Two photons having

Options

A. equal wavelengths have equal linear momenta
B. equal energies have equal linear momenta
C. equal frequencies have equal linear momenta
D. equal linear momenta have equal wavelengths

Answer

To solve the problem, we need to analyze the relationship between the linear momentum of photons and their wavelengths. Here's a step-by-step breakdown:

### Step 1: Understand the relationship between momentum and wavelength
The linear momentum \( P \) of a photon can be expressed using the equation:
\[ P = \frac{h}{\lambda} \]
where:
- \( P \) is the linear momentum,
- \( h \) is Planck's constant,
- \( \lambda \) is the wavelength of the photon.

### Step 2: Set up the equation for two photons
Given that we have two photons with equal wavelengths, we can denote their momenta as \( P_1 \) and \( P_2 \):
\[ P_1 = \frac{h}{\lambda_1} \quad \text{and} \quad P_2 = \frac{h}{\lambda_2} \]

### Step 3: Equate the momenta
Since the problem states that the two photons have equal linear momenta, we can set \( P_1 \) equal to \( P_2 \):
\[ \frac{h}{\lambda_1} = \frac{h}{\lambda_2} \]

### Step 4: Simplify the equation
Since \( h \) is a constant and non-zero, we can cancel it from both sides of the equation:
\[ \frac{1}{\lambda_1} = \frac{1}{\lambda_2} \]

### Step 5: Conclude the relationship between wavelengths
From the above equation, we can deduce that:
\[ \lambda_1 = \lambda_2 \]
This means that if two photons have equal linear momenta, they must also have equal wavelengths.

### Step 6: Analyze the reverse scenario
Now, let's consider the reverse situation: if \( \lambda_1 = \lambda_2 \), does it imply that \( P_1 = P_2 \)?
- While the magnitudes of the momenta will be equal, the directions of the momenta may differ since momentum is a vector quantity. Therefore, equal wavelengths do not guarantee equal linear momenta in terms of direction.

### Conclusion
Based on the analysis, we conclude that:
- Equal linear momenta imply equal wavelengths.
- Equal wavelengths do not necessarily imply equal linear momenta due to the vector nature of momentum.

Thus, the correct option is:
**Option D: Equal linear momenta have equal wavelengths.**