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## Main Question or Discussion Point

I have seen discussions which suggests that there is no solution for the interval after the step in a step potential where E = V0.

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

Wave equation for interval 2:

-ħ

v(x)Ψ(x)

So, d

In order for d

d/dx Ψ(x)

So,

Ψι = A

Ψ

Continuity conditions:

Ψι = Ψ

d/dx Ψ

So,

A

at interval boundary x = 0

So.

A

A

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

d/dx [A

ikA

at interval boundary x = 0

ikA

A

So,

ik[A

0 = A

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

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}= 0In order for d

^{2}/dx^{2}Ψ(x)_{2}to equal 0d/dx Ψ(x)

_{2}= A_{2}and Ψ(x)_{2}= A_{2}x + CSo,

Ψι = A

_{1}e^{ikx}+ B_{1}e^{-ikx}Ψ

_{2}= A_{2}xContinuity conditions:

Ψι = Ψ

_{2}d/dx Ψ

_{1}= d/dx Ψ_{2}So,

A

_{1}e^{ikx}+ B_{1}e^{-ikx}= A_{2}xat interval boundary x = 0

So.

A

_{1}+ B_{1}= 0A

_{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.