The electron in a hydrogen atom makes a transition from `n=n_(1)` to `n=n_(2)` state. The time period of the electron in the initial state `(n_(1))` is eight times that in the final state `(n_(2))`. The possible values of `n_(1)` and `n_(2)` are

Options

A. `n_(1)=8,n_(2)=1`
B. `n_(1)=4, n_(2)=2`
C. `n_(1)=2,n_(2)=4`
D. `n_(1)1, n_(2)=8`

Explanation

To solve the problem, we need to analyze the relationship between the time period of the electron in different energy states of a hydrogen atom.

### Step-by-Step Solution:

1. **Understanding the Time Period Relation**:
The time period \( T \) of an electron in a hydrogen atom is related to the principal quantum number \( n \). The time period can be expressed as:
\[ T \propto n^3 \]
This means that the time period is directly proportional to the cube of the principal quantum number.

2. **Setting Up the Equation**:
According to the problem, the time period of the electron in the initial state \( n_1 \) is eight times that in the final state \( n_2 \):
\[ T_1 = 8 T_2 \]
Using the proportional relationship, we can write:
\[ T_1 = k n_1^3 \quad \text{and} \quad T_2 = k n_2^3 \]
where \( k \) is a constant.

3. **Substituting the Time Periods**:
Substituting these expressions into the time period equation gives:
\[ k n_1^3 = 8 (k n_2^3) \]
Simplifying this, we get:
\[ n_1^3 = 8 n_2^3 \]

4. **Taking the Cube Root**:
Taking the cube root of both sides, we find:
\[ n_1 = 2 n_2 \]

5. **Finding Possible Values**:
Now we need to find integer values for \( n_1 \) and \( n_2 \) that satisfy this relationship. Since \( n_1 \) must be twice \( n_2 \), we can set \( n_2 = x \) and \( n_1 = 2x \).

The principal quantum numbers must be positive integers. Thus, we can choose:
- If \( n_2 = 1 \), then \( n_1 = 2 \times 1 = 2 \)
- If \( n_2 = 2 \), then \( n_1 = 2 \times 2 = 4 \)
- If \( n_2 = 3 \), then \( n_1 = 2 \times 3 = 6 \)
- And so on...

However, since the problem does not specify a maximum limit for \( n \), we can conclude that the pairs \((n_1, n_2)\) can be:
- \( (2, 1) \)
- \( (4, 2) \)
- \( (6, 3) \)
- etc.

6. **Conclusion**:
The possible values of \( n_1 \) and \( n_2 \) that satisfy the condition given in the problem are:
- \( n_1 = 4 \) and \( n_2 = 2 \) (as one of the valid pairs).