An electron in a one-dimensional infinite potential well of length L has ground-state energy `E_(1)`. The length is changed to L' so that the new ground-state energy is `E'_(1) = 0.500E_(1)`. What is the ratio `L'//L`?

Explanation

To find the ratio \( \frac{L'}{L} \) given that the new ground-state energy \( E'_1 = 0.500 E_1 \), we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Energy Formula**:
The energy levels of an electron in a one-dimensional infinite potential well are given by the formula:
\[
E_n = \frac{n^2 h^2}{8 m L^2}
...