Question

Consider the motion of a quantum particle of mass \(m\) and energy \(E\) under the influence of a step potential of height \(V_0\). If \(R\) denotes the reflection coefficient, which one of the following statements is true? 

• A. If \(E = \frac{4}{3}V_0, R = 1\)
• B. If \(E = \frac{4}{3}V_0, R = 0\)
• C. If \(E = \frac{1}{2}V_0, R = 1\)
• D. If \(E = \frac{1}{2}V_0, R = 0.5\)

Hint:

For \(E > V_0\), partial transmission always occurs; \(R\) decreases as \(E\) increases.

Answer 1

The Correct Option is C

Quantum step potential problem:

A particle with energy $E$ encounters a step potential of height $V_0$ at $x = 0$.

Analysis by energy regime:

Case 1: $E > V_0$ (particle has more energy than barrier)

The particle can classically pass over the barrier. Quantum mechanically, there's still some reflection due to the change in potential, but transmission is allowed. The reflection coefficient $R < 1$.

Case 2: $E < V_0$ (particle has less energy than barrier)

Classically, the particle cannot pass. Quantum mechanically, the wave function decays exponentially in the barrier region (evanescent wave), but the particle is reflected. For a step potential (not a finite barrier), $R = 1$ (total reflection).

Case 3: $E = V_0$ (boundary case)

At exactly $E = V_0$, the wave number in the barrier region becomes zero, and the reflection coefficient can be calculated.

Reflection coefficient formula:

For a step potential, when $E < V_0$: $$R = \left|\frac{k_1 - ik_2}{k_1 + ik_2}\right|^2 = 1$$

where $k_1 = \sqrt{2mE}/\hbar$ and $k_2 = \sqrt{2m(V_0 - E)}/\hbar$

When $E > V_0$: $$R = \left|\frac{k_1 - k_3}{k_1 + k_3}\right|^2$$

where $k_3 = \sqrt{2m(E - V_0)}/\hbar$

Evaluating the options:

(A) If $E = \frac{4}{3}V_0$, $R = 1$:

Since $E > V_0$, we have $R < 1$, not equal to 1. FALSE 

(B) If $E = \frac{4}{3}V_0$, $R = 0$:

Even when $E > V_0$, there's always some reflection at a step. FALSE 

(C) If $E = \frac{1}{2}V_0$, $R = 1$:

Since $E < V_0$, the particle cannot penetrate a step potential (infinite extent), so total reflection occurs: $R = 1$. TRUE 

(D) If $E = \frac{1}{2}V_0$, $R = 0.5$:

For $E < V_0$ at a step potential, $R = 1$, not 0.5. FALSE 

Answer: (C) If $E = \frac{1}{2}V_0$, $R = 1$