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There is an upper limit for k. Just as there is a state with highest value
for m, namely l,l , there must be a state with lowest value for m. This occurs
| i
when k=2l in the above equation, corresponding to
L l, l =0,
βˆ’| βˆ’ i
which is (4.77) for k = 2l. So we learn that the spectrum for the operator
L ranges in unit steps from a maximum weight of l to a minimum weight of
z
l. Furthermore, since k is an integer, and 2l k must be zero for some k, the
βˆ’ βˆ’
possible values of l are 0,1/2,1,3/2,2,... .
{ }
Finally, putting l k =m we arrive at (4.74).
βˆ’
4.3.1 Spin 1/2
Inquantumphysics,thereisadistinctionmadebetweenangularmomentumand
spin. Angular momentum refers to, just as in classical physics, to rotational
motion in space. Spin, on the other hand, refers to an intrinsic property of
a particle. Though it can be thought of as some kind of rotation around an
internal axis of the particle, just as one would do for a spinning classical body,
thereisactuallynoneedfor,andnoexperimentalbasisforsuchapicture. Spin
is best regarded as an intrinsic quantity, along with other intrinsic quantities
such as mass, electric charge et cetera.
The simplest non-trivial, and for quantum computation, most important
example, is the case ofspin 1/2,or l = 1. An example ofa particle having spin
2
1/2 is the electron.
The abstract theory of angular momentum given in the preceding section,
canbemadeconcretebyrepresentingthestates l,m andoperatorsLbyvectors
| i
andmatrices. Thinkingoftheprecedingabstracttheoryasaninterface,wewill
now provide an implementation. We will do it in the case l = 1 as that will be
2
needed subsequently when discussing quantum computation.
Implementation of spin one half
With l = 1, the spectrum of m is 1, 1 with the highest weight state 1,1
2 {2 βˆ’2} |2 2i
andthe lowestweightstate 1, 1 . This notationis to heavy,andcustomarily
|2 βˆ’2i
one uses a leaner syntax 1 and 1 . These two states are implemented by
|2i |βˆ’ 2i
2-dimensional vectors unit vectors as
94
1 1
= (4.84)
|2i (cid:18)0(cid:19)
1 0
= . (4.85)
|βˆ’ 2i (cid:18)1(cid:19)
The components of the angular momentum operators (spin operators) are
given by
1 0 1
L = (4.86)
x 2(cid:18)1 0(cid:19)
1 0 i
L y = 2(cid:18)i βˆ’ 0 (cid:19) (4.87)
1 1 0
L = , (4.88)
z 2(cid:18)0 1(cid:19)
βˆ’
But we are more interested in L and L which are easily calculated from
+
βˆ’
L and L
x
βˆ’
0 1
L = (4.89)
+ (cid:18)0 0(cid:19)
0 0
L = . (4.90)
βˆ’ (cid:18)1 0(cid:19)
Employingthisconcreterealizations,almosttrivialmatrixcalculationsyield
1 0 1 1
L = =0 (4.91)
+ |2i (cid:18)0 0(cid:19)(cid:18)0(cid:19)
1 0 1 0 1 1
L = = = (4.92)
+ |βˆ’ 2i (cid:18)0 0(cid:19)(cid:18)1(cid:19) (cid:18)0(cid:19) |2i
1 0 0 1 0 1
L = = = (4.93)
βˆ’|2i (cid:18)1 0(cid:19)(cid:18)0(cid:19) (cid:18)1(cid:19) |βˆ’ 2i
1 0 0 0
L = =0, (4.94)
βˆ’|βˆ’ 2i (cid:18)1 0(cid:19)(cid:18)1(cid:19)
thus implementing the raisingandloweringoperationsofL andL . Compar-
+
ing to the previous section, it can also be checked that the normalizβˆ’ ations ΞΆ(1)
2
and ΞΆ( 1) come out right.
βˆ’2
Next we have to check the action of L
z
1 1 1 0 1 1 1 1 1
L = = = (4.95)
z |2i 2(cid:18)0 1(cid:19)(cid:18)0(cid:19) 2(cid:18)0(cid:19) 2|2i
βˆ’
1 1 1 0 0 1 0 1 1