De Broglie wavelength `lambda` is proportional to

Options

A. `1/sqrtE` for photons and 1/E for particles
B. 1/E for photons and `1/sqrtE` for particles
C. 1/E for both photons and particles in motion
D. `1/sqrtE` for both photons and particles.

Answer

To solve the problem regarding the De Broglie wavelength, we will follow these steps:

### Step 1: Understand the De Broglie Wavelength Formula
The De Broglie wavelength (λ) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is the Planck constant and \( p \) is the momentum of the particle.

### Step 2: Relate Momentum to Energy
For a particle, the momentum \( p \) can be expressed in terms of kinetic energy \( E \):
\[ p = mv \]
where \( m \) is the mass and \( v \) is the velocity of the particle. The kinetic energy \( E \) is given by:
\[ E = \frac{1}{2} mv^2 \]
From this, we can express \( v \) in terms of \( E \):
\[ v = \sqrt{\frac{2E}{m}} \]

### Step 3: Substitute Momentum in the Wavelength Formula
Now substituting \( p \) into the De Broglie wavelength formula:
\[ \lambda = \frac{h}{mv} \]
Substituting \( v \) from the kinetic energy equation:
\[ \lambda = \frac{h}{m \sqrt{\frac{2E}{m}}} = \frac{h \sqrt{m}}{\sqrt{2E}} \]

### Step 4: Analyze the Relationship
From the above equation, we can see that:
\[ \lambda \propto \frac{1}{\sqrt{E}} \]
This indicates that the De Broglie wavelength is inversely proportional to the square root of the energy.

### Step 5: Conclusion
Thus, the De Broglie wavelength \( \lambda \) is inversely proportional to the square root of energy \( E \) for a particle.

### Final Answer
The De Broglie wavelength \( \lambda \) is proportional to \( \frac{1}{\sqrt{E}} \).

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