Question

could someone please help me with part c?

Consider a one-dimensional harmonic oscillator, at a timet - 0 in the state, where In〉 is the nth energy eigenstate with energy eigenvalues En = (n + ,wo a) Write out an expression, in terms of the energies En, for the state vector at a time t. That is, write down what ^(t)) is equal to. b) Calculate the expectation value of the energy of the state vector. c) Calculate the expectation value for the position r. That is calculate (x(t)). (HINT Note that the operator X - 2mwo

Answer 1

Since d(x(t))/dt = 1/i hbar [X^^, H^^] H^^= p^^2/2m + 1/2 m omega^2 x^^2 [X^^, H^^] = [X^^, P^^2/2m] + [X^^, 1/2 m omega^2 X^^2] = [X^^, P^^2/2m] = 2i hbar p/2m d/dt < x(t) > = 1/i hbar middot 2i hbar p/2m = < p > /m = > d^2/dt^2 < x(t) > = 1/m d < p > /dt d < p > /dt = 1/i hbar < P^^, H^^ > = 1/i hbar middot 1/2 m omega^2 [P^^, x^2] = -m omega^2 < x > Therefore d/dt < x(t) > + omega^2 < x(t) > = 0 \r\n Therefore < X^^(t) > = A cos(omega t) + c At t = 0, A = x(0) < x(t) > = x(0) cos omega t + c