The angular momentum of an electron in a hydrogen atom is proportional to (where n is principle quantum number )

Options

A. n
B. `n^2`
C. `n^3`
D. `sqrtn`

Answer

To solve the question regarding the angular momentum of an electron in a hydrogen atom and its proportionality to the principal quantum number \( n \), we can follow these steps:

### Step 1: Understand the Concept of Angular Momentum in Quantum Mechanics
In quantum mechanics, particularly in the Bohr model of the hydrogen atom, the angular momentum \( L \) of an electron is quantized. This means that it can only take on certain discrete values.

**Hint:** Recall that angular momentum in quantum mechanics is not continuous but quantized.

### Step 2: Recall the Formula for Angular Momentum
According to the Bohr model, the angular momentum \( L \) of an electron in a hydrogen atom is given by the formula:
\[ L = n \frac{h}{2\pi} \]
where:
- \( L \) is the angular momentum,
- \( n \) is the principal quantum number (which can take values 1, 2, 3,...),
- \( h \) is Planck's constant.

**Hint:** Remember that \( h \) is a constant, and \( n \) represents the energy level of the electron.

### Step 3: Analyze the Proportionality
From the formula \( L = n \frac{h}{2\pi} \), we can see that the angular momentum \( L \) is directly proportional to the principal quantum number \( n \). This means that as \( n \) increases, the angular momentum increases linearly.

**Hint:** Look for relationships in equations where one variable is multiplied by a constant; this indicates direct proportionality.

### Step 4: Conclusion
Thus, we conclude that the angular momentum of an electron in a hydrogen atom is proportional to the principal quantum number \( n \).

**Final Answer:** The angular momentum of an electron in a hydrogen atom is proportional to \( n \).

### Summary of Steps:
1. Understand the quantization of angular momentum.
2. Recall the formula for angular momentum in the Bohr model.
3. Analyze the relationship to determine proportionality.
4. Conclude the relationship between angular momentum and the principal quantum number.