Quantum Mechanics for Mathematicians

for some constantsB 12 ,B 13 ,B 23. This can be written

h=

1

2

∑^3

j,k=1

Ljkξjξk

whereLjkare the entries of the matrix

L=





0 B 12 B 13

−B 12 0 B 23

−B 13 −B 23 0





The equations of motion on generators will be

d

dt

ξj(t) ={ξj,h}+=−{h,ξj}+

which, sinceL=−LT, by theorem 30.1 can be written

d

dt

ξj(t) =Lξj(t)

with solution
ξj(t) =etLξj(0)

This will be a time-dependent rotation of theξjin the plane perpendicular to

B= (B 23 ,−B 13 ,B 12 )

at a constant speed proportional to|B|.

30.3.2 The pseudo-classical fermionic oscillator

We have already studied the fermionic oscillator as a quantum system (in section
27.2), and one can ask whether there is a corresponding pseudo-classical system.
For the case ofdoscillators, such a system is given by taking an even dimensional
fermionic phase spaceV=R^2 d, with a basis of coordinate functionsξ 1 ,···,ξ 2 d
that generate Λ∗(R^2 d). On generators the fermionic Poisson bracket relations
come from the standard choice of positive definite symmetric bilinear form

{ξj,ξk}+=δjk

As shown in theorem 30.1, quadratic productsξjξkact on the generators by
infinitesimal rotations in thejkplane, and satisfy the commutation relations of
so(2d).
To get a pseudo-classical system corresponding to the fermionic oscillator
one makes the choice

h=

1

2

∑d

j=1

(ξ 2 jξ 2 j− 1 −ξ 2 j− 1 ξ 2 j) =

∑d

j=1

ξ 2 jξ 2 j− 1