24.1. Supersymmetric quantum mechanics Consider two quantum mechanics Hamiltonians H+ = h2 2m d2 dx2 +V+(x),

24.1. Supersymmetric quantum mechanics Consider two quantum mechanics Hamiltonians H+ = −

¯h2 2m d2ψ

dx2

+V+(x), H− = −

¯h2 2m d2ψ

dx2

+V−(x).

where the potential terms are obtained in terms of a superpotential W(x) as V±(x) = W2(x)∓

¯h √

2m W

(x).

a. Show that H± can be written as H+ = A†A, H− = AA†.

in terms of some differential operators A and A† of the first order in the derivative.

Identify these operators.

b. Assume we have chosenW(x) such that the ground state energy ofH+ is identically zero. Show that the eigenvalues of the two Hamiltonians H± are related as E(−)

n

= E(+)

n+1, E(+)

0

= 0.

Find the relation between the relative eigenfunctions of the two Hamiltonians.

c. Show that, introducing the operators Q = 0 0 A 0

, Q† = 0 A†

0 0

, and grouping the two Hamiltonians as H = H+ 0 0 H−



we have the following relations

{Q,Q†} = H, {Q,Q} = {Q†,Q†} = 0,

[H,Q] = [H,Q†] = 0.

d. Suppose that the potentials V±(x) are finite as x→−∞ or as x→+∞ or both and let us define W1 = limx→+∞W(x) and W2 = limx→−∞W(x). Show that the reflection and transmission coefficients of the two potentials are related as R+(k) = W2 +ik W2 −ik  R−(k), T+(k) = W1 −ik
W2 −ik
 T−(k), where k and k are given by k = E −W2 2 , k = E −W2 1 .
Analyse the implication of these relations and their analytic structure.