In which of the following cases the havier of the two particles has a smaller de-Broglie wavelength ? The two particles

Options

A. move with the same speed
B. move with the same linear momentum
C. move with the same kinetic energy
D. have the same change of potential energy in a conservative field

Explanation

To determine in which of the given cases the heavier of the two particles has a smaller de-Broglie wavelength, we need to analyze the relationship between mass, momentum, and de-Broglie wavelength.

### Step-by-Step Solution:

1. **Understanding de-Broglie Wavelength**:
The de-Broglie wavelength (\( \lambda \)) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

2. **Momentum and Kinetic Energy**:
The momentum (\( p \)) of a particle can also be expressed in terms of its mass (\( m \)) and velocity (\( v \)):
\[ p = mv \]
The kinetic energy (\( K \)) of a particle is given by:
\[ K = \frac{1}{2} mv^2 \]
From this, we can express velocity in terms of kinetic energy:
\[ v = \sqrt{\frac{2K}{m}} \]

3. **Substituting into de-Broglie Wavelength**:
By substituting the expression for momentum in terms of kinetic energy into the de-Broglie wavelength equation, we get:
\[ \lambda = \frac{h}{mv} \]
Substituting \( v \):
\[ \lambda = \frac{h}{m \sqrt{\frac{2K}{m}}} = \frac{h}{\sqrt{2Km}} = \frac{h}{\sqrt{2K}} \cdot \frac{1}{\sqrt{m}} \]
This shows that the de-Broglie wavelength is inversely proportional to the square root of the mass when kinetic energy is constant.

4. **Analyzing Each Case**:
- **Case 1: Both particles moving with the same speed**:
- Here, \( \lambda \) is inversely proportional to \( m \). Hence, the heavier particle will have a smaller wavelength.
- **Case 2: Both particles have the same momentum**:
- If both have the same momentum, they will have the same de-Broglie wavelength. Thus, this case is incorrect.
- **Case 3: Both particles have the same kinetic energy**:
- From our earlier derivation, if kinetic energy is the same, the heavier particle has a smaller de-Broglie wavelength.
- **Case 4: Both particles have the same change in potential energy**:
- This does not provide enough information about their initial and final kinetic energies. Thus, we cannot conclude which particle has a smaller wavelength.

5. **Conclusion**:
The cases where the heavier particle has a smaller de-Broglie wavelength are:
- Case 1: Both particles moving with the same speed.
- Case 3: Both particles have the same kinetic energy.

### Final Answer:
The heavier of the two particles has a smaller de-Broglie wavelength in cases 1 and 3.

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