Question

An electron is confined to a cubic \(3 \mathrm{D}\) infinite well \(1 \mathrm{~nm}\) on a side. (a) Whut ure the three lowest different energies possible? (b) To how many different states do these three energies correspond?

Short answer

The three lowest different energies possible for an electron confined to a cubic 3D infinite well of side length 1 nm are obtained when quantum numbers \(n_x, n_y,\) and \(n_z\) are (1,1,1), (1,1,2), and (1,2,2) respectively. These energy levels correspond to 1, 3, and 3 different states, respectively.

Step-by-step solution

Find the Equation for Energy Levels

The energy levels of an electron confined in a cubic 3D infinite well can be calculated using the following equation derived from the solutions to the 3D Schrödinger equation: \(E=\frac{h^2(n_x^2+n_y^2+n_z^2)}{8mL^2}\), where \(h\) is Planck’s constant, \(m\) is the mass of an electron, \(L\) is the side length of the cube, and \(n_x, n_y,\) and \(n_z\) are quantum numbers that can take on any positive integer values.

Calculate the Three Lowest Energies

The three lowest energy levels occur when the sum of the squares of the quantum numbers \(n_x, n_y,\) and \(n_z\) are at their smallest: (1^2+1^2+1^2), (1^2+1^2+2^2), (1^2+2^2+2^2). Substituting these into the energy equation will give the three lowest different energy levels.

Determine the Number of States For Each Energy Level

The number of states corresponding to each energy level depends on the possible combinations of quantum numbers \(n_x, n_y,\) and \(n_z\) that satisfy the sum of their squares. So, for energy level with (1^2+1^2+1^2), there is only 1 way to arrange these numbers (all 1s). For energy level with (1^2+1^2+2^2), there are 3 ways to arrange these numbers. And for energy level with (1^2+2^2+2^2), there are 3 ways to arrange these numbers.

Key concepts

3D Infinite Potential Well

The concept of a 3D infinite potential well is foundational in quantum mechanics. Here, a particle such as an electron is confined within a box-like structure, which it cannot escape. The walls of this box are considered 'infinite' because the potential energy outside this space is infinitely large, making it impossible for the particle to exist beyond its boundaries.
This model is essential because it helps physicists understand how particles behave when restricted to a finite region of space.
In our case, the potential well has a cubic shape, meaning the dimensions along each x, y, and z-axis are equal. Therefore, each side of the cube is denoted by a length L. For the infinite potential well set at 1 nm, the electron's motion is quantized, meaning it can only have specific discrete energy levels, unlike in free space where a particle's energy levels are continuous.

Energy Levels

In a 3D infinite potential well, the energy levels are defined by a specific equation that outlines their quantization. These energy levels are determined by quantum numbers and the physical dimensions of the potential well.
The formula to calculate a particle's energy level inside this well is:

• \(E=\frac{h^2(n_x^2+n_y^2+n_z^2)}{8mL^2}\)

Here, \(h\) is Planck's constant which is a fundamental constant in quantum mechanics, \(m\) is the particle's mass, and \(L\) is the cube's side length.
The quantum numbers \(n_x, n_y, n_z\) define the state of the particle. Each combination of these numbers corresponds to a distinct energy level. Lower combinations of these numbers lead to lower energy levels and vice versa.
This process of calculating energy levels helps in understanding phenomena at the atomic scale, such as electron behavior inside atoms or molecules.

Quantum Numbers

Quantum numbers are sets of values that describe the state of a quantum mechanical system. In the context of a 3D infinite potential well, they are valuable for determining the allowed energy levels of a particle like an electron.
Each quantum number (\(n_x, n_y, n_z\)) can be any positive integer, reflecting the quantized nature of quantum systems. They represent the quantization of motion along the x, y, and z-directions. By using these, we can calculate electron energy levels within the potential well.
For example, the smallest values for these numbers are 1, meaning the lowest energy level is found when \(n_x = 1, n_y = 1,\) and \(n_z = 1\).
These numbers not only define the energy of the electron but also the states it can occupy within the confines of the potential well. Hence, when discussing the different possible states for a given energy level, it refers to how these quantum numbers can vary while still providing the same energy result.