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why is SUSY QM important ? i mean for each one DImensional Hamiltonian , can we write 'H' as

$ H= A.A^{+}+C$ (or similar constant)

here $ A= \frac{d}{dt}+A(x)$ and $ A^{+}= -\frac{d}{dt}+A(x)$

1) can we always express 'x' and 'p' as a combination of A and its adjoint?

2) does SUSY QM makes easier to solve the Hamiltonian

3) can we apply second quantizatio formalism with operators $ A,A^{+}$ in the same way we did for the Harmonic oscillator

4) let be $ Ay(x)=\lambda _{n} y(x) $ an eigenfunction of the anhinilation operator, then is it true that $ E_{n} =C+ |\lambda _{n}|^{2} $ ¿what happens for the eigenfunctions ??

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Here we will not answer all the subquestions, but just mention that there are at least two theoretical reasons why SUSY QM is important:

  1. Witten's derivation of Atiyah-Singer Index Theorem using SUSY QM.

  2. There is a big overlap between SUSY QM systems, and QM systems that can be analytically solved.

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First, this decomposition is due to Schrodinger, not Witten. The operator/adjoint decomposition is from a late paper in the 1940s, especially the way it is written above. Witten published papers on the supersymmetry structure of this decomposition in the 1980s, but it was already known. There is no reason to attribute to Witten things not due to Witten.

Schrodinger discovered it as part of his link between imaginary time Schrodinger equation and stochastic diffusion. The stochastic description is for a particle moving in a potential W, executing a biased Brownian motion:

$$ {dx\over dt} - {\partial W \over \partial x} = \eta(t) $$

Where W is the superpotential (traditionally the gradient of W is called this, but this is a terrible convention), and $\eta$ is a white statistical noise. The path integral description is

$$ e^{-\int \eta^2}$$

and changing variables to x produces SUSY QM. The Boltzmann probability distribution is

$$ e^{-W}$$

and this is invariant under time evolution, so this function is the ground state with exactly zero energy, whenever W gives an integrable weight, which is Witten's condition for unbroken SUSY.

The supersymmetry relates the potential W to its inverted partner -W, and gives rise to amazing theorems--- like the theorem that the tunneling out of a local minimum of a potential W which goes from infinity to minus infinity is the same as the tunneling out of -W. These properties and others are discussed at length at Junkers: Supersymmetric methods...

The answer to your questions are from Schrodinger's 1940s paper:

  1. Any potential with a stable ground state can be expressed in supersymmetric form with unbroken SUSY
  2. Yes. A stochastic equation can be simulated directly.
  3. Yes. This is the generalized creation and annihilation operators of Schrodinger.
  4. No, this is not the way you step up--- you need a different functional form for the creation and annihilation operator for each excited state. This is called "shape invariance" in the literature.
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