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 .. math:: [H,Q_i] = 0, for all charges :math:Q_i and .. math:: {Q_i,Q_j} = \delta_{ij} H. In the 2 charge case, I can define two charges, .. math:: Q_1 & = \frac{1}{2} (\sigma_1 p + \sigma_2 W(x) )\\ Q_2 & = \frac{1}{2} (\sigma_2 p - \sigma_1 W(x) ). .. admonition:: Harmonic Oscillators :class: note The harmonic oscillators can be solved using ladder operators, .. math:: 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). This is a hint why we define the charges in that way. With these charges, we can solve the state that is annihlated by :math:Q_1. .. math:: Q\psi_0 = 0, which is the ground state. The result is .. math:: \psi_0(x) = exp(\int_0^x dy W(y)\sigma_3/\hbar) \psi_0(0).