# 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 . 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

λ

2

2

ψ(x)=|λ|

0

-|λx|

d

dx

d

2

dx

2

1/|x|

s

e

-Zr

For , free particles are scattered by a delta function potential. The positive-energy solutions can be written , 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(x)=e+e

±

k

1

2π

±ikx

λ

ik-λ

k|x|

E=k/2

2

k

2

k+λ

2

2

λ

2

k+λ

2

2

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.