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Rotation operators in spin space
Let us, for the moment, forget about the spatial position of the particle,
and concentrate on its spin state. A general
spin state is represented by the ket

(443) 
in spin space.
In Sect. 5.3, we were able to construct an operator
which
rotates the system by an angle about the axis in position
space. Can we also construct an operator
which rotates the
system by an angle about the axis in spin space? By analogy
with Eq. (358), we would expect such an operator to take the form

(444) 
Thus, after rotation, the ket becomes

(445) 
To demonstrate that the operator (444) really does rotate the spin of the system,
let us consider its effect on
. Under rotation, this
expectation value changes as follows:

(446) 
Thus, we need to compute

(447) 
This can be achieved in two different ways.
First, we can use the explicit formula for given in Eq. (431). We find
that Eq. (447) becomes

(448) 
or

(449) 
which reduces to

(450) 
where use has been made of Eqs. (431)(433).
A second approach is to use the so called BakerHausdorff lemma. This
takes the form
where is a Hermitian operator, and is a real parameter. The proof
of this lemma is left as an exercise. Applying the BakerHausdorff lemma
to Eq. (447), we obtain

(452) 
which reduces to

(453) 
or

(454) 
where use has been made of Eq. (421). The second
proof is more general than the first, since it only uses the fundamental
commutation relation (421), and is, therefore, valid for systems with spin
angular momentum higher than onehalf.
For a spin onehalf system, both methods imply that

(455) 
under the action of the rotation operator (444). It is straightforward to
show that

(456) 
Furthermore,

(457) 
since commutes with the rotation operator. Equations (455)(457)
demonstrate that
the operator (444) rotates the expectation value of by an
angle
about the axis. In fact, the expectation value
of the spin operator behaves like a classical vector under rotation:

(458) 
where the are the elements of the conventional rotation matrix
for the rotation in question. It is clear, from our second derivation of
the result (455), that this property is not restricted to the spin operators of
a spin onehalf system. In fact, we have effectively demonstrated that

(459) 
where the are the generators of rotation, satisfying the fundamental
commutation relation
,
and the rotation operator about the th axis is written
.
Consider the effect of the rotation operator (444) on the state ket (443).
It is easily seen that

(460) 
Consider a rotation by radians. We find that

(461) 
Note that a ket rotated by radians differs from the original ket by a
minus sign. In fact, a rotation by radians is needed to transform a ket
into itself. The minus sign does not affect the expectation value of
, since is sandwiched between and ,
both of which change sign. Nevertheless, the minus sign does give rise to
observable consequences, as we shall see presently.
Next: Magnetic moments
Up: Angular momentum
Previous: Wavefunction of a spin
Richard Fitzpatrick
20060216