Standard Finite Well Problem (Solve without symmetry)

[IndiaNut92]

Homework Statement

Solve the energy eigenvalue problem for the finite square well without using the symmetry assumption and show that the energy eigenstates must be either even or odd.

Homework Equations

The finite well goes-a to a and has a potential V0 outside the box and a potential of 0 inside the box.

The Attempt at a Solution

I understand the book's solution of the problem which utilizes symmetry but am a little stuck as to how to proceed without it. Basically, one plugs the potentials into the energy eigenvalue equation (Schrödinger's Equation) and solves to find ψ(x)={ A e^(q x)+ B e^-(q x) for x<-a; C Sin(kx) + D Cos(kx) when -a<x<a; F e^(q x)+G e^-(q x) for x>a.

Once this ψ is found (it's a piecewise function) boundary conditions reveal that B=F=0 in order for the function to be normalizable. This gives ψ={A e^(q x for x<-a; C Sin(kx) + D Cos(kx) when -a<x<a; G e^-(q x) for x>a.

At this point the book invokes symmetry to split the function into ψeven which carries the Cos term and ψodd which carries the Sin term. It then proceeds to derive the transcendental equations.

I am uncertain as to how to get to the transcendental equations without invoking the symmetry argument.

Any help that can be offered would be appreciated.

Thanks!

 

[vela]

What equations do you get from requiring continuity of $\psi$ and its derivative at x=a and x=-a?