Super-Symmetric Quantum Mechanics

Here is a note on this.

The idea of supersymmetric quantum mechanics is to introduce a hamiltonian related to supercharge, which is defined through

\[[H,Q_i] = 0,\]

for all charges \(Q_i\) and

\[{Q_i,Q_j} = \delta_{ij} H.\]

In the 2 charge case, I can define two charges,

\[\begin{split}Q_1 & = \frac{1}{2} (\sigma_1 p + \sigma_2 W(x) )\\ Q_2 & = \frac{1}{2} (\sigma_2 p - \sigma_1 W(x) ).\end{split}\]

Harmonic Oscillators

The harmonic oscillators can be solved using ladder operators,

\[\begin{split}a &= \sqrt{i/2\hbar}(\hat p/\sqrt{m\omega} -i\sqrt{m\omega}\hat x) \\ a^\dagger & = \sqrt{-i/2\hbar}(\hat p/\sqrt{m\omega} + i\sqrt{m\omega}\hat x).\end{split}\]

This is a hint why we define the charges in that way.

With these charges, we can solve the state that is annihlated by \(Q_1\).

\[Q\psi_0 = 0,\]

which is the ground state.

The result is

\[\psi_0(x) = exp(\int_0^x dy W(y)\sigma_3/\hbar) \psi_0(0).\]