- #1

hnicholls

- 49

- 1

The set up is a potential step where E = V0, with an interval 1 defined as x < 0 before the step and an interval 2 as x > 0 after the step.

Is the following correct?

Wave equation for interval 1: Ψι = A

_{1}e

^{ikx}+ B

_{1}e

^{-ikx}

Wave equation for interval 2:

-ħ

^{2}/2m d

^{2}/dx

^{2}Ψ

_{2}(x) + v(x)Ψ(x)

_{2}= EΨ(x)

v(x)Ψ(x)

_{2}= EΨ(x)

So, d

^{2}/dx

^{2}Ψ(x)

_{2}= 0

In order for d

^{2}/dx

^{2}Ψ(x)

_{2}to equal 0

d/dx Ψ(x)

_{2}= A

_{2}and Ψ(x)

_{2}= A

_{2}x + C

So,

Ψι = A

_{1}e

^{ikx}+ B

_{1}e

^{-ikx}

Ψ

_{2}= A

_{2}x

Continuity conditions:

Ψι = Ψ

_{2}

d/dx Ψ

_{1}= d/dx Ψ

_{2}

So,

A

_{1}e

^{ikx}+ B

_{1}e

^{-ikx}= A

_{2}x

at interval boundary x = 0

So.

A

_{1}+ B

_{1}= 0

A

_{1}= B

_{1}

i.e. full reflection of the incident wave function.

d/dx [A

_{1}e

^{ikx}+ B

_{1}e

^{-ikx}] = d/dx [A

_{2}x]

ikA

_{1}e

^{ikx}+ ikB

_{1}e

^{-ikx}= A

_{2}

at interval boundary x = 0

ikA

_{1}- ikB

_{1}= A

_{2}

A

_{1}= B

_{1}

So,

ik[A

_{1}- A

_{1}] = A

_{2}

0 = A

_{2}

i.e. no transmission of the incident wave function.