Question

A neutron moves between rigid walls \(8.4 \mathrm{fm}\) apart. What is the energy of its \(n=1\) state?

Short answer

Answer: The energy of the neutron in its first energy state is \(3.52 \times 10^{-29}\) Joules.

Step-by-step solution

List the known values

We know the following values: Distance between rigid walls, \(a = 8.4 \, \text{fm} = 8.4 \times 10^{-15} \, \text{m}\) Mass of a neutron, \(m = 1.675 \times 10^{-27} \, \text{kg}\) Reduced Planck constant, \(\hbar = 1.054 \times 10^{-34} \, \text{Js}\) Quantum number for the first state, \(n = 1\)

Plug known values into the formula for energy

Now that we have all the necessary values, we can plug them into the formula for the energy of a particle in a one-dimensional box, \(E_n = \frac{n^2 \pi^2 \hbar^2}{2ma^2}\), to determine the energy in the n=1 state: \(E_1 = \frac{1^2 \pi^2 (1.054 \times 10^{-34})^2}{2(1.675 \times 10^{-27})(8.4 \times 10^{-15})^2}\)

Calculate the energy

Calculate the energy of the neutron in its first state: $E_1 = \frac{(9.87 \times 10^{-70})}{(2.805 \times 10^{-41})} = 3.52 \times 10^{-29}\, \text{J}$ So the energy of the neutron in its n=1 state is \(3.52 \times 10^{-29}\) Joules.