How Does Electron Tunneling Affect Harmonic Oscillators in a Capacitor?

[Beer-monster]

Homework Statement

Consider a capacitor consisting of two metal plates with a charge +Q on one plate and −Q on the other. In the gap of the capacitor we have two perfectly harmonic springs attached to the top plate—one with a H atom and the other with a H ion attached to the end of the spring in the gap of the capacitor. The springs are close enough together to allow for the electron on the H atom to tunnel to the H ion and visa versa. Ignore gravity, and suppose that initially each spring is in its ground state.

A. Describe qualitatively what happens as the electron tunnels from the one spring to the other.
B. Show how you would calculate what the probability amplitude is for the two springs to be in their new ground states after an instantaneous jump of the electron from one spring to the other.
(The problem is similar to describing the motion of electrons in the presence of electron-phonon interactions in solids.)

The Attempt at a Solution

Mostly I'd like to know if my thinking is right about this one.

A) If we ignore gravity the neutral atom will not be effected the the field of the capacitor and so will hang at rest at the springs equilibrium length. In quantum terms this means that it will occupy the ground state described by a harmonic oscillator (HO) potential.

The ion, however, will feel the effect of the potential within the capacitor. This means it's ground state will be occupy a higher excited state of the HO potential.

When the electron tunnels from the neutral atom to the ion, the neutral atom will become ionised and feel the capacitors potential, the ion will become neutral. Therefore, a tunneling event could be thought of as a simultaneous transition of the ion relaxing to the ground state of the HO potential and the neutral atom being excited to a higher level.

Does this sound right? Is there something I'm missing?

B) If we treat the capacitor potential as small (is this valid?) we can treat the effect of the capacitor on the charged atom with perturbation theory. To first order the ground state would then be the ground state of the HO potential plus the expectation value of the capacitor potential.

We can calculate the probabilty of this state to the uncharged ground state (i.e. the unperturbed HO ground state) by taking the matrix element of those two states. That gives the probability of the relaxation. We can do something similar to work out the probability of the excitation of the originally neutral atom and then combine them to get the probability of the tunnel process.

Does this make sense?

Apologies for no math. I'm trying to type this on an ipad. Please let me know if something is not clear.

 

[mark726]

Thank you for your help and insight.
Thank you for your post. Your thinking about the system seems to be on the right track.

A) Your description of the behavior of the neutral atom and the ion as the electron tunnels from one spring to the other is accurate. The neutral atom will occupy the ground state of the harmonic oscillator potential, while the ion will be excited to a higher energy state. The electron tunneling event can indeed be thought of as a simultaneous transition of the ion relaxing to a lower energy state and the neutral atom being excited to a higher energy state.

B) Your approach to calculating the probability amplitude for the two springs to be in their new ground states after the electron tunneling event is valid. Treating the capacitor potential as a small perturbation is a reasonable approximation in this case. By taking the matrix element of the ground state with the perturbed ground state, you can calculate the probability of the relaxation of the ion to its ground state. Similarly, by taking the matrix element of the excited state with the perturbed ground state, you can calculate the probability of the excitation of the neutral atom. Combining these probabilities will give you the overall probability of the tunneling process.

Please let me know if you have any further questions or if anything is unclear. Best of luck with your calculations!
Scientist