Schrödinger Equation for a One-Dimensional Delta Function Potential

bound state

continuum

λ

0

k

1

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:

-

1

2

ψ''(x)+λδ(x)ψ(x)=Eψ(x)

(in units with

ℏ=m=1

). For an attractive potential, with

λ<0

, there is exactly one bound state, with

E

0

=-

2

λ

2

and

ψ

0

(x)=λ

-|λx|



. Note that

d

dx

x=sign(x)

and

2

d

2

x

x=2δ(x)

. Since the delta function has dimensions of

1/|x|

, 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

s

orbital

-Zr

e

.

For

E>0

, free particles are scattered by a delta function potential. The positive-energy solutions can be written

±

ψ

k

(x)=

1

2π

±ikx

e

+

λ

ik-λ

k|x|

e

, with

E=

2

k

/2

. The amplitudes of the transmitted and reflected waves are accordingly given by

2

k

2

k

+

2

λ

and

2

λ

2

k

+

2

λ

, respectively. Note that these are the same for attractive and repulsive delta funtion potentials, independent of the sign of

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.