Question

Consider a system with a pair of observable quantities A and B, whose commutation relations with the Hamiltonian take the form [H, A] = iwB, [H, B] = −iwA, where w is some real constant. Suppose that the expectation values of A and B are known at time t = 0. Give formulas for the expectation values of A and B as a function of time.

Answer 1

We will use the Heisenberg picture. In Heisenberg picture, the time evolution of operator isdetermined by Heisenberg equation of motion.

(i)

We have two commutation relations

[H, A] =iωB, (ii)

[H, B] =-iωA. (iii)

Combine these relations,we can get

(iv)

(v)

Taking second derivatives,

(vi)

(vii)

Since A and B are obsrvable quantiry, hence their expectation values are real.

Hence the solution of the above two equations (vi) and (vii):

(viii)

and

(ix)

Taking derivative of (viii),

from eqn (iv),

hence

C1 = -D2 and C2 = D1

Thus, the expectation values of A and B are given as

values of C1 and C2 can be found from expectation values of A and B at t=0