Question

In terms of the ground-state energy${\mathbf{E}}_{\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}}$ what is the energy of the highest level occupied by an electron when 10 electrons are placed into a cubical box?

Short answer

The energy of the highest level occupied by an electron when 10 electrons are placed into a cubical box is $3{\mathrm{E}}_{1,1,1}$.

Step-by-step solution

Define the energy level.

In a three-dimensional cubical box, the energy level of a particle is written as:

${\mathbf{E}}_{{\mathbf{n}}_{\mathbf{x}}\mathbf{,}{\mathbf{n}}_{\mathbf{y}}\mathbf{,}{\mathbf{n}}_{\mathbf{z}}}\mathbf{=}\frac{\left(n_x^2+n_y^2+n_z^2\right){\mathbf{\pi }}^{\mathbf{2}}{\mathbf{ħ}}^{\mathbf{2}}}{\mathbf{2}{\mathbf{mL}}^{\mathbf{2}}}$………………….(1)

Where,${\mathbf{n}}_{\mathbf{X}}\mathbf{,}{\mathbf{n}}_{\mathbf{Y}}$ and${\mathbf{n}}_{\mathbf{z}}$ are the quantum numbers.

Determine the energy of the highest level.

The quantum numbers of the ground level$({\mathrm{n}}_{\mathrm{X}},{\mathrm{n}}_{\mathrm{Y}},{\mathrm{n}}_{\mathrm{Z}})=(1,1,1)$ with energy ${\mathrm{E}}_{1,1,1},$, the quantum numbers of the first level$(n_X,n_Y,n_Z)=(2,1,1)(1,2,1)(1,1,2)$ with energy $2{\mathrm{E}}_{1,1,1},$, and the quantum numbers of the second level$({\mathrm{n}}_{\mathrm{X}},{\mathrm{n}}_{\mathrm{Y}},{\mathrm{n}}_{\mathrm{Z}})=(2,2,1)(1,2,2)(2,1,2)$ with energy $3{\mathrm{E}}_{1,1,1},$.

According to Pauli’s exclusion principle, the two electrons cannot occupy the same quantum-mechanical state.

The ground state has only a single set of quantum numbers, so it can occupy 2 electrons, the second state has three sets of quantum numbers, so it can occupy 6 electrons and the third state have three sets of quantum numbers, so it can occupy 6 electrons

Therefore, out of 10 electrons 2 are in the ground state, 6 in the second state, and the left 2 electrons are occupied by the third state.

Hence, the energy of the highest level occupied by an electron when 10 electrons are placed into a cubical box is $3{\mathrm{E}}_{1,1,1,}$.