The ratio of de Broglie wavelengths of a proton and a neutron moving with the same velocity is nearly-

Options

A. `1`
B. `sqrt2`
C. `1/sqrt2`
D. none of the above

Answer

To find the ratio of the de Broglie wavelengths of a proton and a neutron moving with the same velocity, we can follow these steps:

### Step 1: Write the formula for de Broglie wavelength
The de Broglie wavelength (\( \lambda \)) is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is the Planck constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

### Step 2: Write the de Broglie wavelength for the proton
For a proton, the de Broglie wavelength (\( \lambda_p \)) is:
\[ \lambda_p = \frac{h}{m_p v} \]
where \( m_p \) is the mass of the proton.

### Step 3: Write the de Broglie wavelength for the neutron
For a neutron, the de Broglie wavelength (\( \lambda_n \)) is:
\[ \lambda_n = \frac{h}{m_n v} \]
where \( m_n \) is the mass of the neutron.

### Step 4: Take the ratio of the de Broglie wavelengths
Now, we take the ratio of the de Broglie wavelengths of the proton and neutron:
\[ \frac{\lambda_p}{\lambda_n} = \frac{\frac{h}{m_p v}}{\frac{h}{m_n v}} \]

### Step 5: Simplify the ratio
By simplifying the above expression, we can cancel \( h \) and \( v \) (since they are the same for both particles):
\[ \frac{\lambda_p}{\lambda_n} = \frac{m_n}{m_p} \]

### Step 6: Use the approximate masses of the proton and neutron
The mass of the neutron (\( m_n \)) is approximately equal to the mass of the proton (\( m_p \)):
\[ m_n \approx m_p \]
Thus, the ratio becomes:
\[ \frac{\lambda_p}{\lambda_n} \approx \frac{m_p}{m_p} = 1 \]

### Conclusion
Therefore, the ratio of the de Broglie wavelengths of a proton and a neutron moving with the same velocity is nearly:
\[ \frac{\lambda_p}{\lambda_n} \approx 1 \]