Classical Mechanics Level 3

All energy eigen states (stationary states) have the form \({\psi}(x,t) = {\phi}(x) e^{-i {\omega}t}\), where \({\omega}\) is the angular frequency so that \(\left |{\psi}(x,t) \right |^{2} = \left |{\phi}(x) \right |^{2}\) implying that probability is independent of time.

Let \({\psi}_{1}(x)\) and\( {\psi}_{2}(x)\) be two non-degenerate states such that follow above condition \({\psi}_{1}(x,t) = {\phi}_{1}(x) e^{-i {\omega}_{1}t}\)

\({\psi}_{2}(x,t) = {\phi}_{2}(x) e^{-i {\omega}_{2}t}\)

Let \({\psi}_{'}(x,t)\) = \({\psi}_{1}(x,t)\) + \({\psi}_{2}(x,t)\)

Question: Is the wave function \({\psi}_{'}(x)\) a stationary state. Meaning is \(\left |{\psi}_{'}(x,t) \right |^{2}\)=\(\left |{\phi}_{'}(x) \right |^{2}\)


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