What Does Isotropy Mean in a Two-Dimensional Harmonic Oscillator?

[cks]

Find the find the eigenfunctions and eigenvalues of a two-dimensional isotropic harmonic oscillator.

I don't understand what does isotropic here mean.

isotropic can be defined "not changing" when the coordinate change to any other position. Am I correct?

Like mass, pressure, magnitude...

but how to imagine a two-dimensional isotropi harmonic oscillator? IF you were me, how would you imagine it?

 

[HallsofIvy]

"isotropic" does not mean "not changing". It means "identical in all directions". Here it just means that the "spring constant" is the same in both directions. You can just ignore it- most harmonic oscillator problems assume "isotropic" without saying it.

 

[sir-pinski]

An isotropic harmonic oscillator is a physical system that exhibits the same behavior in all directions. In other words, it is not influenced by the direction in which it is observed or measured. This can be seen in systems such as an ideal gas or a perfectly spherical object.

In the case of a two-dimensional isotropic harmonic oscillator, we can imagine a system in which a particle is moving in a circular motion, with the same frequency and amplitude in all directions. This can be represented mathematically by the equation:

H = (1/2m)(p1^2 + p2^2) + (1/2)mω^2(x1^2 + x2^2)

Where H is the Hamiltonian operator, m is the mass of the particle, p1 and p2 are the momenta in the two dimensions, ω is the frequency of the oscillator, and x1 and x2 are the position coordinates in the two dimensions.

To find the eigenfunctions and eigenvalues of this system, we can solve the Schrödinger equation:

Hψ = Eψ

where ψ is the wave function and E is the energy. This equation can be solved using separation of variables, resulting in two independent equations for each dimension:

(1/2m)(p1^2 + p2^2)ψ = (1/2)mω^2(x1^2 + x2^2)ψ + Eψ

Solving these equations will give us the eigenfunctions and eigenvalues of the system, which will depend on the frequency of the oscillator and the mass of the particle. These eigenfunctions will describe the probability distribution of the particle in the two dimensions, and the eigenvalues will determine the allowed energy levels of the system.

In summary, an isotropic harmonic oscillator is a physical system that is not influenced by the direction in which it is observed, and a two-dimensional isotropic harmonic oscillator can be represented by a particle moving in a circular motion with equal frequency and amplitude in all directions. The eigenfunctions and eigenvalues of this system can be found by solving the Schrödinger equation and will depend on the frequency and mass of the oscillator.