How can I use determinants to find energy levels in a finite square potential?

[pepsicola]

Hi, I am in the process of learning QM.
I am looking at this problem regarding to a finite square potential well.
I have derived psi(x) and the k's for the 3 domains,

psi1(x) = Ae^kx => k = sqrt(2m(V-E)/h^2)
psi2(x) = Ce^jkx + De^-jkx => k=sqrt(2mE)/h
psi3(x) = He^-kx => same k as in domain 1

and then what I did was to take the boundry conditions and substitue into the equations and make a system of 4 equations with so that I can solve the unknowns.

What I don't understand is the way I've been told to do this from the above.

I've been told to calculate the determinant of the system of the 4 equations, and by scanning E. When the determinant goes to 0, I will get the energy level for E.
Any one can explain to me why this will give the answer?

And also, it seems like I will need to get the answer by ploting the determinant from its imaginary component. Why?

And when I have the 3 different energy levels that satisfy the conditions, how do I find the unknowns?

Thanks for any help.

 

[maverick280857]

The boundary conditions on the wavefunction and its first derivative give you the system of equations you are referring to.

This is a system of equations linear in the unknown coefficients. So the condition you stated is the condition for the existence of a nontrivial solution for this system.

And also, it seems like I will need to get the answer by ploting the determinant from its imaginary component. Why?

Not sure I understand this. The determinant is a complex number. Setting it equivalent to zero is equivalent to setting its real and imaginary parts to zero separately. Perhaps what you want to do is treat the determinant as a function of the parameter E and plot it as a function of E, to determine the zero crossings, which will give you the eigenvalues.

 

[pepsicola]

And when I get the energy states Es, how do I find out the unknowns?

 

[maverick280857]

And when I get the energy states Es, how do I find out the unknowns?

You must have obtained a transcendental equation..just solve for the eigenvalues numerically.