An electron is confined to move in the x y plane in a...

An electron is confined to move in the $x y$ plane in a rectangle whose dimensions are $L_{x}$ and $L_{y}$ . That is, the electron is trapped in a two-dimensional potential well having lengths of $L_{x}$ and $L_{y}$ . In this situation, the allowed energies of the electron depend on two quantum numbers $n_{x}$ and $n_{y}$ and are given by $$ E=\frac{h^{2}}{8 m}\left(\frac{n_{x}^{2}}{L_{x}^{2}}+\frac{n_{y}^{2}}{L_{y}^{2}}\right) $$ (a) Assuming $I_{x}=I_{y}=L,$ find the energies of the lowest four energy levels for the electron. (b) Construct an energy-level diagram for the electron and determine the energy difference between the second excited state and the ground state.

An electron is confined to move in the $x y$ plane in a rectangle whose dimensions are $L_{x}$ and $L_{y}$ . That is, the electron is trapped in a two-dimensional potential well having lengths of $L_{x}$ and $L_{y}$ . In this situation, the allowed energies of the electron depend on two quantum numbers $n_{x}$ and $n_{y}$ and are given by
$$
E=\frac{h^{2}}{8 m}\left(\frac{n_{x}^{2}}{L_{x}^{2}}+\frac{n_{y}^{2}}{L_{y}^{2}}\right)
$$
(a) Assuming $I_{x}=I_{y}=L,$ find the energies of the lowest four energy levels for the electron. (b) Construct an energy-level diagram for the electron and determine the energy difference between the second excited state and the ground state.

Key Concepts

Energy-Level Diagrams and Degeneracy

An energy-level diagram visually represents the allowed energy states of a quantum system. In symmetric systems like a square two-dimensional well, different combinations of quantum numbers can lead to the same energy, a phenomenon known as degeneracy. Understanding energy-level diagrams helps in interpreting the transitions between states, energy differences, and the effects of system symmetry on the spectral properties.

Energy Quantization in Confined Systems

Energy quantization is the phenomenon where only discrete energy values are allowed for a particle in a confined region, resulting from the imposition of boundary conditions on its wavefunction. In a two-dimensional well, the energy expression depends on the square of the quantum numbers, leading to distinct energy levels that are separated by specific energy differences. This quantization explains the discrete spectral lines observed in various quantum systems.

Two-Dimensional Infinite Potential Well

The two-dimensional infinite potential well is a model system in quantum mechanics where a particle, such as an electron, is confined within a perfectly rigid boundary in two spatial dimensions. The potential inside the well is zero while it is infinitely high outside, ensuring the particle remains inside. This model provides insight into how spatial confinement in different dimensions alters the energy spectrum of the system.

Quantum Confinement

Quantum confinement refers to the restriction of a particle’s motion to a very small region of space, which leads to quantized energy levels. In confined systems like potential wells, the particle cannot possess an arbitrary energy but instead only specific allowed values. This concept is fundamental in understanding how materials and nanoscale structures exhibit unique electronic properties.

Quantum Numbers

Quantum numbers are integral values that arise from applying boundary conditions to the wavefunction solutions in a confined system. In a two-dimensional potential well, two independent quantum numbers (commonly denoted as n_x and n_y) uniquely determine the state of the particle. They are essential in labeling energy eigenstates and corresponding probability distributions.

Transcript

00:01 In this question we are given a general formula for the energy of an electron that is trapped in a two -dimensional particle, sorry two -dimensional infinite square well.

00:15 So right now we are considering the case when the length of the box in both the x and y direction are the same.

00:25 So lx is equals to ly, equals to a general l, which we'll replace all of them with.

00:36 So what we get over here is just 8 over atm l square, nx square plus ny square.

00:46 So now it is the energy is actually controlled by two quantum numbers, nx and ny.

00:56 We ought to find what is the energy for the lowest four energy levels.

01:04 And all of this is actually controlled by nx and y.

01:09 So we start off with the lowest possible, nx plus n y square, nx squared plus n y square that is when nx equals to 1 which is also the same for n y.

01:27 So the ground state is most likely given as 1 square plus 1 square which is just 2 so that's h over 4 m l square.

01:47 Now moving all up one energy level.

01:51 Well the possible case is where we have nx increased by one, so now nx is 2, while ny remains the same, ny equals to 1, or on the other hand, x can remain the same, while ny increases to 2.

02:05 Both will give the same result of nx square plus ny square, to give us 2 square plus 1 square, which is just 5.

02:20 E2, you can first 5 over 4 times h over ml square.

02:29 Sorry, not 4.

02:31 So this is 8.

02:38 Now for the third excited state, there's a few possibilities we can consider.

02:44 You can consider either nx further increasing up to 3, and my remaining at 1, or we can have both of them at 2.

03:02 So for this case, this will be 4 plus 4, so nx squared plus n y square will give us 8...

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