Supersymmetry (SUSY) was originally constructed as a non-trivial unification of space-time and internal symmetries within four-dimensional relativistic quantum field theory. In 1981, Witten introduced SUSY quantum mechanics that has since developed as an alternative to the factorization method of Schrödinger, Infeld, and Hull.
Starting with the time independent one-dimensional Schrödinger equation:
we can subtract the ground state energy and obtain:
which we want to factor as:
where W is called the superpotential. The above equation implies that:
which we can easily solve for the wavefunction,
Alternatively, we also have
which gives the wavefunction as
In order for these wavefunction to be true zero modes and actually represent something physical, they must lie in the Hilbert space. If at least one of the groundstate wavefunctions is in the Hilbert space, then SUSY is said to be unbroken. If neither is in the Hilbert space, then SUSY is broken. When the superpotential is non-periodic on the real line, being in the Hilbert space results if the zero-mode wavefunctions are normalizable. The issue of normalization can be reduced to one of three conditions on W. If the asymptotic limits of W differ in sign, if the number of zeros (non-tangent) of W is odd, or if W is an odd function, then one of the zero-mode wavefunctions will be normalizable.
When the superpotential is periodic, the zero modes must be periodic functions with the same period as the superpotential. These zero-modes will be in the Hilbert space if they are Bloch functions. The necessary requirement is thus that
This condition yields in interesting consequences. If the integral is not equal to zero, then it can be made to equal zero by subtracting off an appropriate constant. Also, even if the integral is already zero, we have freedom to shift the superpotential by any constant, as long as the condition is maintained.
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