# 2D isotropic quantum harmonic oscillator: polar coordinates

• CharlieCW
In summary, the conversation discusses finding the eigenfunctions and eigenvalues of the isotropic bidimensional harmonic oscillator in polar coordinates. The hamiltonian for this system is given, and the attempt at a solution involves using separation of variables and plugging into the Schrodinger equation. The solution for the angular part is given, but the radial part is not fully developed. A possible approach to solving the radial part is suggested, involving taking care of the limits and using an ansatz to simplify the differential equation. Useful references for this problem are also mentioned.
CharlieCW

## Homework Statement

Find the eigenfunctions and eigenvalues of the isotropic bidimensional harmonic oscillator in polar coordinates.

## Homework Equations

$$H=-\frac{\hbar}{2m}(\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \phi^2})+\frac{m\omega^2}{2m}r^2$$

## The Attempt at a Solution

The hamiltonian of the 2D isotropic harmonic oscillator is:

$$H=-\frac{\hbar}{2m}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})+\frac{m\omega^2}{2}(x^2+y^2)$$

I can easily solve the N-dimensional case in cartesian coordinates as we can separate the hamiltonian in independent oscillators for each coordinate. For the polar case in two dimensions, we can rewrite:

$$H=-\frac{\hbar}{2m}(\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \phi^2})+\frac{m\omega^2}{2m}r^2$$

With ##r^2=x^2+y^2## and ##\phi=arctan(y/x)##.

Using separation of variables ##\psi(r,\phi)=R(r)\Phi(\phi)## and plugging into the Schrodinger equation, we can easily solve for the angular part ##\Phi=e^{im\phi}##, where ##m\in \mathbb{Z}##.

Plugging back into the Schrodinger equation, for the radial part, we get:

$$r^2R''+rR'+(r^2E-m^2-\omega^2r^4)=0$$

While I have an idea for the solution by making an analogy with the 3D case (where we get Laguerre polynomials), I'm not sure how to correctly proceed from here (never saw this in undergraduate). I tried plugging into Wolfram Alpha but I just get a sum of logarithms,

$$R(r)=c_1 log(r)+c_2+\frac{1}{2}m^2 log^2(r)+\frac{\omega^2 r^4}{16}-\frac{e r^2}{4}$$

which is not satisfactory as not only the solution lacks the form of an harmonic oscillator but I also have the impression the solution should be in terms of special functions.

I appreciate any input or even useful references* (all of the references I've found deal with the 3D case, which I have no problem solving since it's just spherical harmonics).*I read somewhere else that this problem is treated in the book of "Wave Mechanics" from Pauli, but unfortunately it isn't available on neither of my campus libraries nor online (it's only available for purchase abroad and I lack the funds to buy it).

Last edited:
The way I would do it is to first take care of the two limits: ##r \rightarrow 0## and ##r \rightarrow \infty##. (By the way, it's a little confusing for you to use "m" to mean both mass and angular momentum quantum number)

So you could try the ansatz: ##R(r) = r^\alpha e^{-\lambda r^2} Q(r)##. Then you can look near ##r \approx 0## to find out what the exponent ##\alpha## must be, and look near ##r \rightarrow \infty## to find out what the constant ##\lambda## must be. Hopefully, then you can get a differential equation for ##Q(r)## that can be solved using polynomials (or something simple).

Keith_McClary
I found this by searches such as this and this.

Keith_McClary said:
I found this by searches such as this and this.

Thanks. That was one the articles I found but it only deals with the calculation of the angular component (p. 15), but the radial part is just given without further development, and I don't know to arrive at that expression without making the analogy to the 3D case.

## 1. What is a 2D isotropic quantum harmonic oscillator in polar coordinates?

A 2D isotropic quantum harmonic oscillator in polar coordinates is a mathematical model used to describe the behavior of a particle in a two-dimensional system that is confined by a central force. It is characterized by a potential energy function that is proportional to the square of the distance from the center and can be solved using the Schrödinger equation.

## 2. How is the potential energy of a 2D isotropic quantum harmonic oscillator calculated in polar coordinates?

The potential energy of a 2D isotropic quantum harmonic oscillator in polar coordinates is calculated using the radial component of the Hamiltonian operator, which is proportional to the square of the radial distance from the center. This potential energy function is also known as the harmonic potential and is represented by a parabola in two dimensions.

## 3. What is the significance of polar coordinates in the 2D isotropic quantum harmonic oscillator?

Polar coordinates are used in the 2D isotropic quantum harmonic oscillator because they provide a more natural representation of the system's symmetry. The potential energy function in polar coordinates is also simpler compared to Cartesian coordinates, making it easier to solve the Schrödinger equation and analyze the behavior of the system.

## 4. How are the energy levels of a 2D isotropic quantum harmonic oscillator in polar coordinates determined?

The energy levels of a 2D isotropic quantum harmonic oscillator in polar coordinates are determined by solving the Schrödinger equation using separation of variables. This results in a set of equations, known as the radial and angular quantum numbers, which determine the possible energy states of the system. The energy levels are quantized and can only take on certain discrete values.

## 5. What are some real-world applications of the 2D isotropic quantum harmonic oscillator in polar coordinates?

The 2D isotropic quantum harmonic oscillator in polar coordinates is a fundamental model used in quantum mechanics to describe the behavior of particles in confined systems. It has applications in various fields, such as solid-state physics, atomic and molecular physics, and quantum optics. It is also used in the development of technologies such as lasers, semiconductors, and superconductors.

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