Super-Symmetric Quantum Mechanics
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`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).