A particle of mass m moves in the potential V(x, y)=(1)/(2) m ω^2(x^2+. .y^2), where ωis a const

step 1 In this problem, we have a quantum harmonic oscillator in which its electrostatic potential is e

A particle of mass m moves in the potential V(x, y)=(1)/(2)...

A particle of mass $m$ moves in the potential $V(x, y)=\frac{1}{2} m \omega^{2}\left(x^{2}+\right.$ $\left.y^{2}\right)$, where $\omega$ is a constant. Show that the Hamiltonian can be written as the sum $H_{x}+H_{y}$ of the Hamiltonians of two identical one-dimensional harmonic oscillators. Write down the particle's energy spectrum. Write down kets for two stationary states in the first-excited level in terms of the stationary states $\left|n_{x}\right\rangle$ of $H_{x}$ and $\left|n_{y}\right\rangle$ of $H_{y} .$ Show that the $n^{\text {th }}$ excited level is $n+1$ fold degenerate. The oscillator is disturbed by a small potential $H_{1}=\lambda x y .$ Show that this perturbation lifts the degeneracy of the first excited level, producing states with energies $2 \hbar \omega \pm \lambda \hbar / 2 m \omega$. Give expressions for the corresponding kets. The mirror operator $M$ is defined such that $\langle x, y|M| \psi\rangle=\langle y, x \mid \psi\rangle$ for any state $|\psi\rangle$. Explain physically the relationship between the states $|\psi\rangle$ and $M|\psi\rangle .$ Show that $\left[M, H_{1}\right]=0 .$ Show that $M H_{x}=H_{y} M$ and thus that $[M, H]=0 .$ What do you infer from these commutation relations?

A particle of mass $m$ moves in the potential $V(x, y)=\frac{1}{2} m \omega^{2}\left(x^{2}+\right.$ $\left.y^{2}\right)$, where $\omega$ is a constant. Show that the Hamiltonian can be written as the sum $H_{x}+H_{y}$ of the Hamiltonians of two identical one-dimensional harmonic oscillators. Write down the particle's energy spectrum. Write down kets for two stationary states in the first-excited level in terms of the stationary states $\left|n_{x}\right\rangle$ of $H_{x}$ and $\left|n_{y}\right\rangle$ of $H_{y} .$ Show that the $n^{\text {th }}$ excited level is $n+1$ fold degenerate.

The oscillator is disturbed by a small potential $H_{1}=\lambda x y .$ Show that this perturbation lifts the degeneracy of the first excited level, producing states with energies $2 \hbar \omega \pm \lambda \hbar / 2 m \omega$. Give expressions for the corresponding kets.
The mirror operator $M$ is defined such that $\langle x, y|M| \psi\rangle=\langle y, x \mid \psi\rangle$ for any state $|\psi\rangle$. Explain physically the relationship between the states $|\psi\rangle$ and $M|\psi\rangle .$ Show that $\left[M, H_{1}\right]=0 .$ Show that $M H_{x}=H_{y} M$ and thus that $[M, H]=0 .$ What do you infer from these commutation relations?

Key Concepts

Symmetry Operators and Commutation Relations

Symmetry operators, such as a mirror or exchange operator, play a crucial role in quantum mechanics by reflecting underlying geometric or physical symmetries. When such an operator commutes with the Hamiltonian, it indicates that there is a conserved quantity, and the Hamiltonian and the symmetry operator can be simultaneously diagonalized. This leads to further constraints on the eigenstates and elucidates their transformation properties under symmetry operations.

Hamiltonian Decomposition

In many quantum systems, a multidimensional Hamiltonian can often be expressed as a sum of independent one-dimensional Hamiltonians when the potential is separable. This allows one to reduce a complex multidimensional problem to the study of simpler, well?understood one-dimensional systems, with the overall energy being the sum of individual energies.

Quantum Harmonic Oscillator and Energy Quantization

The quantum harmonic oscillator is a fundamental model in quantum mechanics where the energy levels are quantized, with each level corresponding to an integer number of excitations (or quanta). In a two-dimensional oscillator, the total energy is the sum of the energies along each independent coordinate, leading to a spectrum that reflects contributions from two sets of quantized oscillations.

Degeneracy of Energy States

Degeneracy occurs when two or more distinct quantum states have the same energy. In symmetric systems, such as a two-dimensional harmonic oscillator, energy levels can be degenerate because there are multiple ways to distribute the quantum numbers among the degrees of freedom. The presence and degree of degeneracy are closely tied to the symmetries of the Hamiltonian.

Perturbation Theory

Perturbation theory is an approximation method used to analyze how small modifications to a Hamiltonian affect the energy levels and eigenstates of a system. It is especially important for understanding how external influences or additional potentials can lift degeneracies or otherwise alter the spectrum without solving the full, modified Hamiltonian exactly.

Degenerate Perturbation Theory

When a system has degenerate energy levels, standard perturbation theory must be modified to consider the mixing of degenerate states. Degenerate perturbation theory involves diagonalizing the perturbation within the subspace of degenerate states to determine the new energy levels and find the proper linear combinations of the original states, revealing how the degeneracy is partially or completely lifted.

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