The quantum harmonic oscillator in the plane

This post investigates some solutions of the (or rather a) quantum harmonic oscillator in the plane. Here the plane is \mathbb{C} which will be identified with \mathbb{R}^2 through z=x+\textrm{i} y. Let \partial_x and \partial_y denote the partial derivative operators with respect to x and y respectively. Then the Laplace operator \nabla^2 is \partial_x^2 + \partial_y^2. The time independent form of the Schrödinger equation for the quantum harmonic oscillator considered here is

\displaystyle (-\tfrac{1}{4} \nabla^2 + |z|^2) \varphi = E \varphi

where \varphi is a complex function and E a real eigenvalue (related to energy in physical terms). The operator -\tfrac{1}{4} \nabla^2 + |z|^2 is called the Hamiltonian and will be denoted by H. Only radial solutions \varphi will be considered. A complex function is called radial when it is of the form

\displaystyle \varphi(z) = f(|z|^2) z^m or \displaystyle \varphi(z) = f(|z|^2) \overline{z}^{\,m}

for some real analytic function f and some m \in \mathbb{N}. Radial functions of the first and second form are said to be of degree m and -m respectively. If \varphi is a radial function of degree m \in \mathbb{Z} and |\omega| = 1 then \varphi(\omega z) = \omega^m \varphi(z).

Working over \mathbb{C} it is convenient to rewrite the Laplace operator as follows. Let \partial = \tfrac{1}{2}(\partial_x - \textrm{i}\partial_y) and \overline{\partial} = \tfrac{1}{2}(\partial_x + \textrm{i} \partial_y). If f is a complex differentiable function then \overline{\partial} f = 0 and \partial f = f' while \overline{\partial}\, \overline{f} = \overline{f'} and \partial \overline{f} = 0. (These are the Cauchy-Riemann equations.) The operators \partial and \overline{\partial} commute and are related to the Laplace operator by

\displaystyle \partial \overline{\partial} = \overline{\partial} \partial = \tfrac{1}{4} \nabla^2.

The Hamiltonian is therefore -\partial \overline{\partial} + |z|^2 and we will use it in this form. For a radial function f(|z|^2) z^m

\displaystyle \partial \overline{\partial} f(|z|^2) z^m = \partial f'(|z|^2) z^{m+1} = |z|^2 f''(|z|^2) z^m + (m+1) f'(|z|^2) z^m.

Applying the Hamiltonian results in

H f(|z|^2) z^m = |z|^2 \left(f(|z|^2) - f''(|z|^2)\right) z^m - (m+1) f'(|z|^2) z^m

and in particular if f(x) = e^{-x} then H e^{-|z|^2} z^m = (m+1) e^{-|z|^2} z^m. So for each degree m \geq 0 the function e^{-|z|^2} z^m is an eigenfunction of H with eigenvalue E = m + 1. Complex conjugation shows that the same holds for e^{-|z|^2} \overline{z}^{\,m} of degree -m. For each degree m \in \mathbb{Z} we found a radial solution for the harmonic oscillator of degree m and eigenvalue |m| + 1. These examples do not exhaust all such solutions however. Others can be found by a clever trick that was already known in this context by Schrödinger and Dirac.

This trick is known as the factorisation or algebraic method and the auxiliary operators that appear are called ladder operators or annihilation and creation operators. The following facts can be readily verified:

  1. The operators -\partial+\overline{z} and \partial + \overline{z} lower the degree of a radial function by 1.
  2. The operators -\overline{\partial} + z and \overline{\partial} + z raise the degree of a radial function by 1.
  3. If \varphi is a radial function of degree m \in \mathbb{Z} then (z \partial - \overline{z} \, \overline{\partial}) \varphi = m \varphi and

    \displaystyle (-\partial + \overline{z})(\overline{\partial} + z) \varphi = (H - 1 - m) \varphi
    \displaystyle (\overline{\partial} + z)(-\partial + \overline{z}) \varphi = (H + 1 - m) \varphi
    \displaystyle (-\overline{\partial} + z)(\partial + \overline{z}) \varphi = (H - 1 + m) \varphi
    \displaystyle (\partial + \overline{z})(-\overline{\partial} + z) \varphi = (H + 1 + m) \varphi

Combining these observations we find for a radial function \varphi of degree m:

\displaystyle \begin{matrix}  H(-\partial + \overline{z})\varphi &=& ((-\partial + \overline{z})(\overline{\partial} + z) + m)(-\partial + \overline{z})\varphi\\   &=& (-\partial + \overline{z})((\overline{\partial} + z)(-\partial + \overline{z}) + m)\varphi\\   &=&(-\partial + \overline{z})(H + 1)\varphi.  \end{matrix}

In particular if \varphi \neq 0 is a solution with eigenvalue E then (-\partial + \overline{z})\varphi is a solution of degree m-1 and eigenvalue E+1. In this case (\overline{\partial} + z)(-\partial + \overline{z})\varphi = (E + 1 - m) \varphi and so (-\partial + \overline{z})\varphi \neq 0 if E \neq m - 1. Under the assumption that E \geq |m| + 1 we can assemble the following table of solutions based on \varphi:

\displaystyle \begin{matrix}  \textrm{solution} & \textrm{degree} & \textrm{eigenvalue} & \textrm{remark}\\[1ex]  (-\partial + \overline{z})\varphi & m - 1 & E + 1 & \neq 0\\  (-\overline{\partial} + z)\varphi & m + 1 & E + 1 & \neq 0\\  (\partial + \overline{z})\varphi & m - 1 & E - 1 & \neq 0 \textrm{ unless } E = 1 - m\\  (\overline{\partial} + z)\varphi & m + 1 & E - 1 & \neq 0 \textrm{ unless } E = 1 + m  \end{matrix}

The first two operators in this table are called raising or creation operators: they raise the eigenvalue or create energy. The last two are called lowering or annihilation operators for similar reasons. Starting from the solution \varphi(z) = e^{-|z|^2} of degree m=0 and eigenvalue E=1 and repeatedly applying the operators -\partial + \overline{z} and -\overline{\partial} + z we find non-zero solutions for all pairs (m, E) for which m + E is odd and E \geq |m|+1. This process results in the solutions below up to scalar multiples. Solutions for negative degrees can be obtained by complex conjugation.

\displaystyle \begin{matrix}  m & E & \textrm{solution}\\[1ex]  0 & 1 & e^{-|z|^2} \\  1 & 2 & z \, e^{-|z|^2}\\  0 & 3 & (1 - 2|z|^2) \, e^{-|z|^2}\\  2 & 3 & z^2 \, e^{-|z|^2}\\  1 & 4 & z (1 - |z|^2) \, e^{-|z|^2}\\  3 & 4 & z^3 \, e^{-|z|^2}  \end{matrix}

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