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Problem 53 Derive Heisenberg equations for operators , àt andb, in the following Hamiltonian appearing Commutation relations for these operators are as follows:

Answer 1

Heisenberg equation of for operators a^, a^t, b^, b^t partial differential a^/partial differential t = -i/h [a^, H] implies from the commutation relation [a, a^t] = 1 other commutation are zero so, implies - i/h [[a^, h w a^t a^] + [a^, k a^t, b^]] = -i/k [h w a^t k b^] = -I w a^- ki/h b^partial differential a^t/partial differential t = -i/h [a^t, H] similarly from above. = -i/h[[a^t, h w a^^+ a^] [a^t kb^= a^]] = -i/k [- h wa^t + (- kb^t)] = i/k [h w a^+ kb^t] = I w a^t + ki/hb^t partial differential b^/partial differential t = -i/h [b^, H] similarly from above [b, b^=] = 1 = -i/k [[a^, h ohm b^t, b^] + [b^, k d^t a^]] = -i/h [h ohm b^+ ka^] = -ohm i b^- I k/h a^[b^t, b] = -1 partial differential b^^t/partial differential t = -i/k [b^t, H] = -i/h [[b^t, h ohm b^t b^] + [b^t, ka^t b6]] = -i/h [- h ohm b^t + (- ka^t)] = ohm ib^t + ki/h a^t