Answer to Question #87926 in Quantum Mechanics for Sridhar

Question

A particle is placed inside a one
dimensional box with rigid walls. The first
excited state energy of the particle is
4 eV. The ground state energy is
Answer 1eV

Answer 1

The energy levels of a particle in a 1-D box with rigid walls is given by

$$E_n = \frac{\hbar^2 \pi^2}{2 m a^2} n^2 \equiv E_0 n^2,$$

where m is the particle's mass, a is the length of the box, n = 1,2,... is the energy quantum number. Thus, the energy of the first and the second levels can be defined as follows:

$$E_1 = E_0 \cdot 1^2 = E_0,\\ E_2 = E_0 \cdot 2^2 = 4 E_0.$$

Hence,

$$E_1 = \frac{E_2}{4} = \frac{4 \, eV}{4} = 1 \, eV.$$

Answer: 1 eV.