# ‘Supersymmetrizing’ an arbitrary quantum-mechanical potential

To my understanding, it is not possible to $$\text{supersymmetrize}"$$ an arbitrary quantum-mechanical system unless one knows how to represent the corresponding Hamiltonian in the form $$H = A^\dagger A \quad,$$ which can be as difficult as solving the Schrödinger equation. (I guess, in QM we can generate SUSY potentials by playing with arbitrary superpotentials $$W(x)$$.)

So, I realize that my question is not very rigorous. Anyways,

Is there ANY way to construct a supersymmetric Hamiltonian $$H_S$$ from a given one, $$H$$? (without the necessity to solve Schrödinger equation for $$H$$ or doing some equally hard things)

What I mean here by $$\text{construct}"$$ is not necessarily what people typically mean by this word in SUSY QM (see e.g. here). The only thing I want is SOME SUSY potential whose construction would somehow involve using $$H$$.

P.S.

Note that we know how to $$\text{supersymmetrize}"$$ nearly any field theory. This suggests that a similar thing should be possible in QM.

• What would you mean by "some susy potential"? In the end, you want the Hamiltonian to be an anticommutators of two (somehow) odd-graded operators? In QFT you have real fermions and the boson part of their yukawa coupling is related to the potential through simple relations traceable to the respective Lorentz properties of the fields. A zero-dimensional limit of such QFTs would still possess fermonic field operators. – Cosmas Zachos Feb 2 at 21:43
• I don't need the original Hamiltonian to be represented as such an anti-commutator. I'm looking for SOME Hamiltomian of this form, constructing which would involve using the original one. – mavzolej Feb 7 at 7:38

1. Let there be given a 1D self-adjoint Hamiltonian of the form $$H~=~\frac{p^2}{2m}+V(x), \qquad p~=~\frac{\hbar}{i}\frac{d}{dx}.\tag{1}$$

2. A necessary condition for supersymmetrization (in the sense of Refs. 1-2) is that the self-adjoint Hamiltonian $$H\geq C$$ is bounded from below, where $$C\in\mathbb{R}$$ can be interpreted as a choice of zero-point energy.

3. Case of spontaneously unbroken SUSY: Here we assume that the system has a normalizable ground state $$\psi_0(x)$$ with energy-eigenvalue $$E_0$$. Since $$\psi_0(x)$$ is the ground state, it should have no nodes.

By shifting the energy-level, we may & shall assume that $$E_0=0$$ from now on. We can now choose a superpotential $$W~:=~-\frac{\hbar}{\sqrt{2m}}\frac{d}{dx}\ln|\psi_0|.\tag{2}$$

4. Both cases of spontaneously unbroken & broken SUSY: More generally, we are looking for a global solution $$W$$ to the Riccati equation $$V~=~W^2-\frac{\hbar W^{\prime}}{\sqrt{2m}}, \tag{3}$$ possibly after shifting the the energy-level. It follows that the Hamiltonian is semipositive $$H~\stackrel{(1)+(3)}{=}~A^{\dagger}A, \qquad A~:=~\frac{ip}{\sqrt{2m}}+W(x).\tag{4}$$ [It is easy to see that eq. (2) satisfies the Riccati eq. (3). Conversely, given a $$W$$-solution to the Riccati eq. (3), we can reverse eq. (2) to defined a lowest energy eigen-function $$\psi_0$$, which may not be normalizable. The unnormalizable case corresponds to spontaneously broken SUSY, cf. Refs. 1-2.]

5. The construction of the SUSY QM model proceed in the standard fashion: $$H_{\rm SUSY}~:=~\begin{pmatrix} A^{\dagger}A & 0 \cr 0 & AA^{\dagger} \end{pmatrix}~\stackrel{(6)}{=}~\{Q,Q^{\dagger} \}_+~\stackrel{(4)}{=}~(\frac{p^2}{2m}+W^2)\mathbb{1}_{2\times 2}-\frac{\hbar W^{\prime}}{\sqrt{2m}}\sigma_z ,\tag{5}$$ with supercharge $$Q~:=~ \begin{pmatrix} 0 & 0 \cr A & 0 \end{pmatrix}~=~A\sigma_-,\qquad \sigma_{\pm}:= \frac{\sigma_x\pm i\sigma_y}{2},\qquad Q^2~=~0,\tag{6}$$ cf. Refs. 1-2. Here we have rewritten the $$2\times2$$ matrices in terms of Pauli matrices.

6. Alternatively, we may replace the $$2\times 2$$ Pauli matrices with a Grassmann-odd/fermionic creation & annilation-operator $$H_{\rm SUSY}~\stackrel{(5)+(8)}{\leftrightarrow}~(\frac{p^2}{2m}+W^2)\mathbb{1}-\frac{\hbar W^{\prime}}{\sqrt{2m}}[b^{\dagger},b]_-, \qquad Q ~\stackrel{(6)+(8)}{\leftrightarrow}~Ab,\tag{7}$$ under the identification $$\sigma_-~\leftrightarrow~b, \qquad \sigma_+~\leftrightarrow~b^{\dagger}, \qquad \sigma_z~\leftrightarrow~[b^{\dagger},b]_-, \qquad \mathbb{1}_{2\times 2}~\leftrightarrow~\mathbb{1}~=~\{b,b^{\dagger}\}_+ .\tag{8}$$

References:

1. F. Cooper, A. Khare, & U. Sukhatme, Supersymmetry and Quantum Mechanics, Phys. Rept. 251 (1995) 267, arXiv:hep-th/9405029; Chapter 2.

2. P. Uttayarat, Supersymmetrizing a QM System, J. Phys.: Conf. Ser. 901 (2017) 012052.

• The problem with this approach is that constructing a supersymmetric model is as complicated as solving the Schrödinger equation (we need $\psi_0$!). I was looking for some alternative way of doing this, probably more in the QFT spirit. – mavzolej Feb 2 at 18:22
• Well, it is still interesting that the SUSY algebra exists in principle. – Qmechanic Feb 2 at 20:45