2D isotropic quantum harmonic oscillator: polar coordinates

[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).

 

[stevendaryl]

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.

 

[CharlieCW]

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.