- #1
Flenzil
- 10
- 0
Homework Statement
Consider a one-dimensional time-independent Schrodinger equation for an electron in a double quantum well separated by an additional barrier. The potential is modeled by:
V (x) = -γδ(x - a) -γδ(x + a) + βδ(x)
Find algebraic equations which determine the energies (or k-values) of electron bound states for γ > 0 and arbitrary real β (positive or negative). Describe the symmetry of their wave functions in terms of even and odd
solutions. How many bound states do you expect for this system?
Homework Equations
You may nd it useful to work in units ħ = 1 and m = ½, and to introduce k defined as E = -k2, where E < 0 is the energy of a bound state, so that k is real.
The Attempt at a Solution
I've tried to solve this for arbitrary E, since I don't understand what happens at the barrier for E<0. So I split the problem into 4 parts: x<-a, -a<x<0, 0<x<a and x>a. This results in the wavefunctions:
ψ1 = AeiE½x + Be-iE½x
ψ2 = CeiE½x + De-iE½x
ψ3 = FeiE½x + Ge-iE½x
ψ4 = HeiE½x
ψ2 = CeiE½x + De-iE½x
ψ3 = FeiE½x + Ge-iE½x
ψ4 = HeiE½x
Using the conditions of continuity and differential continuity results in the following conditions for the coefficients:
Ce-iE½a + DeiE½a = Ae-iE½a + BeiE½a
F+G=C+D
HeiE½a= Fe-iE½a + GeiE½a