Math Infinitum

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Intuition

Send a particle at a step in the potential. Classically, if it has more energy than the step it rolls over without a hitch, and if it has less it bounces straight back. Quantum mechanically a wave partly reflects even when it has energy to spare, and when it does not, it still reaches a little way into the step before turning round.

Light passing from air into glass mostly goes through but partly reflects off the surface, which is why a window shows a faint reflection. A matter wave meeting a potential step does the same.

A wave of energy $E=2V_{0}$ meeting a step of height $V_{0}$ , drawn on its energy level at one instant, in units with $\hbar^{2}/2m=V_{0}=1$ . Beyond the step it has less kinetic energy and a longer wavelength; about 3% of it is reflected even though it has twice the energy to pass.

Reflection and transmission at a step

For $V=0$ at $x<0$ and $V=V_{0}$ at , with , the solution is an incoming and a reflected wave on the left and a transmitted wave on the right. Matching and at the step fixes the amplitudes; the probabilities of reflection and transmission are ratios of currents.

\begin{aligned}R&=\Bigl(\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\Bigr)^{2},\qquad R+T=1\\ T&=\frac{4k_{1}k_{2}}{(k_{1}+k_{2})^{2}}\end{aligned}

Properties

$k_{1}=\sqrt{2mE}/\hbar$ and .

The reflected and transmitted amplitudes

\begin{aligned}&\psi=\begin{cases}e^{ik_{1}x}+re^{-ik_{1}x},&x<0\\ te^{ik_{2}x},&x>0\end{cases}\\ &\Longrightarrow\ r=\frac{k_{1}-k_{2}}{k_{1}+k_{2}},\ \ t=\frac{2k_{1}}{k_{1}+k_{2}}\end{aligned}

Match the value and the slope at $x=0$ . That gives two linear equations in $r$ and $t$ ; adding a suitable multiple eliminates $r$ and gives $t$ , and then .

Proof steps

1+r=t

$\psi$ is continuous at the step.

ik_{1}(1-r)=ik_{2}t

$\psi'$ is continuous at a finite step.

\begin{aligned}&k_{1}(1-r)+k_{1}(1+r)=k_{2}t+k_{1}t\\ &\Rightarrow\ t=\frac{2k_{1}}{k_{1}+k_{2}}\end{aligned}

Add $k_{1}$ times the first equation to the second, divided by $i$ .

r=t-1=\frac{k_{1}-k_{2}}{k_{1}+k_{2}}

From the first equation.

R=|r|^{2},\qquad T=\frac{k_{2}}{k_{1}}|t|^{2}

Ratios of the reflected and transmitted currents to the incoming one.

Applications

Practice

Reflection Above the Step

Even with more energy than the step, part of the wave reflects. The fraction depends on how different the wavenumbers on the two sides are.

R=\Bigl(\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\Bigr)^{2}

Try it

A particle with $E=\tfrac43V_{0}$ meets a step of height $V_{0}$ . What is $R$ ? Give three decimal places.