The magnitude of an orbital angular momentum vector of an electron is `sqrt(6) (h)/(2pi)` into how many components will the vector split if an external field is applied to it ?

Options

A. 3
B. 5
C. 7
D. 10

Explanation

To solve the problem, we need to determine how many components the orbital angular momentum vector of an electron will split into when an external field is applied. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the given information
The magnitude of the orbital angular momentum vector \( L \) is given as:
\[ L = \sqrt{6} \frac{h}{2\pi} \]

### Step 2: Use the formula for orbital angular momentum
The formula for the magnitude of the orbital angular momentum is:
\[ L = \sqrt{l(l + 1)} \frac{h}{2\pi} \]
where \( l \) is the azimuthal quantum number.

### Step 3: Set the two expressions for \( L \) equal to each other
We can equate the two expressions for \( L \):
\[ \sqrt{l(l + 1)} \frac{h}{2\pi} = \sqrt{6} \frac{h}{2\pi} \]
Cancelling \( \frac{h}{2\pi} \) from both sides gives:
\[ \sqrt{l(l + 1)} = \sqrt{6} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides results in:
\[ l(l + 1) = 6 \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ l^2 + l - 6 = 0 \]

### Step 6: Factor the quadratic equation
To factor the quadratic equation, we look for two numbers that multiply to \(-6\) and add to \(1\):
\[ (l + 3)(l - 2) = 0 \]

### Step 7: Solve for \( l \)
Setting each factor to zero gives:
\[ l + 3 = 0 \quad \Rightarrow \quad l = -3 \quad (\text{not valid, as } l \text{ cannot be negative}) \]
\[ l - 2 = 0 \quad \Rightarrow \quad l = 2 \]

### Step 8: Determine the magnetic quantum number \( m_l \)
For \( l = 2 \), the magnetic quantum number \( m_l \) can take values from \(-l\) to \(+l\):
\[ m_l = -2, -1, 0, +1, +2 \]
This gives us a total of \( 5 \) possible values.

### Step 9: Conclusion
Thus, when an external field is applied, the orbital angular momentum vector will split into \( 5 \) components.

### Final Answer
The number of components into which the vector will split is **5**.

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