Discussion
Consider the normalized superposed state \(\psi = c_{0}\phi_{o} + c_{1}\phi_{1}\), where \(\phi_{0}\) and \(\phi_{1}\) are the ground and first excited states of a simple harmonic oscillator, respectively. \(c_{0}\) and \(c_{1}\) are imaginary superposition coefficients. Which of the following options is correct for the expectation value of \((\langle x \rangle + i \langle p \rangle)\)?
Answer: B
The quantity \(\langle x \rangle + i\langle p \rangle\) is real.
Write the imaginary coefficients as \(c_0 = ia\) and \(c_1 = ib\) with \(a, b\) real. For the oscillator, \(\langle 0|x|1\rangle = x_{01}\) is real and \(\langle 0|p|1\rangle = i p_{01}\) is purely imaginary, with \(x_{01}, p_{01}\) real; diagonal elements of \(x\) and \(p\) vanish.
\[ \begin{aligned} \langle x \rangle &= c_0^{*}c_1 x_{01} + c_1^{*}c_0 x_{01} = 2ab\,x_{01} \\ \langle p \rangle &= ab(ip_{01}) + ab(-ip_{01}) = 0 \end{aligned} \]
So \(\langle x \rangle + i\langle p \rangle = 2ab\,x_{01}\), a real number, non-zero whenever both states are present. With purely imaginary coefficients the relative phase is zero, so the packet is displaced but momentarily at rest.
It is not zero in general, and since \(\langle p \rangle\) vanishes there is no imaginary part, which rules out the imaginary and complex options.
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