In Bohr's model of the hydrogen atom, which of the following pairs of quantities are quantized?

Options

A. Energy and linear momentum
B. Linear and angular momentum
C. Energy and angular momentum
D. None of the above

Answer

To solve the question regarding which pairs of quantities are quantized in Bohr's model of the hydrogen atom, we can follow these steps:

### Step-by-Step Solution

1. **Understanding Quantization**: In physics, quantization refers to the concept that certain properties, such as energy or angular momentum, can only take on discrete values rather than a continuous range.

2. **Bohr's Model Basics**: According to Bohr's model of the hydrogen atom, electrons move in circular orbits around the nucleus, and these orbits correspond to specific energy levels.

3. **Angular Momentum Quantization**:
- Bohr proposed that the angular momentum (L) of an electron in a given orbit is quantized and is given by the formula:
\[ L = n \frac{h}{2\pi} \]
where \( n \) is the principal quantum number (1, 2, 3, ...), and \( h \) is Planck's constant.
- Since \( n \) can only take certain integer values, the angular momentum is quantized.

4. **Energy Quantization**:
- The energy (E) of an electron in a specific orbit is also quantized and can be expressed as:
\[ E_n = -\frac{13.6 \text{ eV}}{n^2} \]
- Here, \( E_n \) represents the energy of the electron in the nth orbit. As \( n \) changes, the energy changes in discrete steps, indicating that energy is also quantized.

5. **Evaluating the Options**:
- **Option 1**: Energy and linear momentum - Linear momentum is not quantized in the same way as angular momentum.
- **Option 2**: Linear and angular momentum - Linear momentum is not quantized.
- **Option 3**: Energy and angular momentum - Both are quantized as explained.
- **Option 4**: None of these - This is incorrect since we have established that energy and angular momentum are quantized.

6. **Conclusion**: The correct answer is option 3: **Energy and Angular Momentum**.