Quantum Mechanics II
Phys 712 — Spring 2015 — Assignment 1
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The time evolution of a quantum state is described by
\[i\hbar\frac{\partial}{\partial t}\lvert \psi(t) \rangle = \hat{H} \lvert \psi(t) \rangle.\]In the case where the Hamiltonian has no time dependence of its own, this differential equation has a formal solution
\[\lvert \psi(t) \rangle = \exp\bigl(-i\hat{H}t/\hbar\bigr) \lvert \psi(0) \rangle.\]Prove that this is true. Proceed by expanding the exponential as a power series.
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Imagine instead that the Hamiltonian has a contribution that increases linearly in time:
\[\hat{H} = \hat{H}_0 + \hat{H}_1\frac{t}{t_1}.\]Here, \(\hat{H}_0\) and \(\hat{H}_1\) are time-independent operators with units of energy, and \(t_1\) is a constant with units of time.
(a) Show explicitly that
\[\lvert \psi(t) \rangle = \exp\biggl[-i\Bigl(\hat{H}_0t + \frac{1}{2t_1}\hat{H}_1t^2\Bigr)/\hbar\biggr] \lvert \psi(0) \rangle.\]is not a general solution to
\[i\hbar\frac{\partial}{\partial t}\lvert \psi(t) \rangle = \Bigl(\hat{H}_0 + \hat{H}_1\frac{t}{t_1}\Bigr) \lvert \psi(t) \rangle.\](b) What relationship must hold between \(\hat{H}_0\) and \(\hat{H}_1\) so that it is a solution.
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Consider a spin-half object coupled to a time-dependent external field:
\[\hat{H} = \mathbf{B}(t) \cdot \hat{\mathbf{S}}.\]The field \(\mathbf{B}(t) = (2J_0/\hbar) \cos(\omega_0t) \mathbf{e}_x\) is x-directed and oscillates with angular frequency \(\omega_0\). The constant \(J_0\) has units of energy.
(a) Show that the evolution operator is as follows.
\[\hat{U}(t) = \cos\biggl(\frac{J_0}{\hbar \omega_0}\sin\omega_0t\biggr) -i \sigma_x \sin\biggl(\frac{J_0}{\hbar \omega_0}\sin\omega_0t\biggr).\](b) Suppose that \(\lvert \psi(0) \rangle = \lvert \uparrow \rangle\); in other words, that the system is prepared at time \(t=0\) so that the spin is aligned with the positive z direction. Compute the probability \(P_\uparrow(t) = \lvert \langle \uparrow\!\vert\,\psi(t) \rangle \rvert^2\) of finding the spin in the spin-up configuration. Make a plot of \(P_\uparrow(t)\) with \(t\) running over four complete cycles of the field oscillation. Show example curves for the case of fast (\(\omega_0 \gg J_0/\hbar\)) and slow (\(\omega_0 \gg J_0/\hbar\)) oscillation.