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

Step 1: Reflection coefficient formula.
For a step potential, \[ R = \left( \frac{k_1 - k_2}{k_1 + k_2} \right)^2 \] where \( k_1 = \sqrt{\frac{2mE}{\hbar^2}} \) and \( k_2 = \sqrt{\frac{2m(E - V_0)}{\hbar^2}} \) for \(E > V_0\).

Step 2: Analyze the given case.
If \(E = \frac{4}{3}V_0 > V_0\), then both \(k_1\) and \(k_2\) are real and positive. \[ R = \left( \frac{\sqrt{E} - \sqrt{E - V_0}}{\sqrt{E} + \sqrt{E - V_0}} \right)^2 \] Substituting \(E = \frac{4}{3}V_0\): \[ R = \left( \frac{\sqrt{\frac{4}{3}V_0} - \sqrt{\frac{1}{3}V_0}}{\sqrt{\frac{4}{3}V_0} + \sqrt{\frac{1}{3}V_0}} \right)^2 = 0 \] Thus, \(R = 0\).

Step 3: Conclusion.
When \(E = \frac{4}{3}V_0\), reflection does not occur, so \(R = 0\).