Electron on Helium Surface

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z<0 } :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(z) = \begin{cases} -\frac{Q^2e^2}{z} &\mbox{if} \qquad z>0\\ \infty &\mbox{if} \qquad else \end{cases} }

Note: the potential is infinite when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z<=0} 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(z) = \begin{cases} eE_0ze^{t/\tau} &\mbox{if} \qquad t>=0\\ 0 &\mbox{if} \qquad else \end{cases} }

If the electron is initially in its ground state, find the probability makes a transition to its first excited state for times Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t >> \tau } .

Solution...

(a) Solve the Schrödinger equation.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = -\frac{\hbar^2}{2m}\nabla^2 + V(z) }

The Schrödinger equation for when $z>0$ is:

$\left[{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)-{\frac {Q^{2}e^{2}}{z}}\right]\psi (\mathbf {r} )=E\psi (\mathbf {r} )$

Using separation of variables:

$\psi (x,y,z)=X(x)Y(y)Z(z)$

For X and Y we get place waves.

$X(x)Y(y)=Ae^{i(k_{x}x+k_{y}y)}$

This corresponds to motion parallel to the helium surface.

For z-component the Schroedinger equation becomes:

$\left[{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}-{\frac {Q^{2}e^{2}}{z}}\right]Z(z)=$

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:

$Z(z)=zR_{n0}(z)$

where

$R_{10}=\left({\frac {1}{a_{0}}}\right)^{3/2}ze^{-{\frac {z}{a_{0}}}}$

$R_{20}=\left({\frac {1}{2a_{0}}}\right)^{3/2}\left(2-{\frac {z}{a_{0}}}\right)e^{-{\frac {z}{2a_{0}}}}$

The total wave function and energies are:

$\psi =Ae^{i(k_{x}x+k_{y}y)}zR_{n0}(z)$

$E_{k_{x}K_{y}K_{z}}={\frac {\hbar ^{2}}{2m}}\left(k_{x}^{2}+k_{y}^{2}\right)-{\frac {Q^{4}e^{4}m}{2\hbar ^{2}n^{2}}}$

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

$a_{0}={\frac {hbar^{2}}{mQ^{2}e^{2}}}$

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

The electric field introduces a perturbation to the hamiltonian:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^'(t>0) = E_0ze^{t/\tau} }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} P_{1 \rightarrow 2}(t \rightarrow \infty) &= |\langle n|\psi(t)\rangle|^2 \\&= \left|\frac{1}{i\hbar}\int_{0}^{\infty}dt' e^{\frac{i}{\hbar}(E_2 - E_1)t'} \langle \psi_{n=2}|V_{t'}| \psi_{n=1}\rangle\right|^2 \\&= \end{align}}