Problem 6 A particle of mass $m$ moves i... [FREE SOLUTION] (2024)

Chapter 6: Problem 6

A particle of mass $m$ moves in the three-dimensional potential $$ V(x, y, z)=\left\\{\begin{array}{ll} \frac{1}{2} m \omega^{2} z^{2} & \text { for } 00\\\ +\infty, & \text { elsewhere. } \end{array}\right. $$ (a) Write down the time-independent Schrödinger equation for this particle andreduce it to a set of familiar one-dimensional equations; then find thenormalized wave function $\psi_{n_{x} n_{v} n_{z}}(x, y, z)$ (b) Find the allowed eigenenergies of this particle and show that they can bewritten as: $E_{n_{x} n_{y} n_{z}}=E_{n_{x} n_{y}}+E_{n_{z}}$ (c) Find the four lowest energy levels in the $x y$ plane (i.e.,$\left.E_{n_{x} n_{y}}\right)$ and their corresponding degeneracies.

Short Answer

Expert verified

The normalized wave function is $\psi_{n_x n_y n_z}(x, y, z) = X(x)Y(y)Z(z)$. The allowed eigenenergies are $E_{n_x n_y n_z}=E_{n_x}+E_{n_y}+E_{n_z}$. The four lowest energy levels in the XY plane exhibit degeneracies: the first and third levels are non-degenerate, and the second level is doubly degenerate due to energy states $(1, 2)$ and $(2, 1)$ having the same energy.

Step by step solution

Writing the Time-Independent Schrödinger Equation

For a particle of mass $m$ in a three-dimensional potential, the time-independent Schrödinger equation can be written as: $$-\frac{\hbar^2}{2m}abla^2\psi(x,y,z) + V(x,y,z)\psi(x,y,z) = E\psi(x,y,z)$$Since the potential $V$ is an infinite square well in the $x$ and $y$ directions, and a harmonic oscillator in the $z$ direction, the equation can by symmetry be separated into three one-dimensional equations for each spatial coordinate.

Separating Variables

We can look for solutions of the form $\psi(x,y,z) = X(x)Y(y)Z(z)$. Then we substitute into the Schrödinger equation and separate variables, giving us three independent equations, one for each coordinate:$$-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}X(x) = E_x X(x),$$$$-\frac{\hbar^2}{2m}\frac{d^2}{dy^2}Y(y) = E_y Y(y),$$$$-\frac{\hbar^2}{2m}\frac{d^2}{dz^2}Z(z) + \frac{1}{2}m\omega^2z^2Z(z) = E_z Z(z).$$Each of these is a one-dimensional Schrödinger equation for a particle in a box (first two) and a harmonic oscillator (third).

Solving the One-Dimensional Equations and Normalization

We now solve these one-dimensional equations. For the infinite square well potentials $X(x)$ and $Y(y)$, the boundary conditions dictate that the wave functions must vanish at the walls, and we can use the standard solutions for the particle in a box:$$X(x) = \sqrt{\frac{2}{a}}\sin\left(\frac{n_x \pi x}{a}\right), Y(y) = \sqrt{\frac{2}{a}}\sin\left(\frac{n_y \pi y}{a}\right).$$For the harmonic oscillator potential in the $z$-direction, $Z(z)$, the solutions are Hermite polynomials multiplied by a Gaussian:$$Z(z) = A_n z^ne^{-\frac{m\omega}{2\hbar}z^2},$$where $A_n$ is a normalization constant. The full wave function will then be:$$\psi_{n_x n_y n_z}(x, y, z) = X(x)Y(y)Z(z)$$

Finding the Allowed Eigenenergies

For a particle in a one-dimensional box of width $a$, the energy eigenvalues are given by$$E_{n_x} = \frac{n_x^2 \pi^2 \hbar^2}{2m a^2},E_{n_y} = \frac{n_y^2 \pi^2 \hbar^2}{2m a^2}.$$For a quantum harmonic oscillator, the energy eigenvalues are$$E_{n_z} = \left(n_z + \frac{1}{2}\right)\hbar\omega.$$By the superposition principle of energy in quantum mechanics, the total energy can be written as the sum of the individual energies for each direction. Therefore, the total energy is$$E_{n_x n_y n_z}=E_{n_x}+E_{n_y}+E_{n_z}.$$

Finding the Four Lowest Energy Levels in the XY Plane

The two-dimensional box energies for the XY plane are the sum of the one-dimensional energies, $E_{n_x}+E_{n_y}$. We shall calculate for the lowest four states, which correspond to $n_x, n_y$ equal to $(1, 1), (1, 2), (2, 1),$ and $(2, 2)$. The corresponding energies are:$$E_{1,1}, E_{1,2} = E_{2,1}, E_{2,2}.$$Notice that $E_{1,2}$ and $E_{2,1}$ are degenerate, meaning they have the same energy. To find the degeneracies, count the number of different states that result in the same energy level.

