# 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.