Schrödinger Equation for a One-Dimensional Delta Function Potential
Schrödinger Equation for a One-Dimensional Delta Function Potential
After the free particle, the most elementary example of a one-dimensional time-independent Schrödinger equation is conceptually that of a particle in a delta function potential: (in units with ). For an attractive potential, with , there is exactly one bound state, with =- and (x)=λ. Note that x=sign(x) and x=2δ(x). Since the delta function has dimensions of , this solution is considered the one-dimensional analog of a hydrogen-like atom. The bound state, in fact, resembles a cross section of a 1 orbital .
-ψ''(x)+λδ(x)ψ(x)=Eψ(x)
1
2
ℏ=m=1
λ<0
E
0
2
λ
2
ψ
0
-|λx|
d
dx
2
d
d
2
x
1/|x|
s
-Zr
e
For , free particles are scattered by a delta function potential. The positive-energy solutions can be written (x)=+, with . The amplitudes of the transmitted and reflected waves are accordingly given by + and +, respectively. Note that these are the same for attractive and repulsive delta funtion potentials, independent of the sign of .
E>0
±
ψ
k
1
2π
±ikx
e
λ
ik-λ
k|x|
e
E=/2
2
k
2
k
2
k
2
λ
2
λ
2
k
2
λ
λ
For continuum states, the graphic shows a wave incident from the left. The transmitted wave is shown on the right in blue and the reflected wave, on the left in red, with opacities indicating relative wave amplitudes.