A free electron in motion along the x axis has a localized wave function. The uncertainty in its momentum is decreased if

Options

A. The wave function is made more narrow
B. The wave function is made less narrow
C. The wave function remains the same but the energy of the electron is increased
D. The wave function remains the same but the energy of the electron is decreased

Answer

To solve the question regarding the decrease in uncertainty of momentum for a free electron with a localized wave function, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Wave Function**:
- A wave function describes the quantum state of a particle. For a free electron, the wave function can be localized, meaning it has a specific position in space.

2. **Heisenberg Uncertainty Principle**:
- According to the Heisenberg Uncertainty Principle, there is a fundamental limit to how precisely we can know both the position (Δx) and momentum (Δp) of a particle. The principle is mathematically expressed as:
\[ Δx \cdot Δp \geq \frac{\hbar}{2} \]
- Here, Δx is the uncertainty in position, Δp is the uncertainty in momentum, and \(\hbar\) is the reduced Planck's constant.

3. **Effect of Localizing the Wave Function**:
- When the wave function of the electron is localized (narrowed), the uncertainty in position (Δx) decreases. According to the uncertainty principle, if Δx decreases, then Δp must increase, meaning the uncertainty in momentum increases.

4. **Making the Wave Function Less Narrow**:
- To decrease the uncertainty in momentum (Δp), we need to make the wave function less narrow. A less narrow wave function implies a greater spread in position, which leads to a smaller uncertainty in momentum.

5. **Conclusion**:
- Therefore, the uncertainty in momentum of the free electron decreases if the wave function is made less narrow.

### Final Answer:
The uncertainty in momentum of a free electron in motion along the x-axis decreases if the wave function is made less narrow.

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