Supersymmetry is a proposed extension of the symmetries of space-time beyond that of just familiar Lorentz transformations and translations. A supersymmetry

transformation interchanges fermions and bosons, which are particles of different spins and correspondingly different statistical properties. While there is no evidence for supersymmetry in Nature, it is nevertheless an interesting proposal that has produced new insights into familiar phenomena.
Supersymmetric quantum mechanics brings some of the ideas of the particle physics supersymmetry to non-relativistic quantum mechanics. \({ }^{5}\) In this problem, we will study features of this formulation of quantum mechanics. The first step is to identify two operators \(A\) and \(A^{\dagger}\) such that\[\begin{equation*}\hat{A}^{\dagger} \hat{A}=\hat{H}-E_{0} \tag{6.125}\end{equation*}\]
where \(\hat{H}\) is the familiar Hamiltonian and \(E_{0}\) is its ground-state energy (i.e., just a number). \(\hat{A}\) and \(\hat{A}^{\dagger}\) are Hermitian conjugates of one another and in general can be expressed as\[\begin{equation*}\hat{A}=\frac{i}{\sqrt{2 m}} \hat{p}+W(\hat{x}), \quad \hat{A}^{\dagger}=-\frac{i}{\sqrt{2 m}} \hat{p}+W(\hat{x}), \tag{6.126}\end{equation*}\]
where \(\hat{p}\) and \(\hat{x}\) are the familiar momentum and position operators. Here, \(W(\hat{x})\) is called the superpotential and is a real function of \(\hat{x}\).
(a) With these relationships, determine the potential \(V(\hat{x})\) in terms of the superpotential \(W(\hat{x})\).
(b) With this definition of \(\hat{A}\) and \(\hat{A}^{\dagger}\), what is their commutator, \(\left[\hat{A}, \hat{A}^{\dagger}\right]\) ?
(c) We can instead consider the anti-commutator \(\left\{\hat{A}, \hat{A}^{\dagger}\right\}\) defined as
\[\begin{equation*}\left\{\hat{A}, \hat{A}^{\dagger}\right\}=\hat{A} \hat{A}^{\dagger}+\hat{A}^{\dagger} \hat{A} \tag{6.127}\end{equation*}\]
What is this?
The anti-commutator is present in the algebra that defines supersymmetry and its nice properties with these definitions is why this is called "supersymmetric quantum mechanics." The algebra of supersymmetry consists of both commutation and anti-commutation relations and is referred to as a "graded Lie algebra." We'll discuss Lie algebras in great detail starting in a few chapters.
(d) Now, connecting back to the harmonic oscillator, what is its superpotential? When we constructed the raising and lowering operators \(\hat{a}^{\dagger}\) and \(\hat{a}\), we just used their commutator \(\left[\hat{a}, \hat{a}^{\dagger}\right]\). Why could we get away with that, and didn't need to use their anti-commutator?
(e) Challenging! What is the superpotential for the infinite square well?