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## Homework Statement

We know that a particle in SHM is in a state such that measurements of the energy will yield either [itex]E_0[/itex] or [itex]E_1[/itex] (and nothing else), each with equal probability. Show that the state must be of the form

[tex] \psi = \frac{1}{\sqrt2} \psi_0 + \frac{e^{i \phi}}{\sqrt2} \psi_1 [/tex]

where [itex] \psi_ [/itex] and [itex] \psi_1 [/itex] are the ground and first excited state, respectively.

## Homework Equations

For a Hamiltonian with discrete energy spectrum, the probability of measuring the particular eigenvalue associated with the orthonormalized eigenfunction [itex]f_n[/itex] is [itex] \mid c_n \mid ^2 [/itex].

## The Attempt at a Solution

Since we are just as likely to measure [itex]E_0[/itex] as we are to measure [itex]E_1[/itex], we know that the wave function must look like

[tex] \psi = c_1 \psi_0 + c_2 \psi_1 [/tex]

where

[tex] \mid c_1 \mid ^2 + \mid c_2 \mid ^2 =1 \rightarrow \mid c_n \mid ^2 = \frac{1}{2} [/tex]

I have no idea where the factor of [itex]e^{i \phi}[/itex] comes from in the final answer.

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