In the superpotential wiki article there are definitions of two supersymmetric operators: $$Q_1=\frac{1}{2}\left[(p-iW)b+(p+iW)b^\dagger\right] \\ Q_2=\frac{i}{2}\left[(p-iW)b-(p+iW)b^\dagger\right] $$

Then the following Hamiltonian is defined and presented:

$$H=\{Q_1,Q_1\}=\{Q_2,Q_2\}=\frac{p^2}{2}+\frac{W^2}{2}+\frac{W'}{2}(bb^\dagger-b^\dagger b)$$

Where $W' = \frac{dW(x)}{dx}$ and $\{b,b^\dagger\}=1$ and $[b,b^\dagger]=0$.

  1. Why does $=\{Q_1,Q_1\}=\{Q_2,Q_2\}=\frac{p^2}{2}+\frac{W^2}{2}+\frac{W'}{2}(bb^\dagger-b^\dagger b)$ hold?

  2. What's the motivation/justification to define the superpotential as such? Also why do $Q_1, Q_2$ map "bosonic" states into "fermionic" states and vice versa?

  3. Lastly, why does it take the form: $H = \frac{p^2}{2}+\frac{W^2}{2} \pm \frac{W'}{2}$?


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