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: