Quantum Mechanics for Mathematicians

on the spaceHB=L^2 (R^3 ) of square-integrable functions of the position

coordinates. The Hamiltonian operator is

H=

1

2 m

|P|^2 =−

1

2 m

(

∂^2

∂q^21

+

∂^2

∂q 22

+

∂^2

∂q^23

)

• The spin^12 quantum system, discussed first in chapter 7 and later in
  section 31.1.1. This had a pseudo-classical fermionic phase spaceR^3 with
  coordinatesξ 1 ,ξ 2 ,ξ 3 which after quantization became the operators
  1
  √
  2

σ 1 ,

1

√

2

σ 2 ,

1

√

2

σ 3

on the state spaceHF=C^2. For this system we considered the Hamilto-

nian describing its interaction with a constant background magnetic field

H=−

1

2

(B 1 σ 1 +B 2 σ 2 +B 3 σ 3 ) (34.1)

It turns out to be an experimental fact that fundamental matter particles are
described by a quantum system that is the tensor product of these two systems,
with state space
H=HB⊗HF=L^2 (R^3 )⊗C^2 (34.2)

which can be thought of as two-component complex wavefunctions. This system
has a pseudo-classical description using a phase space with six conventional
coordinatesqj,pj and three fermionic coordinatesξj. On functions of these
coordinates one has a generalized Poisson bracket{·,·}±which provides a Lie
superalgebra structure on such functions. On generators, the non-zero bracket
relations are
{qj,pk}±=δjk, {ξj,ξk}±=δjk
For now we will take the background magnetic fieldB= 0. In chapter 45 we
will see how to generalize the free particle to the case of a particle in a general
background electromagnetic field, and then the Hamiltonian term 34.1 involving
theBfield will appear. In the absence of electromagnetic fields the classical
Hamiltonian function will still be

h=

1

2 m

(p^21 +p^22 +p^23 )

but now this can be written in the following form (using the Leibniz rule for a
Lie superbracket)

h=

1

2 m

{

∑^3

j=1

pjξj,

∑^3

k=1

pkξk}±=

1

2 m

∑^3

j,k=1

pj{ξj,ξk}±pk=

1

2 m

∑^3

j=1

p^2 j

Note the appearance of the functionp 1 ξ 1 +p 2 ξ 2 +p 3 ξ 3 which now plays a role
even more fundamental than that of the Hamiltonian (which can be expressed