- #1
maverick280857
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Hi
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
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