An electron with energy E is incident on a potential energy barrier of height `E_(pot)` and thickness L. The probability of tunneling increases if

Options

A. E decreases without any other changes
B. `E_(pot)` increases without any other changes
C. L decreases without any other changes
D. E and `E_(pot)` increase by the same amount

Answer

To solve the problem regarding the tunneling probability of an electron encountering a potential energy barrier, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have an electron with energy \( E \) approaching a potential energy barrier of height \( E_{\text{pot}} \) and thickness \( L \).
- We need to determine the conditions under which the probability of tunneling through this barrier increases.

2. **Condition for Tunneling**:
- Tunneling occurs when the energy of the electron \( E \) is greater than the potential energy barrier \( E_{\text{pot}} \). Thus, for tunneling to happen, we require:
\[ E > E_{\text{pot}} \]

3. **Transmission Coefficient**:
- The probability of tunneling can be quantified using the transmission coefficient \( T \). According to the WKB approximation, the transmission coefficient is given by:
\[ T \propto e^{-2\gamma L} \]
- Where \( \gamma \) is a function of the mass of the electron \( m \) and the difference between the energy of the electron and the potential barrier:
\[ \gamma = \sqrt{\frac{2m(E_{\text{pot}} - E)}{\hbar^2}} \]

4. **Effect of Barrier Thickness \( L \)**:
- From the equation \( T \propto e^{-2\gamma L} \), we can see that as the thickness \( L \) increases, the exponent becomes more negative, leading to a decrease in \( T \).
- Conversely, if \( L \) decreases, \( T \) increases.

5. **Conclusion**:
- The probability of tunneling increases if the thickness \( L \) of the barrier decreases. Therefore, the answer to the question is:
\[ \text{The probability of tunneling increases if } L \text{ decreases.} \]