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Physics LibreTexts

6.2: Potential step

( \newcommand{\kernel}{\mathrm{null}\,}\)

Consider a potential step

V(x)={V0x<aV1x>a

step1.png

Figure 6.2.1: The step potential discussed in the text

Let me define

k0=2m2(EV0),k1=2m2(EV1).

I assume a beam of particles comes in from the left,

ϕ(x)=A0eik0xfor x<0.

At the potential step the particles either get reflected back to region I, or are transmitted to region II. There can thus only be a wave moving to the right in region II, but in region I we have both the incoming and a reflected wave,

ϕI(x)=A0eik0x+B0eik0x,ϕII(x)=A1eik1x.

We define a transmission (T) and reflection (R) coefficient as the ratio of currents between reflected or transmitted wave and the incoming wave, where we have canceled a common factor

R=|B0|2|A0|2T=k1|A1|2k0|A0|2.

Even though we have given up normalisability, we still have the two continuity conditions. At x=0 these imply, using continuity of ϕ and ddxϕ,

A0+B0=A1,ik0(A0B0)=ik1A1.

We thus find

A1=2k0k0+k1A0B0=k0k1k0+k1A0,

and the reflection and transmission coefficients can thus be expressed as

R=(k0k1k0+k1)2,T=4k1k0(k0+k1)2.

Notice that R+T=1!

imageedit_4_8071888387.png

Figure 6.2.2: The transmission and reflection coefficients for a square barrier.

In Figure 6.2.2 we have plotted the behaviour of the transmission and reflection of a beam of Hydrogen atoms impinging on a barrier of height 2 meV.


This page titled 6.2: Potential step is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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