Electron on Helium Surface

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An electron close to the surface of liquid helium experiences an attractive force due to the electrostatic polarization of the helium and a repulsive force due to the exclusion principle(hard core). To a reasonable approximation for the potential when helium fills the space where :

Note: the potential is infinite when because the cannot penetrate the helium surface.


(a) Solve the Schrödinger equation. Find the Eingenenergies and Eigenvalues.

(b) An electric field is turned on at t=0 which produces the perturbation:

If the electron is initially in its ground state, find the probability makes a transition to its first excited state for times .

Solution...

(a) Solve the Schrödinger equation.

The Schrödinger equation for when is:

Using separation of variables:

For X and Y we get place waves.

This corresponds to motion parallel to the helium surface.

For z-component the Schroedinger equation becomes:

This has the same form as the hydrogin atom with l=0 (s-wave). Since similar equations have similar solutions, the solution to the z-component is:

where


The total wave function and energies are:

where n = 1,2,... is the quantum number for the z-direction and the bohr radius has become


(b) Turn on electric field at t=0.

The electric field introduces a perturbation to the hamiltonian:

From expression 2.1.10 in Time Dependent Perturbation Section of the PHY5646 page:


where the quantity inside the brackets is the normalization for and , respectively.

let

So the probablity of a transition from ground state to the first excited state is: