Step potential with E<V: reflection coefficient and probability current

[hnicholls]

I can show that for a step potential and E < V, that the wave function is fully reflected and has no transmission into the potential interval (interval 2), x =0 at interval boundary, by

Wave equation for interval 1: Ψ₁ = A₁e^ik₁x + B₁e^-ik₁x

Wave equation for interval 2: Ψ₂ = A₂e^κ₂x

where B₁ = A₁ (k₁ - iκ₂)/(k₁ + iκ₂)

and

A₂ = A₁ (2k₁)/(k₁ + ik₂)

Further B₁/A₁ = (k₁ - iκ₂)/(k₁ + iκ₂)

|B₁|/|A₁| = (k₁² - iκ²₂)^1/2/(k²₁ + iκ²₂)^1/2 = 1

and |B₁| = |A₁| by which we can conclude full reflection.

However, if I calculate the reflection and transmission value by the probability current density, I find,

For interval 1

j₁x = ħk₁/m (|A₁|² -|A₁|²|ρ(E)|²)

where

|ρ(E)| = (k₁ - iκ₂)/(k₁ + iκ₂)

But with this value for |ρ(E)| the probability current density in interval 1 does not match the probability current density of the incident wave, i.e.

j_incidentx = ħk₁/m (|A₁|²)

But these must match for full reflection.

Not sure how I am getting this result.

 

[hideelo]

I'm not entirely sure I got your question, but if E<V then you should be getting full reflection, however as best as I can imagine it, not all of them reflect at the barrier (step), some of them go a little further before coming back. So you will find some probability density out past the step and correspondingly you won't find all of your particles on the same side as your source. But, you will also find that they all reflect (eventually).

 

[hnicholls]

Thank you. I believe my confusion relates to the interpretation of

j₁x = ħk₁/m (|A₁|² -|A₁|²|ρ(E)|²)


I see when |ρ(E)|² = 1

then

j₁x = ħk₁/m (|A₁|² -|A₁|²|)

reflects the identical probability current density flux proceeding to the right and then to the left, i.e. full reflection.