Quantum Mechanics for Mathematicians

where the “H” subscripts indicate the Heisenberg picture choice for the treat-
ment of time-dependence. It can easily be seen that the physically observable
quantities given by eigenvalues and expectations values are identical in the two
pictures:

H〈ψ(t)|OH|ψ(t)〉H=〈ψ(t)|U(t)(U

− (^1) (t)OU(t))U− (^1) (t)|ψ(t)〉=〈ψ(t)|O|ψ(t)〉
In the Heisenberg picture the dynamics is given by a differential equation
not for the states but for the operators. Recall from our discussion of the adjoint
representation (see equation 5.1) the formula
d
dt
(etXY e−tX) =

(

d

dt

(etXY)

)

e−tX+etXY

(

d

dt

e−tX

)

=XetXY e−tX−etXY e−tXX

Using this with
Y=O, X=iH

we find
d
dt
OH(t) = [iH,OH(t)] =i[H,OH(t)]

and this equation determines the time evolution of the observables in the Heisen-
berg picture.
Applying this to the case of the spin^12 system in a magnetic field, and taking
for our observableS(theSj, taken together as a column vector) we find

d

dt

SH(t) =i[H,SH(t)] =i

eg

2 mc

[SH(t)·B,SH(t)] (7.3)

We know from the discussion above that the solution will be

SH(t) =U(t)SH(0)U(t)−^1

for
U(t) =e−it

ge|B|

2 mcS·|BB|

By equation 6.5 and the identification there of vectors and 2 by 2 matrices, the
spin vector observable evolves in the Heisenberg picture by rotating about the
magnetic field vectorBwith angular velocityge 2 mc|B|.

7.4 Complex projective space

There is a different possible approach to characterizing states of a quantum
system withH=C^2. Multiplication of vectors inHby a non-zero complex
number does not change eigenvectors, eigenvalues or expectation values, so ar-
guably has no physical effect. Thus what is physically relevant is the quotient
space (C^2 − 0 )/C∗, which is constructed by taking all non-zero elements ofC^2
and identifying those related by multiplication by a non-zero complex number.