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Suppose we a pair of symmetric wells of finite potential and the particle is given to be in the initial state

[tex]|\psi(0)\rangle = \frac{1}{\sqrt{2}}(|\psi_{s}\rangle + |\psi_{a}\rangle)[/tex]

(a = antisymmetric state, s = symmetric state)

For t > 0, we have

[tex]|\psi(t)\rangle = \frac{1}{\sqrt{2}}e^{-iE_{S}t/\hbar}(|\psi_{s}\rangle + e^{-it/\tau}|\psi_{a}\rangle)[/tex]

where [itex]\tau = \hbar\pi/(E_{a}-E_{s})[/itex]

We see that the particle oscillates between the two wells, but the expectation value of the energy

[tex]\langle\psi(t)|H|\psi(t)\rangle[/tex]

is constant and equals [itex](E_{s}+E_{a})/2[/itex].

I have two questions:

1. What is the physical significance of this?

2. Is this due to the specific initial state given?

Thanks in advance.

Cheers

Vivek

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# Particle oscillating between two wells

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