Introduction to Quantum Mechanics · Problem 3.41

The key is recognizing that a superposition of two energy eigenstates can have a time-dependent expectation value of momentum due to interference between the states. The momentum oscillates as the relative phase between $|0\rangle$ and $|1\rangle$ evolves, reaching its maximum when you choose the superposition coefficient phases optimally—specifically, when they differ by $\pi/2$. Finding $\langle p \rangle_{\max}$ requires computing the matrix element $\langle 1 | p | 0 \rangle$ and choosing phases to align the complex amplitude constructively, then writing the full wavefunction with that optimal phase relationship and letting time evolution handle the rest.

The key is recognizing that a superposition of two energy eigenstates can have a time-dependent expectation value of momentum due to interference between the states. The momentum oscillates as the relative phase between $|0\rangle$ and $|1\rangle$ evolves, reaching its maximum when you choose the superposition coefficient phases optimally—specifically, when they differ by $\pi/2$. Finding $\langle p \rangle_{\max}$ requires computing the matrix element $\langle 1 | p | 0 \rangle$ and choosing phases to align the complex amplitude constructively, then writing the full wavefunction with that optimal phase relationship and letting time evolution handle the rest.