Select the correct relation between azimuthal quantum number (l) and magnetic quantum (m)

Options

A. `l=(m-1)/2`
B. `m=1+2`
C. `m=2l^2+1`
D. `l=2m+1`

Answer

To solve the question regarding the relationship between the azimuthal quantum number (l) and the magnetic quantum number (m), we can follow these steps:

### Step 1: Understand the Definitions
- The **azimuthal quantum number (l)** determines the shape of the orbital and can take on integer values from 0 to n-1, where n is the principal quantum number.
- The **magnetic quantum number (m)** describes the orientation of the orbital in space and can take on integer values from -l to +l, including 0.

### Step 2: Determine the Range of m
- For a given value of l, the magnetic quantum number m can take values:
\[ m = -l, -l+1, ..., 0, ..., l-1, l \]
- This means that the total number of possible values for m is given by:
\[ \text{Total values of } m = 2l + 1 \]

### Step 3: Establish the Relationship
- From the above, we can derive that for any given l:
\[ m \text{ can take values from } -l \text{ to } +l \]
- Therefore, the relationship can be expressed as:
\[ l = \frac{m - 1}{2} \]
or alternatively,
\[ m = 2l + 1 \]

### Step 4: Conclusion
- Based on the derived relationships, we can conclude that the correct relation between the azimuthal quantum number (l) and the magnetic quantum number (m) is:
\[ m = 2l + 1 \]

### Step 5: Check Options
- After establishing the relationship, we can check the provided options to find the correct one. According to the solution, Option A is the correct choice.

### Final Answer
- The correct relation between the azimuthal quantum number (l) and the magnetic quantum number (m) is:
\[ m = 2l + 1 \]