Question

Discuss what happens to the energy levels for an electron trapped in a one-dimensional box as the length of the box increases.

Short answer

Energy levels become closely spaced with an increase in the length of the box.

Step-by-step solution

Boundary conditions for particles present in a one-dimensional box.

Below mentioned are boundary conditions for particles present in a one-dimensional box.

1. Particle is always bounded to be present inside the box only and it cannot be a present outside box.

2. Total probability to find the particle in the box is always one.

3. Wave function must be continuous.

The mathematical expression for a particle in a one-dimensional box is as follows:

$\mathit{E}\mathbf{=}\frac{{\mathbf{n}}^{\mathbf{2}}{\mathbf{h}}^{\mathbf{2}}}{\mathbf{8}\mathbf{m}{\mathbf{L}}^{\mathbf{2}}}$

Where,

-E is the energy of a particle in a 1-D box.

-n is an integer.

-h is Planck's constant.

-m is the mass of the particle.

-L is the length of the box.

Determine the change in energy levels for an electron trapped in a one-dimensional box with an increase in length of the box.

The mathematical expression for a particle in a one-dimensional box is as follows:

$E=\frac{n^2{h}^2}{8m{L}^2}$

From the above expression, it is clear that the energy of a particle in a one-dimensional box is inversely related to the square of the length of the box. As for the length of the box increases, the energy of a particle in one-dimensional box decreases. Since the difference in energy levels is governed directly by energy, the gap or space between energy levels decreases with a decrease in energy. Therefore, energy levels become closely spaced with an increase in the length of the box.