This post investigates some solutions of the (or rather a) quantum harmonic oscillator in the plane. Here the plane is which will be identified with through . Let and denote the partial derivative operators with respect to and respectively. Then the Laplace operator is . The time independent form of the Schrödinger equation for the quantum harmonic oscillator considered here is
where is a complex function and a real eigenvalue (related to energy in physical terms). The operator is called the Hamiltonian and will be denoted by . Only radial solutions will be considered. A complex function is called radial when it is of the form
for some real analytic function and some . Radial functions of the first and second form are said to be of degree and respectively. If is a radial function of degree and then .
Working over it is convenient to rewrite the Laplace operator as follows. Let and . If is a complex differentiable function then and while and . (These are the Cauchy-Riemann equations.) The operators and commute and are related to the Laplace operator by
The Hamiltonian is therefore and we will use it in this form. For a radial function
Applying the Hamiltonian results in
and in particular if then . So for each degree the function is an eigenfunction of with eigenvalue . Complex conjugation shows that the same holds for of degree . For each degree we found a radial solution for the harmonic oscillator of degree and eigenvalue . 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:
- The operators and lower the degree of a radial function by .
- The operators and raise the degree of a radial function by .
- If is a radial function of degree then and
Combining these observations we find for a radial function of degree :
In particular if is a solution with eigenvalue then is a solution of degree and eigenvalue . In this case and so if . Under the assumption that we can assemble the following table of solutions based on :
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 of degree and eigenvalue and repeatedly applying the operators and we find non-zero solutions for all pairs for which is odd and . This process results in the solutions below up to scalar multiples. Solutions for negative degrees can be obtained by complex conjugation.