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z + + z + z
βˆ’ βˆ’ βˆ’ βˆ’
9This is standard practice in quantum mechanics. It is actually possible to reinstate hΒ―
in the formulas at a later stage via so called dimensional analysis. Another way to look at
this isto rescalethe operators, or absorbfactors of √¯h. Thisis actually what wedidinthe
precedingsectionwhendefiningthecreationandannihilationoperators.
90
To understand the action of the operator L , perform the following calcu-
+
lation
L L l,m = [L ,L ]+L L l,m =
z + z + + z
| i | i
(cid:0) (cid:1) (cid:0) (cid:1)
L +mL l,m =(m+1)L l,m .
+ + +
| i | i
(cid:0) (cid:1)
This is a calculation 10 of the L -eigenvalue for the state L l,m and it
z +
| i
shows that as compared to the state l,m , the state L l,m has L l,m
+ +
| i | i | i
eigenvalue m+1, i.e. the eigenvalue is one unit larger. In the same way, it can
be shown that the state L l,m has L -eigenvalue m 1 as compared to the
z
βˆ’| i βˆ’
state l,m . For this reason, the operators L and L are called raising and
+
| i βˆ’
lowering operators respectively. It now remains to calculate the spectrum of
eigenstates for L and L2.
z
The intuition is the following. We have a physical system with a certain
angular momentum, somehow parameterized by the quantum number l. The
quantum number m corresponds to the z-component. Obviously, if the total
angular momentum has a finite value, the z-component must also be bounded.
Evenclassically,no component ofL canbe largerthan L itself. Then, thinking
of L as an operator that increases z-component of the angular momentum, it
+
makessensetopostulatethe existenceofastatewiththehighestpossiblevalue
for m, denoted by l,l , that is annihilated by L , or
+
| i
L l,l =0. (4.71)
+
| i
Thisequationplaysasimilarroletothatofequation(4.51)fortheharmonic
oscillatorasprovidingabasecaseforbuildingthespectrumofstates. Thestate
l,l is called the highest weight state.
| i
Actingonthestate l,l withtheloweringoperatorL shouldyieldthestate
| i βˆ’
l,l 1 , having one unit lowerm-value. It is perhaps tempting then to assume
| βˆ’ i
that L l,l = l,l 1 , but that is wrong. There are normalization factors to
βˆ’| i | βˆ’ i
take into account. Instead, a detailed analysis will show
L l,m =ΞΆ(m)l,m 1 (4.72)
βˆ’| i | βˆ’ i
as well as
L l,m =ΞΆ(m+1)l,m+1 (4.73)
+
| i | i
with normalization factors ΞΆ(m). These factors have the following form
ΞΆ(m)= l(l+1) m(m 1). (4.74)
βˆ’ βˆ’
p
Applying L repeatedly, we must at some stage arrive at a state with low-
βˆ’
est possible value for the L -eigenvalue. A careful analysis shows that this
z
lowest weight is m. So for a given l, the spectrum of the operator L is
z
βˆ’
l, l+1,...,l 1,l yielding in total 2l+1 states. Now, the eigenvalues of
{βˆ’ βˆ’ βˆ’ }
the operator L2 can also be calculated. First rewrite L2 as
10Notetheuseofthestandardtrick,AB=[A,B]+BA,totakeadvantage ofthecommu-
tationrelationsforapairofoperators AandB.
91
1
L2 = L L +L L +L2, (4.75)
2 + βˆ’ βˆ’ + z
(cid:0) (cid:1)
using the definitions (4.68) and (4.69). Calculating the eigenvalues of L2 is
now a simple matter of applying L2 to a state l,m and using the appropriate
| i
equations. The result is
L2 l,m =l(l+1)l,m , (4.76)
| i | i
Thus the eigenvalue of L2 in a state l,m is l(l+1).
| i
Derivation of the normalization conditions
In this section, we will prove that the normalization coefficients has the form
given by equation (4.74). We will do it in the form of a proof by induction. To
simplify, we will assume them to be real, i.e. ΞΆ = ΞΆ. This is in fact a choice
βˆ—
that is always possible to make.