# The simplest 1D Superpotential Hamiltonian derivation

+ 3 like - 0 dislike
120 views

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

and then somehow the following Hamiltonian is defined and derived: $$H=\{Q_1,Q_1\}=\{Q_2,Q_2\}=\frac{p^2}{2}+\frac{W^2}{2}+\frac{W'}{2}(bb^\dagger-b^\dagger b)$$

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

Why does $=\{Q_1,Q_1\}=\{Q_2,Q_2\}=\frac{p^2}{2}+\frac{W^2}{2}+\frac{W'}{2}(bb^\dagger-b^\dagger b)$? I can't see how they got this expression.

What's the motivation to define the superpotential is such away? Also why does $Q_1, Q_2$ map "bosonic" states into "fermionic" states and vice versa? Lastly, why does it take the form: $H = \frac{p^2}{2}+\frac{W^2}{2} \pm \frac{W'}{2}$

This post imported from StackExchange Physics at 2017-11-18 23:19 (UTC), posted by SE-user 0x90

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysics$\varnothing$verflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.