If a proton and electron have the same de Broglie wavelength, then

Options

A. kinetic energy of electron `lt` kinetic energy of proton
B. kinetic energy of electron = kinetic energy of proton
C. momentum of electron = momentum of proton
D. momentum of electron `lt` momentum of proton

Explanation

To solve the question, "If a proton and electron have the same de Broglie wavelength, then...", we can follow these steps:

### Step-by-Step Solution:

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

2. **Setting Up the Equation for Both Particles**:
Since the problem states that the proton and electron have the same de Broglie wavelength, we can set up the equations for both:
\[ \lambda_{e} = \frac{h}{p_{e}} \quad \text{(for electron)} \]
\[ \lambda_{p} = \frac{h}{p_{p}} \quad \text{(for proton)} \]

3. **Equating the Wavelengths**:
Since \(\lambda_{e} = \lambda_{p}\), we can equate the two expressions:
\[ \frac{h}{p_{e}} = \frac{h}{p_{p}} \]
This implies that:
\[ p_{e} = p_{p} \]
Therefore, the momentum of the electron is equal to the momentum of the proton.

4. **Relating Momentum to Mass and Velocity**:
The momentum \(p\) is related to mass \(m\) and velocity \(v\) by the equation:
\[ p = mv \]
For the electron and proton, we can write:
\[ p_{e} = m_{e} v_{e} \quad \text{and} \quad p_{p} = m_{p} v_{p} \]

5. **Using the Equality of Momenta**:
Since \(p_{e} = p_{p}\), we have:
\[ m_{e} v_{e} = m_{p} v_{p} \]
Here, \(m_{e}\) is the mass of the electron and \(m_{p}\) is the mass of the proton.

6. **Conclusion About Kinetic Energy**:
The kinetic energy \(K\) of a particle is given by:
\[ K = \frac{1}{2} mv^2 \]
Since the masses are different and the momenta are the same, the velocities must be different. Thus, we cannot determine the kinetic energy directly without knowing the specific velocities.

### Final Answer:
The correct conclusion is that the momentum of the electron is equal to the momentum of the proton.