Question

Which has the lowest (ground-state) energy, an electron trapped in a one-dimensional box of length${\mathbf{10}}^{\mathbf{-}\mathbf{6}}$m or one with length${\mathbf{10}}^{\mathbf{-}\mathbf{10}}\mathit{m}$?

Short answer

Box with a length of ${10}^{-6}m$has the lowest energy as compared to that of length ${10}^{-10}m.$

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:

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

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 whether the box of 10-6m or 10-10m has the lowest energy.  

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 equation, it is clear that the energy of a particle in a one-dimensional box is inversely proportional to the square of the length of the box. This indicates that the lowest energy is possessed by a particle in a one-dimensional box of larger length. So box with a length of ${10}^{-6}m$has the lowest energy as compared to that of length${10}^{-10}m$ .