Supersymmetry for the Square-Well Potential
Supersymmetry for the Square-Well Potential
The most elementary problem in quantum mechanics considers a particle of mass in a one-dimensional infinite square well of width ("particle in a box"). The Schrödinger equation can conveniently be written in the modified form in , such that the ground state energy is rescaled to =0. The eigenstates are then given by =-1], (x)=sin[(n+1)πx/a]. The quantum number is now equal to the number of nodes in the wavefunction. For simplicity, let =1 and . The Schrödinger equation then simplifies to with , =n(n+2), (x)=, .
m
a
-(x)-(x)=(x)
2
ℏ
2m
ψ
n
2
π
2
ℏ
2m
2
a
ψ
n
E
n
ψ
n
0⩽x⩽a
E
0
E
n
2
π
2
ℏ
2m
2
a
2
(n+1)
ψ
n
2
a
n
2
ℏ
2m
a=π
H(x)=(x)
ψ
n
E
n
ψ
n
H=--1
2
d
d
2
x
E
n
ψ
n
2/π
sin[(n+1)x]n=0,1,2,…
The first step is to define the superpotential and two ladder operators and =-+W(x). The original Hamiltonian is then given by . The operator obtained by reversing and , =A, is called the supersymmetric-partner Hamiltonian. More explicitly, =-+(x) and =-+(x), where (x)=-W'(x) and (x)=+W'(x). It can then be shown that if (x) is an eigenfunction of with eigenvalue then is an eigenfunction of with the same eigenvalue: A(x)=A(x)≡const(x). We denote the eigenfunction of by (x) call its eigenvalue . For unbroken supersymmetry, =. Note that , meaning that the ground state of has no superpartner. Correspondingly, we find (x)=const(x). (The constants provide normalization factors.) Note that the operator removes one of the nodes of the wavefunction (x) as it converts it into (x). Conversely, adds a node.
W(x)=-'(x)(x)=-cotx
ψ
0
ψ
0
A=+W(x)
d
dx
+
A
d
dx
H≡=A
H
1
+
A
A
+
A
H
2
+
A
H
1
2
d
d
2
x
V
1
H
2
2
d
d
2
x
V
2
V
1
2
W(x)
V
2
2
W(x)
ψ
n
H
1
E
n
A(x)
ψ
n
H
2
H
2
ψ
n
E
n
ψ
n
ε
n
ϕ
n
H
2
ϕ
n
ε
n
ε
n
E
n
A(x)=0
ψ
0
H
1
H
1
+
A
ϕ
n
E
n
ψ
n
A
ψ
n
ϕ
n
+
A
In this Demonstration, you can plot any of the lowest four square-well eigenfunctions (x)=, on a scale with each origin at the corresponding eigenvalue =n(n+2). On the right are the corresponding eigenfunctions of the supersymmetric partner Hamiltonian , moving in the potential well (x)=2x+1 (compared to (x)=-1). The first three normalized supersymmetric eigenstates are given by (x)=x, =; (x)=x, =; (x)=x, =.
ψ
n
2/π
sin[(n+1)x]n=0,1,2,3
E
n
H
2
V
2
2
cot
V
1
ϕ
1
8/3π
2
sin
ε
1
E
1
ϕ
2
16/π
cosx2
sin
ε
2
E
2
ϕ
3
32/15π
[2+3cos(2x)]2
sin
ε
3
E
3
In particle physics, supersymmetry has been proposed as a connection between bosons and fermions. Although this is a beautiful theory, there is, as yet, no experimental evidence that Nature contains supersymmetry. If it does exist, it must be a massively broken symmetry. It is possible that the Large Hadron Collider will find supersymmetric partners of some known particles.