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 $$E=\frac{h^{2}}{8 m_{e}}\left(\frac{n_{x}^{2}}{L_{x}^{2}}+\frac{n_{y}^{2}}{L_{y}^{2}}\right)$$ Using this information, we wish to find the wavelength of a photon needed to excite the electron from the ground state to the second excited state, assuming $L_{x}=L_{y}=L$. (a) Using the assumption on the lengths, write an expression for the allowed energies of the electron in terms of the quantum numbers $n_{x}$ and $n_{y}$. (b) What values of $n_{x}$ and $n_{y}$ correspond to the ground state? (c) Find the energy of the ground state. (d) What are the possible values of $n_{x}$ and $n_{y}$ for the first excited state, that is, the next-highest state in terms of energy? (e) What are the possible values of $n_{x}$ and $n_{y}$ for the second excited state? (f) Using the values in part (e), what is the energy of the second excited state? (g) What is the energy difference between the ground state and the second excited state? (h) What is the wavelength of a photon that will cause the transition between the ground state and the second excited 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
$$E=\frac{h^{2}}{8 m_{e}}\left(\frac{n_{x}^{2}}{L_{x}^{2}}+\frac{n_{y}^{2}}{L_{y}^{2}}\right)$$
Using this information, we wish to find the wavelength of a photon needed to excite the electron from the ground state to the second excited state, assuming $L_{x}=L_{y}=L$.
(a) Using the assumption on the lengths, write an expression for the allowed energies of the electron in terms of the quantum numbers $n_{x}$ and $n_{y}$. (b) What values of $n_{x}$ and $n_{y}$ correspond to the ground state? (c) Find the energy of the ground state.
(d) What are the possible values of $n_{x}$ and $n_{y}$ for the first excited state, that is, the next-highest state in terms of energy?
(e) What are the possible values of $n_{x}$ and $n_{y}$ for the second excited state?
(f) Using the values in part (e), what is the energy of the second excited state?
(g) What is the energy difference between the ground state and the second excited state?
(h) What is the wavelength of a photon that will cause the transition between the ground state and the second excited state?

Key Concepts

Quantization in a Two-Dimensional Infinite Potential Well

In quantum mechanics, a particle confined within an infinitely deep potential well can only exist at certain discrete energy levels. In a two-dimensional well, such as a rectangular or square box, the energy of the particle depends on its quantum numbers along each spatial dimension. This leads to quantized energy eigenstates where the allowed energies are determined by the squared values of these quantum numbers divided by the square of the respective dimensions of the box.

Quantum Numbers

Quantum numbers are integers that arise due to the boundary conditions imposed on the wavefunction in confined systems. In a 2D potential well, two quantum numbers (commonly denoted n_x and n_y) specify the state of the particle along the x and y dimensions, respectively. They take on positive integer values (1, 2, 3, …) and determine both the shape of the wavefunction and the energy of the associated state.

Energy Quantization Formula

The energy levels for an electron in a 2D rectangular potential well are given by a formula that incorporates the quantum numbers and the dimensions of the well. When the dimensions are equal, the expression simplifies and shows that the energy is a sum of two terms, each proportional to the square of a quantum number divided by the square of the side length. This formula is fundamental in calculating the discrete energy states available to the electron in the system.

Ground State and Excited States

The ground state of a quantum system is the state with the lowest possible energy, typically corresponding to the smallest allowed quantum numbers. Excited states have higher energies and correspond to larger values of the quantum numbers or different combinations thereof. In a two-dimensional well, the energy ordering, including possible degeneracies, depends on the particular combinations of n_x and n_y that yield the next highest energy levels.

Symmetry and Degeneracy

In a 2D potential well with equal side lengths, the system possesses symmetry with respect to the x and y dimensions. This symmetry can lead to degenerate states, where different pairs of quantum numbers produce the same energy. Understanding degeneracy is important when identifying distinct quantum states such as the first and second excited states as they might have multiple configurations that yield identical energy levels.

Photon Transition Energy

When a photon is absorbed or emitted during a transition between two quantized energy states, the energy of the photon directly corresponds to the energy difference between these states. The relationship is given by E = hc/?, where h is Planck's constant, c is the speed of light, and ? is the wavelength of the photon. This concept is crucial in determining the wavelength needed to excite an electron from a lower energy state to a higher one in quantum systems.

Transcript

00:01 For this question, we're looking at an electron being confined to a two -dimensional potential well, rather than a one -dimensional one.

00:11 So this is the expression for its energy, right, for a particular n -x value and particular n -y value.

00:21 So it actually depends on two quantum numbers in this case, n -x and n -y.

00:26 So the first part of this question, we want to write the expression for the energy when lx equals to ly.

00:35 So simplifying this, when lx equals to ly, you can take this out as a factor, right? just call this l square.

00:45 Take this factor out.

00:47 It should be h bar h square.

00:49 Right, it m, l square.

00:53 You get nx square plus ny square, left in the brackets.

00:57 So this expression for the energy.

01:00 Now to find the value of nx and ny for the ground state, right? so ground states, remember that we want the smallest possible energy, right, for the ground state.

01:11 And that is when nx and ny equals to 1.

01:15 So to be when nx and y are both 1.

01:22 So any value that's higher than 1 would give us a larger energy.

01:28 This must be when this is the ground state.

01:33 Now to find the actual energy, we just substitute in.

01:38 Nx and y equals to 1, get 2, so this 2h square, which is actually just equals to square over for m, e, l square...

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