33.2 Supersymmetric quantum mechanics with a superpotential

33.2 Supersymmetric quantum mechanics with a superpotential

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · symmetry group

The supersymmetric oscillator system can be generalized to a much wider class of potentials, while still preserving the supersymmetry of the system. For simplicity, we will here choose constants $\hslash =\omega =1$ . Recall that our bosonic annihilation and creation operators were defined by

$$a_B=\frac{1}{\sqrt{2}}(Q+iP),a_B^{†}=\frac{1}{\sqrt{2}}(Q-iP)$$

Introducing an arbitrary function $W(q)$ (called the “superpotential”) with derivative $W^{\prime }(q)$ we can define new annihilation and creation operators:

$$a_B=\frac{1}{\sqrt{2}}(W^{\prime }(Q)+iP),a_B^{†}=\frac{1}{\sqrt{2}}(W^{\prime }(Q)-iP)$$

Here $W^{\prime }(Q)$ is the multiplication operator $W^{\prime }(q)$ in the Schr¨odinger position space representation on functions of $q.$ . The harmonic oscillator is the special case

$$W(q)=\frac{q^2}{2}$$

We keep our definition of the operators

$$Q_{+}=a_B{a}_F^{†},Q_{-}=a_B^{†}{a}_F$$

These satisfy

$$Q_{+}^2=Q_{-}^2=0$$

for the same reason as in the oscillator case: repeated factors of $a_F$ or $a_F^{†}$ vanish. Taking as the Hamiltonian the same square as before, we find

$$\begin{array}{rl}H& =(Q_{+}+Q_{-}{)}^2\\ & =\frac{1}{2}(W^{\prime }(Q)+iP)(W^{\prime }(Q)-iP)a_F^{†}{a}_F+\frac{1}{2}(W^{\prime }(Q)-iP)(W^{\prime }(Q)+iP)a_F{a}_F^{†}\\ & =\frac{1}{2}(W^{\prime }(Q{)}^2+P^2)(a_F^{†}{a}_F+a_F{a}_F^{†})+\frac{1}{2}(i[P,W^{\prime }(Q)])(a_F^{†}{a}_F-a_F{a}_F^{†})\\ & =\frac{1}{2}(W^{\prime }(Q{)}^2+P^2)+\frac{1}{2}(i[P,W^{\prime }(Q)]){\sigma }_3\end{array}$$

But $iP$ is the operator corresponding to infinitesimal translations in $Q,$ , so we have

$$i[P,W^{\prime }(Q)]=W^{\prime \prime }(Q)$$

and

$$H=\frac{1}{2}(W^{\prime }(Q{)}^2+P^2)+\frac{1}{2}{W}^{\prime \prime }(Q){\sigma }_3$$

For diferent choices of $W$ this gives a large class of quantum systems that can be used as toy models to investigate properties of ground states. All have the same state space

$$\mathcal{H}={\mathcal{H}}_B\otimes {\mathcal{F}}_1^{+}=L^2(\mathbf{R})\otimes {\mathbf{C}}^2$$

(using the Schr¨odinger representation for the bosonic factor). The energy eigenvalues will be non-negative, and energy eigenvectors with positive energy will occur in pairs

$$|\psi ⟩,(Q_{+}+Q_{-})|\psi ⟩$$

For any quantum system, an important question is that of whether it has a unique lowest energy state. If the lowest energy state is not unique, and a symmetry group acts non-trivially on the space of lowest energy states, the symmetry is said to be “spontaneously broken”, a situation that will be discussed in section 39.4. In supersymmetric quantum mechanics systems, thinking in terms of Lie superalgebras, one calls $Q_1$ the generator of the action of a supersymmetry, with $H$ invariant under the supersymmetry in the sense that the commutator of $Q_1$ and $H$ is zero. The question of how the supersymmetry acts on the lowest energy state depends on whether or not solutions can be found to the equation

$$(Q_{+}+Q_{-})|0⟩=Q_1|0⟩=0$$

which will be a lowest energy state with zero energy. If such a solution does exist, one describes the ground state |0⟩ as “invariant under the supersymmet ${\mathrm{ry}}^{\mathsf{y}}$ . If no such solution exists, $Q_1$ will take a lowest energy state to another, diferent, lowest energy stat ${}^{\mathrm{e},}$ in which case one says that one has “spontaneously broken supersymmetry”. The question of whether a given supersymmetric theory has its supersymmetry spontaneously broken or not is one that has become of great interest in the case of much more sophisticated supersymmetric quantum field theories. There, hopes (so far unrealized) of making contact with the real world rely on finding theories where the supersymmetry is spontaneously broken.

In this simple quantum mechanical system, one can try and explicitly solve the equation $Q_1|\psi ⟩=0$ . States can be written as two-component complex functions

$$|\psi ⟩=(\genfrac{}{}{0ex}{}{{\psi }_{+}(q)}{{\psi }_{-}(q)})$$

and the equation to be solved is

$$\begin{array}{l}(Q_{+}+Q_{-})|\psi ⟩\\ =\frac{1}{\sqrt{2}}((W^{\prime }(Q)+iP)a_F^{†}+(W^{\prime }(Q)-iP)a_F)\left(\genfrac{}{}{0ex}{}{{\psi }_{+}(q)}{{\psi }_{-}(q)}\right)\\ =\frac{1}{\sqrt{2}}\left(\left(W^{\prime }(Q)+\frac{d}{dq}\right)\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)+\left(W^{\prime }(Q)-\frac{d}{dq}\right)\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)\right)\left(\genfrac{}{}{0ex}{}{{\psi }_{+}(q)}{{\psi }_{-}(q)}\right)\\ =\frac{1}{\sqrt{2}}\left(W^{\prime }(Q)\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)+\frac{d}{dq}\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\right)\left(\genfrac{}{}{0ex}{}{{\psi }_{+}(q)}{{\psi }_{-}(q)}\right)\\ =\frac{1}{\sqrt{2}}\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\frac{d}{dq}-W^{\prime }(Q){\sigma }_3\right)\left(\genfrac{}{}{0ex}{}{{\psi }_{+}(q)}{{\psi }_{-}(q)}\right)=0\end{array}$$

which has general solution

$$\left(\genfrac{}{}{0ex}{}{{\psi }_{+}(q)}{{\psi }_{-}(q)}\right)=e^{W(q){\sigma }_3}\left(\genfrac{}{}{0ex}{}{c_{+}}{c_{-}}\right)=\left(\genfrac{}{}{0ex}{}{c_{+}{e}^{W(q)}}{c_{-}{e}^{-W(q)}}\right)$$

for complex constants $c_{+},c_{-}$ . Such solutions can only be normalizable if

$$c_{+}=0,\underset{q\to \pm \infty }{\lim }W(q)=+\infty$$

or

$$c_{-}=0,\underset{q\to \pm \infty }{\lim }W(q)=-\infty$$

If, for example, $W(q)$ is an odd polynomial, one will not be able to satisfy either of these conditions, so there will be no solution, and the supersymmetry will be spontaneously broken.

Previous section: 33.1 The supersymmetric oscillator · Next section: 33.3 Supersymmetric quantum mechanics and diferential forms · 中文版本

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 351、352、353、354、355、356、357

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：4bffdfd7f61d0f27874e4d55387956022c3478d2484e219ae76652492608b208

OCR 产物 SHA-256：4bffdfd7f61d0f27874e4d55387956022c3478d2484e219ae76652492608b208