Problem 7 A particle of mass \(m\) moves i... [FREE SOLUTION] (2024)

Expert verified

The concept of a two-dimensional harmonic oscillator extends from the familiar simple harmonic motion we encounter in classical physics. In quantum mechanics, it refers to the behavior of a particle confined in a quadratic potential well that depends on two spatial dimensions.

In mathematical terms, the potential energy for such a system is defined as:
\[ V_1(x, y) = \frac{1}{2} m \omega^2(x^2 + y^2) \].
Here, \( m \) is the particle's mass, and \( \omega \) is the angular frequency of the oscillator. The harmonic oscillator is a fundamental model because it provides a solvable quantum system that can approximate the behavior near the equilibrium points of more complicated systems.

In quantum mechanics, the energy of a two-dimensional harmonic oscillator is quantized, implying that the particle can only occupy certain discrete energy states. The energy levels are given by:
\[ E_{n_x, n_y} = \left(n_x + \frac{1}{2}\right)\hbar\omega + \left(n_y + \frac{1}{2}\right)\hbar\omega \],
where \( \hbar \) is the reduced Planck's constant, and \( n_x \) and \( n_y \) are the quantum numbers associated with the motion in the \( x \) and \( y \) axes, respectively. The corresponding wave functions are products of the one-dimensional oscillator wave functions for each axis.

The particle in a one-dimensional infinite potential box is another cornerstone model of quantum mechanics. It provides a straightforward scenario in which the wave function of a particle is constricted within a region, with the potential outside being infinitely high.

The potential in this model can be described as:
\[ V_2(z) = \begin{cases} 0, & 0 \leq z \leq a \ +\infty, & \text{elsewhere} \end{cases} \].

This condition leads to the particle being confined to a 'box' of width \( a \). The quantization of energy levels arises due to the wave nature of quantum particles, which requires the wave function to have nodes at the boundaries of the box. The distinct energy levels are:
\[ E_n = \frac{n^2\pi^2\hbar^2}{2m a^2} \],
where \( n \) is the principle quantum number and can take positive integer values starting from 1. Each energy state corresponds to a unique wave function given by:
\[ \psi_n(z) = \sqrt{\frac{2}{a}} \sin\left(\frac{n\pi z}{a}\right) \].
These results emerge from applying boundary conditions to the Schrödinger equation, which governs the behavior of quantum systems.

Quantum energy levels are an essential feature of quantum systems. Unlike classical systems where a particle's energy can vary continuously, quantum mechanics stipulates that particles can only have specific, discrete energy values. This is one reason quantum systems are said to be quantized.

The existence of quantum energy levels can be visualized by comparing it to a ladder where a particle can only stand on the rungs and not in between. These levels are determined by solving the Schrödinger equation for the system and are dependent on the boundary conditions and the nature of the potential experienced by the particle.

The specific allowed energy levels for the combined system of a two-dimensional harmonic oscillator and a particle in a 1D infinite potential box are expressed as:
\[ E_{total} = E_{n_x, n_y} + E_n \],
where \( E_{total} \) represents the total energy, \( E_{n_x, n_y} \) comes from the 2D harmonic oscillator, and \( E_n \) from the particle in a 1D box. Each of these energy levels correspond to permissible states the particle can occupy, with transitions between levels typically involving the absorption or emission of energy, such as a photon.

Wave functions in quantum mechanics are fundamental descriptions of the state of a quantum system. They are complex functions that contain all the information about a system's properties, such as position and momentum. The probability density associated with the position of a particle is given by the square of the modulus of the wave function.

For the 2D harmonic oscillator, the wave function is separable into the product of two one-dimensional oscillator wave functions. Similarly, for the particle in a one-dimensional box, the wave function is a sine function that conforms to the boundary conditions of the problem.

Upon combining the state functions for each independent scenario, the complete wave function for a particle subject to both potentials, when applicable, is:
\[ \Psi_{n_x, n_y, n}(x, y, z) = \psi_{n_x}(x)\psi_{n_y}(y)\psi_n(z) \].

This product of wave functions ensures that the combined wave function is still a solution to the Schrödinger equation for the overall system. The wave function's squared modulus represents the probability distribution to find the particle in space.

Quantum degeneracy occurs when two or more quantum states share the same energy level. In classical physics, each state corresponds to a unique energy value, but quantum mechanics allows for the possibility of these states 'piling up' to form degenerate states.

For the two-dimensional harmonic oscillator, the degeneracy is determined by the number of ways the quantum numbers \( n_x \) and \( n_y \) can sum up to a particular value of \( n = n_x + n_y \), where \( n \) is the principal quantum number. When considering only this oscillator, the degeneracy of the nth energy level, denoted as \( g_n \), reflects the number of unique state combinations for a given energy and is given by:
\[ g_n = n + 1 \],
where the lowest energy state \( n = 0 \) is non-degenerate, while higher energy levels exhibit higher degrees of degeneracy. This combinatorial aspect of quantum mechanics is crucial for understanding the filling of energy states in systems such as atoms and solid-state crystals which govern much of their electronic properties and behavior.