Question

Which wavelength is largest de Broglie?

Answer 1

The de Broglie wavelength of a particle is given by the equation:

\[
\lambda = \frac{h}{p}
\]

where:
- \(\lambda\) is the de Broglie wavelength,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34}\, \text{J·s}\)),
- \(p\) is the momentum of the particle (\(p = mv\), where \(m\) is the mass and \(v\) is the velocity).

### Factors Affecting the De Broglie Wavelength:
1. **Momentum (p):**
   - The de Broglie wavelength is inversely proportional to momentum.
   - A particle with **lower momentum** (smaller \(m\) or \(v\)) will have a **larger wavelength**.
   - A particle with **higher momentum** (larger \(m\) or \(v\)) will have a **smaller wavelength**.

2. **Mass (m):**
   - For a given velocity, lighter particles have a smaller momentum and hence a **larger de Broglie wavelength** than heavier particles.

3. **Velocity (v):**
   - For a given mass, slower particles have smaller momentum and hence a **larger wavelength**.

### Largest De Broglie Wavelength:
- To achieve the **largest de Broglie wavelength**, we should consider particles with:
  1. The **smallest mass**.
  2. The **slowest speed**.

### Example of Particles:
1. **Electrons:** Being very light, electrons can have relatively large de Broglie wavelengths, especially if moving slowly.
2. **Protons:** They have larger mass than electrons, so their de Broglie wavelength is smaller at the same speed.
3. **Macroscopic Objects:** Larger masses (e.g., a moving baseball) have extremely small de Broglie wavelengths, making them negligible.

### Conclusion:
The largest de Broglie wavelength is observed for particles with the smallest mass moving at the slowest speed. Among common particles, **slow-moving electrons** typically exhibit the largest de Broglie wavelengths in experimental setups.

Answer 2

The de Broglie wavelength refers to the wavelength associated with a particle moving with a certain momentum. It was proposed by physicist Louis de Broglie in 1924, and it connects the wave-like and particle-like properties of matter, showing that particles such as electrons, protons, or even larger objects can exhibit wave-like behavior.

The de Broglie wavelength \(\lambda\) is given by the following equation:

\[
\lambda = \frac{h}{p}
\]

Where:
- \(\lambda\) is the de Broglie wavelength,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34}\) J·s),
- \(p\) is the momentum of the particle, which is the product of its mass \(m\) and velocity \(v\): \(p = mv\).

### Understanding the Wavelength:

- The **larger the momentum** of the particle, the **smaller** the de Broglie wavelength.
- The **smaller the momentum**, the **larger** the de Broglie wavelength.

Thus, to have the **largest de Broglie wavelength**, a particle must have the **smallest momentum**. This can be achieved in two ways:
1. **Decreasing the particle's mass**: A particle with a very small mass will have a larger wavelength at a given velocity.
2. **Decreasing the particle's velocity**: A particle moving more slowly will have a smaller momentum and thus a larger wavelength.

### Examples:
1. **Light particles (photons)** have zero rest mass, but they can still have a wavelength depending on their energy, which is related to their frequency. Their de Broglie wavelength is much smaller than a macroscopic object’s wavelength because their momentum is relatively large due to their high energy.
   
2. **Heavy particles** like a baseball have a much larger mass than something like an electron, but even at slow speeds, their momentum is large compared to a much lighter particle like an electron, resulting in a much smaller de Broglie wavelength.

### Answer:
The largest de Broglie wavelength corresponds to the smallest momentum. This can occur when a particle has a very small mass or is moving very slowly. For instance, subatomic particles with very small masses (such as neutrinos or extremely slow-moving particles) can have larger de Broglie wavelengths compared to larger, faster-moving objects.