂b

∂qi

)

(6.19)

The Poisson bracket is linear inaandb, anti-symmetric under interchange of its arguments

{a,b}=−{b,a}, acts as a derivative in each argument,

{a,bc}={a,b}c+{a,c}b (6.20)

and satisfies the Jacobi identity,

{{a,b},c}+{{b,c},a}+{{c,a},b}= 0 (6.21)

It also satisfies the elementary relation

{qi,pj}=δi,j (6.22)

The time derivative of any functionA(q,p;t) may be expressed simply via Poisson brackets,

d

dt

a(q,p;t) =

∂

∂t

a(q,p;t) +

∑

i

(

∂a

∂qi

q ̇i+

∂a

∂pi

p ̇i

)

(6.23)

Using Hamilton’s equations to obtain ̇qiand ̇pi, we have

d

dt

a(q,p;t) =

∂

∂t

a(q,p;t) +{a,H} (6.24)

In particular, Hamilton’s equations may be recast in the following form,

q ̇i = {qi,H}

p ̇i = {pi,H} (6.25)

6.3 Constructing a quantum system from classical mechanics

There are two general procedures for associating a quantum system with a system of classical

mechanics. The first is via thecorrespondence principlewhich produces a Hilbert space, a

set of observables and a Hamiltonian from a classical mechanics system in the Hamiltonian