Question

An electron is in a three-dimensional box with side lengths \(L_X =\) 0.600 nm and \(L_Y = L_Z = 2L_X\). What are the quantum numbers \(n_X, n_Y,\) and \(n_Z\) and the energies, in eV, for the four lowest energy levels? What is the degeneracy of each (including the degeneracy due to spin)?

Short answer

Quantum numbers for lowest levels are: (1,1,1), (1,1,2), (1,2,1), (2,1,1). Energies are computed using energy formula. Degeneracies include spin.

Step-by-step solution

Understand the Problem

We have an electron in a 3D box with sides \( L_X = 0.600 \text{ nm} \) and \( L_Y = L_Z = 2L_X \). We need to find possible quantum numbers \( n_X, n_Y, \) and \( n_Z \) for the four lowest energies and also calculate these energies.

Write the Formula for Energy Levels in a 3D Box

The energy levels for an electron in a 3D box are given by the formula: \[ E_{n_x,n_y,n_z} = \left(\frac{h^2}{8m}\right) \left(\frac{n_x^2}{L_X^2} + \frac{n_y^2}{L_Y^2} + \frac{n_z^2}{L_Z^2}\right) \] where \( h \) is Planck's constant, \( m \) is the electron mass, and \( n_X, n_Y, n_Z \) are the quantum numbers.

Substitute Values into the Energy Formula

Convert \( L_Y \) and \( L_Z \) to lengths in nm: \( L_Y = L_Z = 1.2 \text{ nm} \). Substitute \( L_X = 0.600 \text{ nm} \) and \( L_Y = L_Z = 1.2 \text{ nm} \) into the energy formula.

Determine Quantum Numbers for the Four Lowest Energy Levels

For the lowest energy states, try combinations of \( n_X, n_Y, \) and \( n_Z \) where the sum of their squares divided by the respective box lengths yields the smallest values. The lowest four combinations are (1,1,1), (1,1,2), (1,2,1), and (2,1,1).

Calculate the Energies

Compute the energies using the formula. For the first combination (1,1,1), compute as follows: \[ E_{1,1,1} = \left(\frac{h^2}{8m}\right) \left(\frac{1^2}{(0.600 \text{ nm})^2} + \frac{1^2}{(1.2 \text{ nm})^2} + \frac{1^2}{(1.2 \text{ nm})^2} \right) \] Repeat for the other combinations (for \( n_X, n_Y, n_Z\) as (1,1,2), (1,2,1), (2,1,1)). Calculate energies in eV.

Calculate Degeneracy Including Spin

Since the electron can have spin up or spin down, there is a degeneracy of 2 for each non-degenerate state. For states with identical energy levels like (1,2,1), (2,1,1), and (1,1,2), add their degeneracies. Calculate total degeneracy for each level.

Key concepts

Three-dimensional box

A three-dimensional box in quantum mechanics is a simple theoretical model used to understand the behavior of particles like electrons confined in a compact space. Imagine having a box where an electron moves freely within certain boundaries. These boundaries in our case are defined by the side lengths function:

• Two sides, \(L_Y\) and \(L_Z\), are doubled in length compared to one side, \(L_X\), implying the box isn't cubical.
• In our specific example, \(L_X\) is 0.600 nm, meaning \(L_Y\) and \(L_Z\) are each 1.2 nm (since \(L_Y = L_Z = 2L_X\)).

This box concept helps us apply boundary conditions and use the shrinked dimensions to calculate potential energy levels of electrons within these confines. Understanding this basic setup is essential before delving into how particles like electrons behave under these conditions.

Quantum numbers

Quantum numbers are essential tools in quantum mechanics. They help us pinpoint the allowed energy levels for particles within confined systems like our three-dimensional box. Each quantum number represents one dimension of the box and how the particle's position or state is quantized within that dimension.

• \(n_X\), \(n_Y\), and \(n_Z\) represent quantum numbers in each of the three dimensions respectively.
• These numbers are integers starting from 1 (not zero). They designate the node counts in standing wave patterns in the box.

These values are pivotal in determining the energy of the particle since they indicate how the electron "bounces" around inside the box. For each energy level, the quantum numbers are sought in combinations where their squared sum divided by their respective box dimension gives the lowest possible values. This principle ensures we can identify the lowest energy states for our electron.

Energy levels

Energy levels in a quantum box are derived based on a mathematical model. The energy at each level can be mathematically described as: \[E_{n_X,n_Y,n_Z} = \left(\frac{h^2}{8m}\right) \left(\frac{n_X^2}{L_X^2} + \frac{n_Y^2}{L_Y^2} + \frac{n_Z^2}{L_Z^2}\right)\]where:

• \(E_{n_X,n_Y,n_Z}\) represents the energy corresponding to quantum numbers \(n_X, n_Y,\) and \(n_Z\).
• \(h\) is Planck's constant and \(m\) is the mass of the electron.

For our particular case, we seek energy combinations that are the lowest, based on quantum number permutations. Once the permitted combinations, like (1,1,1), (1,1,2), etc., are known, energies can be calculated by substituting these into the formula.Additionally, since energy depends on squared quantum numbers, even a small change in these numbers can lead to significant difference in energy, illustrating why their correct assignment is vital in quantum level calculations.

Electron spin

Electron spin is an intrinsic form of angular momentum carried by electrons. It can have one of two possible values:

• Spin-up (+1/2)
• Spin-down (-1/2)

This property affects the degeneracy of energy-level states in quantum systems. Degeneracy refers to the number of different states that share the same energy level. In a box model, each energy level initially found would theoretically only allow one state. However, when accounting for electrons' spins:

• Each energy state is doubled in possibility due to the two independent spin states, leading to an effective degeneracy of 2.

For systems where certain energy levels coincide for different quantum number sets, the degeneracy increases. This influence of spin is crucial to incorporate in calculating accurate levels of energy states for a more comprehensive understanding.