Question

In: Physics

Consider a system with a pair of observable quantities A and B, whose commutation relations with...

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.

Expert Solution

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

genius_generous answered 2 years ago

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