Quantum Mechanics for Mathematicians

•

Γ =

∏d

j=1

(1− 2 a†FjaFj)

•

Γ =cγ 1 γ 2 ···γ 2 d

for some constantc. Computec.

• γjΓ + Γγj= 0
  for allj.


• 
  

  Γ^2 = 1

  
  


• 
  

  P±=

  
  

1

2

( 1 ±Γ)

are projection operators onto subspacesH+andH−ofHF.

• Show thatH+andH−are each separately representations ofspin(n) (i.e.,
  the representation operators commute withP±).

Problem 4:
Using the fermionic analog of Bargmann-Fock to construct spinors, and the
inner product 31.3, show that the operatorsaFj andaF†j are adjoints with
respect to this inner product.

B.15 Chapters 33 and 34

Problem 1:
Consider a two dimensional version of the Pauli equation that includes a
coupling to an electromagnetic field, with Hamiltonian

H=

1

2 m

((P 1 −eA 1 )^2 + (P 2 −eA 2 )^2 )−

e

2 m

Bσ 3

whereA 1 andA 2 are functions ofq 1 ,q 2 and

B=

∂A 2

∂q 1

−

∂A 1

∂q 2

Show that this is a supersymmetric quantum mechanics system, by finding
operatorsQ 1 ,Q 2 that satisfy the relations 33.1.

Problem 2:
For the three choices of inner product given in section 34.3, show that the

inner product is invariant under the action of the groupE ̃(3) on the space of
solutions.