A two-dimensional harmonic oscillator has energy E= ħω0(nx+ny+1), where nx and ny are integers b

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A two-dimensional harmonic oscillator has energy E=...

A two-dimensional harmonic oscillator has energy $E=$ $\hbar \omega_{0}\left(n_{x}+n_{y}+1\right),$ where $n_{x}$ and $n_{y}$ are integers beginning with zero. ( $a$ ) Justify this result based on the energy of the one-dimensional oscillator. (b) Sketch an energy-level diagram similar to Figure 5.21 , showing the lowest 4 energy levels. For each level, show the value of $E$ (in units of $\hbar \omega_{0}$ ), the quantum numbers $n_{x}$ and $n_{y},$ and the degeneracy. (c) Show that the degeneracy of each level is equal to $n_{x}+n_{y}+1$

Key Concepts

Energy Quantization in Multi-Dimensional Systems

When systems are separable, the total energy is the sum of independently quantized contributions from each degree of freedom. This leads to a quantization of energy that reflects combinations of the quantum numbers associated with each coordinate, thereby establishing the foundational structure for the energy spectrum in higher dimensions.

Degeneracy in Multi-Dimensional Quantum Systems

Degeneracy occurs when different quantum states share the same energy level. In systems like the two-dimensional harmonic oscillator, various combinations of quantum numbers yield the same total energy. The degeneracy can be determined by counting all the possible combinations of the individual quantum numbers that add up to a particular sum, illustrating an important concept in the study of quantum mechanics.

One-Dimensional Quantum Harmonic Oscillator

This is a fundamental quantum system where a particle moves in a quadratic potential. Its energy levels are quantized as E(n) = ??(n + 1/2), with n being a non-negative integer. The equally spaced energy levels (separated by ??) form the basis for constructing more complex multi-dimensional oscillator systems.

Separation of Variables

In the context of multi-dimensional quantum systems, separation of variables is a method that allows the total Hamiltonian to be split into independent parts, each depending on one coordinate. For a two-dimensional harmonic oscillator with a separable potential, the overall energy becomes a sum of the energies from the one-dimensional oscillators along each independent axis.

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