Supersymmetric Quantum Mechanics (SUSYQM) is a method to solve most solvable potentials commonly seen in quanutum mechanics with relative ease. The idea behind this is similar to solving the harmonic oscilator with ladder operators - to factor the Hamiltonian in a day that makes it easy to compute the energy levels and stationary states.

Superpotentials

Let , the overall goal of SUSYQM is to be able to solve for and in the Schrödinger equation:

Using a similar idea as the harmonic oscilator solution, let's define the operators as $$A^\pm=\mp\frac d{dx}+W(x)$$ where is known as the superpotential. Define the Hamiltonians as

Let's suppose we are given a Hamiltonian with a ground state , i.e. . It turns out with only the ground state information, we can recover and with

Let be the normalizable eigenfunction to with eigenvalue , with . Since by definition is a product of conjugates, it is a semi-positive definite operator, so we also have .

We have the following way to relate both set of solutions:

Hence as long as is renormalizable, it is a eigenfunction of with eigenvalue .

The only case when is not renormalizable is when it vanishes. By construction we have , this yields . If we have , this implies that , which is not renormalizable, hence this is not possible and the only nonrenormalizable case is from . Hence we obtain $$E*{n+1}^-=E_n^+$$

Example

Let's consider the simple case where is a infinite square well given by

We have and . Using the result above, we obtain and .

Using the isospectral relations above, the eigenvalues of is given by and eigenfunctions are

Shape invariance

Let's introduce a parameter, into our Hamiltonians, and where the ground state of has energy and let's suppose that there exists some function such that

If such a function exists, the potentials are shape invariant. We can also see this implies another relationship between the energies, now given by and :

and this also implies that the wave functions and are equal.

Furthermore, this lets us compute with just the function . As an example, we can compute with

More generally, , which is significantly easier than solving the eigenvalue problem.

Example

The solution to the hydrogen atom can be simplified down to solving the Schrödinger equation for , setting as well, we have .

Consider the superpotential , with this, we obtain the following potentials.

And we immediately notice the columb potential inside and also the shape invariance with and . With this, we can easily find the energy levels of the hydrogen atom:

References

• Asim Gangopadhyaya, Constantin Rasinariu, and Jeffry V. Mallow - Supersymmetric Quantum Mechanics: An Introduction
• Fred Cooper, Avinash Khare, Uday Sukhatme - Supersymmetry and Quantum Mechanics