If travelling at same speeds, which of the following matter waves have the shortest wavelength?

Options

A. Electron
B. Alpha particle `(He^(2+))`
C. Neutron
D. Proton

Explanation

To determine which matter wave has the shortest wavelength when traveling at the same speed, we can use the de Broglie wavelength formula:

### Step 1: Understand the de Broglie Wavelength Formula
The de Broglie wavelength (λ) is given by the formula:
\[ \lambda = \frac{h}{mv} \]
Where:
- \( h \) = Planck's constant (approximately \( 6.626 \times 10^{-34} \) Joule seconds)
- \( m \) = mass of the particle
- \( v \) = velocity of the particle

### Step 2: Analyze the Relationship Between Wavelength and Mass
From the formula, we can see that the wavelength (λ) is inversely proportional to the mass (m) of the particle when the speed (v) is constant:
\[ \lambda \propto \frac{1}{m} \]
This means that as the mass increases, the wavelength decreases.

### Step 3: List the Masses of the Given Particles
Now, we need to find the masses of the particles mentioned in the question:
- **Electron**: \( 9.1 \times 10^{-31} \) kg
- **Neutron**: \( 1.675 \times 10^{-27} \) kg
- **Proton**: \( 1.6727 \times 10^{-27} \) kg
- **Alpha particle**: Consists of 2 protons and 2 neutrons, so its mass is approximately:
\[ \text{Mass of alpha particle} = 4 \times (1.6727 \times 10^{-27} \text{ kg}) \approx 6.68 \times 10^{-27} \text{ kg} \]

### Step 4: Compare the Masses
Now we compare the masses:
- Electron: \( 9.1 \times 10^{-31} \) kg
- Neutron: \( 1.675 \times 10^{-27} \) kg
- Proton: \( 1.6727 \times 10^{-27} \) kg
- Alpha particle: \( 6.68 \times 10^{-27} \) kg

### Step 5: Identify the Particle with the Highest Mass
From the comparison:
- The alpha particle has the highest mass.

### Step 6: Conclusion
Since the wavelength is inversely proportional to mass, the particle with the highest mass (alpha particle) will have the shortest wavelength when traveling at the same speed.

Thus, the answer to the question is:
**Alpha particle has the shortest wavelength.**

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