1-d SUSY Quantum Mechanics

(Illustration)

Try a W(x):  

Find Bosonic Solution        Find Fermionic Solution

Enter a Superpotential, W(x), to see a SUSY QM Potential, V + or V -, and a ground state wave function (if it has zero energy and is normalizable) .  Try x, sinh(x),  tanh(x), and sin(pi*x), for starters.  The parser can understand the following functions:

sin(x)

cos(x)

 tan(x)

sinh(x)

cosh(x)

tanh(x)

asin(x)

acos(x)

 atan(x)

asinh (x)

acosh(x)

atanh(x)

exp(x)

ln(x)

log(x)

sqr(x)

sqrt(x)

round(x)

abs(x)

sign(x)

floor(x)

ceil(x)

 int(x)

frac(x)

step(x)

random(x)

 

 

 

 

Description

SUSY quantum mechanics is the study of quantum mechanical systems whose Hamiltonians can be factored.  The wave functions and the energy spectrum of such a Hamiltonian can be found analytically either by hand or on a computer.  We use the idea of factorization in quantum mechanics when we solve for the energy levels and wave functions of the quantum harmonic oscillator using ladder (raising/lowering or creation/annihilation) operators instead of explicitly solving a second-order linear differential equation. The algebraic method for determining the energy eigenfunctions for the harmonic oscillator succeeds because the Hamiltonian is almost an exact sum of squares that can be easily factored: 

p2 + x2 = (ip + x)*(-ip + x)+i*(xp - px) = (ip + x)*(-ip + x) +1, 

where i is the square root of -1 and, for clarity, hbar/sqrt(2m) is set equal to 1. Therefore, despite the cool name, supersymmetric methods are not new.  In fact,  factorization methods in physics can be traced back to Dirac, Schrödinger, Infeld, and Hull. General factorization methods can be traced back to Bernoulli (1702) and Cauchy (1827). 

Existing Hamiltonians describing quantum systems can easily be placed into the SUSY framework by subtracting off the known ground state energy, thereby making the new ground state energy zero and the rest of the energy spectrum positive. SUSY methods then yield, from these original Hamiltonians, new partner Hamiltonians that have almost the identical energy spectrum as their partner.  Conversely, starting from an operator called a superpotential, new Hamiltonians can be constructed and exact ground state wave functions, when they exist, can then be found using the standard graphical and integration packages of programs such as Mathematica on a personal computer or a workstation.

Credits

Script by Mario Belloni.
Java applets by Wolfgang Christian.