Question

A white dwarf is the end stage of a star which started with a mass a few times that of our sun. It results from gravitational collapse after the outward pressure due to the heat of fusion has subsided. After collapse, the whole star is like one chunk of metal, all the electrons are free to move anywhere in the star and can be modeled with a simple particle in a well model. To do so, we will ignore nuclei and treat the electrons as particles in a 3d well. To refresh your memory, the energy levels in a well of width L with infinite potential walls are given by:

E = n²h²/8mL²

and the wavefunctions:

?=Asin(n?x/L).

To shift to three dimensions instead of one, we simply multiply the resulting wavefunctions and the energies will add. To see how this works, consider a 2-dimensional well of width L in both directions, x and y with the origin at one corner. The Schr?dinger equation for this well is

$E\Psi=-\frac{\hbar^2}{2m}\nabla^2\Psi=-\frac{\hbar^2}{2m}[\frac{\partial^2}{\partial x^2}\Psi+\frac{\partial^2}{\partial y^2}\Psi]$

$E=E_x+E_y; \Psi(x,y)=X(x)Y(y)$

[a] Make the subtitutions for E and ? the Schr?dinger equation

[b] Split the equation into an x and a y equation

[c] Show that the x equation and y equation have the same solutions as the one-dimensional problem, that is show X(x) = Asin(n_x?x/L) and Y(y) = Asin(n_y?y/L) solve the equations and give the corresponding energies.

Answer 1

Schrodinger equation in one dimension is given by, x(x)= Function for variable x only Y(y)-Funcion for variable y only Rewrite he Schrodinger equation in 3-D Ey(x.y) Omitting z-component atr d'v(x,y) 1 d'y Substitute in equation (1) E Energy constant Sepearate each tem in equation(2)

dY 2m Normalization wave functio n equation for each variable 2. x a. b = width of the box in particular direction Combined wave function Total energy level hn 2