- #1
hnicholls
- 49
- 1
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 = A1eik1x + B1e-ik1x
Wave equation for interval 2: Ψ2 = A2eκ2x
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.
jincidentx = ħk1/m (|A1|2)
But these must match for full reflection.
Not sure how I am getting this result.
Wave equation for interval 1: Ψ1 = A1eik1x + B1e-ik1x
Wave equation for interval 2: Ψ2 = A2eκ2x
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.
jincidentx = ħk1/m (|A1|2)
But these must match for full reflection.
Not sure how I am getting this result.