eigenstate spectrum. It is analogous to the original description of SUSY, which referred to bosons and fermions. We can imagine a "bosonic Hamiltonian", whose eigenstates are the various bosons of our theory. The SUSY partner of this Hamiltonian would be "fermionic", and its eigenstates would be the theory's fermions. Each boson would have a fermionic partner of equal energy—but, in the relativistic world, energy and mass are interchangeable, so we can just as easily say that the partner particles have equal mass. SUSY concepts have provided useful extensions to the WKB approximation in the form of a modified version of the Bohr-Sommerfeld quantization condition. In addition, SUSY has been applied to non-quantum statistical mechanics through the Fokker–Planck equation, showing that even if the original inspiration in high-energy particle physics turns out to be a blind alley, its investigation has brought about many useful benefits. Example: the harmonic oscillator The Schrödinger equation for the harmonic oscillator takes the form where is the th energy eigenstate of with energy . We want to find an expression for in terms of . We define the operators and where , which we need to choose, is called the superpotential of . We also define the aforementioned partner Hamiltonians and as A zero energy ground state of would satisfy the equation Assuming that we know the ground state of the harmonic oscillator , we can solve for as We then find that We can now see that This is a special case of shape invariance, discussed below. Taking without proof the introductory theorem mentioned above, it is apparent that the spectrum of will start with and continue upwards in steps of The spectra of and will have the same even spacing, but will be shifted up by amounts and , respectively. It follows that the spectrum of is therefore the familiar . The SUSY QM superalgebra In fundamental quantum mechanics, we learn that an algebra of operators is defined by commutation relations among those operators. For example, the canonical operators of position and momentum have the commutator . (Here, we use "natural units" where Planck's constant is set equal to 1.) A more intricate case is the algebra of angular momentum operators; these quantities are closely connected to the rotational symmetries of three-dimensional space. To generalize this concept, we define an anticommutator, which relates operators the same way as an ordinary commutator, but with the