Question

The magnitude of energy difference between the energy levels 𝑛=3 and 𝑛=2 of a quantum particle of mass 𝑚 in a box of length L is \(\frac{Xℎ ^2}{ 8𝑚L ^2} .\) 
Then X = _______. 
(rounded off to the nearest integer) 
[Given: ℎ is Planck’s constant and 𝑛 denotes the quantum number]

Answer 1

Correct Answer: 5

For a particle in a box, the energy levels are quantized and given by the formula:

\( E_n = \frac{n^2 h^2}{8mL^2} \)

where:

• \( n \) is the quantum number,
• \( h \) is Planck’s constant,
• \( m \) is the mass of the particle,
• \( L \) is the length of the box.

The energy difference between two energy levels is:

\( \Delta E = E_3 - E_2 = \frac{9h^2}{8mL^2} - \frac{4h^2}{8mL^2} = \frac{5h^2}{8mL^2} \)

Therefore, comparing with the given expression \( \frac{Xh^2}{8mL^2} = \frac{5h^2}{8mL^2} \) , we find that \( X = 5 \).