Degeneracies of the Energy Levels

The lowest energy level $E_{1,1}$ has no degeneracy (non-degenerate). The second energy level has two states that give the same energy $E_{1,2} = E_{2,1}$, so it is doubly degenerate. The third energy level, $E_{2,2}$, is also non-degenerate. No higher energy levels are considered for this particular calculation. Thus, the degeneracies of the four lowest energy levels in the XY plane are as follows: Non-degenerate, doubly degenerate, and non-degenerate for the first, second, and third energy levels respectively.

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Time-Independent Schrödinger Equation

At the very heart of quantum mechanics education is the time-independent Schrödinger equation. It's a critical tool for understanding how quantum systems behave. When dealing with a particle exposed to a potential field, the equation takes the form:
\[\begin{equation}-\frac{\hbar^2}{2m}abla^2\psi(x,y,z) + V(x,y,z)\psi(x,y,z) = E\psi(x,y,z)\end{equation}\]
Here, $ \hbar $ is the reduced Planck's constant, $ m $ is the mass of the particle, $ abla^2 $ is the Laplacian operator representing the sum of second partial derivatives with respect to space coordinates, $ \psi $ is the wave function, $ V $ is the potential energy, and $ E $ is the total energy eigenvalue.
What makes this equation so special is its ability to isolate energy levels and associated wave functions through separation of variables when the potential allows it. This process breaks down a complex 3D problem into manageable 1D equations, each corresponding to a different spatial dimension, much like the one featured in the exercise provided. This simplification makes it possible to tackle quantum physics problems systematically and is foundational for students mastering quantum mechanics.

Quantum Harmonic Oscillator

Another cornerstone concept in quantum mechanics is the quantum harmonic oscillator. This system describes a particle subject to a quadratic potential, typically visualized as a particle in a spring-like force field. For a particle moving in one dimension, the potential energy is given by
\[\begin{equation}V(z) = \frac{1}{2} m \omega^2 z^2\end{equation}\]
where $ m $ is the mass of the particle, $ \omega $ is the angular frequency, and $ z $ is the position. When we insert this potential into the Schrödinger equation, we get a differential equation whose solutions are wave functions involving Hermite polynomials and a Gaussian component. These wave functions are significant in that they represent standing wave patterns of the oscillator, and each wave function corresponds to a distinct energy level, quantized as
\[\begin{equation}E_{n_z} = \left(n_z + \frac{1}{2}\right)\hbar\omega\end{equation}\]
with $ n_z $ being the quantum number associated with the z-direction. Quantum harmonic oscillators are not just theoretical constructs but are essential for understanding molecular vibrations, quantum field theory, and the behavior of photons in a cavity, thus making it imperative for students to grasp this complex, yet fundamental, quantum system.

Quantum Degeneracy

Quantum degeneracy is a phenomenon that occurs when two or more quantum states share the same energy level. For students delving into quantum mechanics, understanding degeneracy is crucial for explaining the physical properties of a wide array of systems. In the context of our exercise, we find that the eigenenergies in the XY plane of a particle in a 2D box can exhibit degeneracy. This happens as some combinations of quantum numbers, which determine the state of the system, lead to the same energy. For example,

\[\begin{equation}E_{1,2} = E_{2,1}\end{equation}\]
illustrates a doubly degenerate state because the energy can result from either a higher quantum number in the x-direction and a lower one in the y-direction, or vice versa. Degeneracy is not just a mathematical quirk; it has physical implications, such as in the spectral lines of atoms and the behavior of electrons in solids. Students of quantum mechanics should appreciate that the concept of degeneracy helps explain why some materials can conduct electricity and others cannot. It is an essential concept that provides insight into the underlying symmetry of the laws of physics.

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Problem 6 A particle of mass \(m\) moves i... [FREE SOLUTION] (2024)