38-8 Reflection from a Potential Step

38-8 Reflection from a Potential Step#

Prompts

  • Describe a potential step: electron in region 1 (\(U=0\)) approaches a boundary at \(x=0\) beyond which \(U = U_b > 0\). If \(E > U_b\), what does classical physics predict? What does quantum physics predict?

  • Write the wave functions in region 1 and region 2. Why do we set \(D=0\) in region 2? What are the boundary conditions at \(x=0\)?

  • Define the reflection coefficient \(R\) and transmission coefficient \(T\). How do we find \(R\) from the wave function coefficients? What is \(T\) in terms of \(R\)?

  • Interpret \(R\): (a) for a single electron, (b) for a beam of many electrons. If \(R = 0.01\), what can we say about 10,000 electrons incident on the step?

  • Why is quantum reflection from a potential step surprising? A classical particle with \(E > U_b\) would never reflect.

Lecture Notes#

Overview#

  • A potential step is a boundary at which the potential energy \(U\) changes abruptly (e.g., from \(U=0\) to \(U = U_b\)).

  • Classically: If \(E > U_b\), the particle has enough energy—it always passes through. No reflection.

  • Quantum: Because electrons are matter waves, some reflect even when \(E > U_b\). Wave reflection at a boundary is natural; particle reflection when “there’s enough energy” is not.

  • The reflection coefficient \(R\) gives the probability of reflection; the transmission coefficient \(T = 1 - R\) gives the probability of transmission.

Setup: potential step#

Consider an electron moving along the \(x\) axis. In region 1 (\(x < 0\)), \(U = 0\). At \(x = 0\) the potential steps up to \(U = U_b > 0\) (region 2, \(x > 0\)). The electron has total energy \(E > U_b\).

Classically, the electron would simply slow down and continue into region 2—its kinetic energy would drop from \(E\) to \(E - U_b\), but it would keep going. No reflection.

Quantum mechanically, we solve Schrödinger’s equation in each region and match the wave functions at the boundary. The result: some electrons reflect.

Why reflection?

Waves reflect at boundaries where the wave speed (or wavelength) changes—like light at a glass surface. The matter wave “sees” a change in the potential energy (and thus in \(k\)). Reflection is a wave phenomenon; the classical particle picture misses it.

Wave functions and boundary conditions#

In region 1 (\(U = 0\)), the angular wave number is \(k = 2\pi\sqrt{2mE}/h\). The general solution is

(375)#\[ \psi_1(x) = A e^{ikx} + B e^{-ikx} \]

The term \(Ae^{ikx}\) represents the incident wave (moving \(+x\)); \(Be^{-ikx}\) represents the reflected wave (moving \(-x\)).

In region 2 (\(U = U_b\)), the angular wave number is \(k_b = 2\pi\sqrt{2m(E-U_b)}/h\) (smaller than \(k\) because kinetic energy is less). The general solution would be

\[ \psi_2(x) = C e^{ik_b x} + D e^{-ik_b x} \]

But there is no electron source to the right—no wave traveling left. So we set \(D = 0\):

(376)#\[ \psi_2(x) = C e^{ik_b x} \]

Boundary conditions at \(x = 0\): the wave function and its derivative must be continuous (matching values and slopes):

(377)#\[ A + B = C \quad \text{(matching values)} \]
(378)#\[ Ak - Bk = Ck_b \quad \text{(matching slopes)} \]

Solving these determines \(B\) and \(C\) in terms of \(A\).

Reflection and transmission coefficients#

The probability density is proportional to \(|\psi|^2\). The incident amplitude is \(|A|^2\); the reflected amplitude is \(|B|^2\). The reflection coefficient is

(379)#\[ R = \frac{|B|^2}{|A|^2} \]

  • Single electron: \(R\) is the probability that this electron will reflect.

  • Beam of many electrons: \(R\) is the average fraction that reflect.

The transmission coefficient is

(380)#\[ T = 1 - R \]

For example, if \(R = 0.01\), then of 10,000 electrons incident on the step, about 100 reflect and 9,900 transmit. We cannot predict which ones—only probabilities.

The quantum surprise#

For \(E > U_b\), classical physics says \(R = 0\) and \(T = 1\). Schrödinger’s equation gives \(R > 0\) and \(T < 1\) (except in special cases). The reflection probability depends on \(E\), \(U_b\), and \(m\)—it follows from the boundary conditions. The key point: electrons are matter waves; they reflect at potential steps just as light reflects at interfaces, even when classically “they have enough energy to pass.”

Summary#

  • Potential step: \(U\) jumps from 0 to \(U_b\) at \(x = 0\); electron with \(E > U_b\).

  • Classical: no reflection. Quantum: some reflection (\(R > 0\)).

  • Wave functions: \(\psi_1 = Ae^{ikx} + Be^{-ikx}\), \(\psi_2 = Ce^{ik_b x}\); boundary conditions match \(A+B=C\) and \(Ak-Bk=Ck_b\).

  • Reflection coefficient \(R = |B|^2/|A|^2\) = probability of reflection; transmission \(T = 1 - R\).