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I'm looking at this wave function:

[tex]\psi(x,t) = \frac{4}{5}{\psi}_{1} + \frac{3}{5}{\psi}_{2}[/tex]

The functions involved here are the typical eigenfunctions for the ground state and first excited level in an infinitely-deep 1-D square well.

Defining

[tex]A = 4/5.\sqrt{2/a}[/tex]

[tex]B = 3/5.\sqrt{2/a}[/tex]

[tex]K = \pi/a[/tex]

I might have rewritten this as

[tex]\psi(x,t) = Asin(Kx).exp(-\frac{iE_1.t}{\hbar}) + Bsin(2Kx).exp(-\frac{iE_2.t}{\hbar})[/tex]

However, the text restates this as

[tex]\psi(x,t) = Asin(Kx) + Bsin(2Kx).exp(-\frac{i\Delta.t}{\hbar})[/tex]

where

[tex]\Delta = E_{2} - E_{1}[/tex]

Can someone tell me how the time element has been attached to only one of the eigenfunctions like that? I expect it's obvious, but I'm just not seeing it at the moment! (It's evidently been done this way to make the expectation value calculation that follows simpler).

Many thanks!

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# Rewriting a wave function superposition

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