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η(n)=√n.
Itiseasytoprovetheseequationsusingmathematicalinductiononequations
(4.59) and (4.63).
A computer scientist cannot fail to register how closely this construction
of the state space of the harmonic oscillator runs to the construction of the
natural numbers as an inductive set. There are however important differences.
This issue will be explored elsewhere.
88
4.3 Angular momentum and spin
In classical physics, angular momentum is a quantity related to rotation. So
for example, a particle of mass m, rotating with the velocity v in a circle at a
distance r from a center, has an angular momentum L given by
L=mvr.
This, however, is an oversimplification. Since the circle of rotation lies in a
certain plane, and the velocity v is a vector (the direction of which is changing
as the particle moves in the circle), it turns out that angular momentum must
be described by a vector quantity L. In the simple case of a particle rotating
with constantvelocity v in a circle, the direction of the vector L coincides with
a vector normal to the plane of the circle.
The angularmomentum, being a vector,canbe written in terms of its com-
ponents (L ,L ,L ) in a rectangular coordinate system. Classically, all of the
x y z
components of L can be determined to arbitrary precision.
Whenangularmomentumforatomicquantumsystemssuchasthehydrogen
atomwasstudied,itturnedoutthatthe situationwasradicallydifferent. Since
atomicsystemscontainrotatingcomponentssuchaselectrons,angularmomen-
tum played a central role in the development of quantum mechanics. Without
going into either the history of the subject,8 or the detailed theory, let us just
record the basic facts.
In quantum mechanics the components of the angular momentum become
hermitean operators (Lˆ ,Lˆ ,Lˆ ). Having said that, we will at once drop the
x y z
’hats’ over the operators. The order of application of these operators on a
quantum state matters. This is recorded in the commutation relations
[L ,L ]=i¯hL , [L ,L ]=i¯hL , [L ,L ]=i¯hL . (4.64)
x y z y z x z x y
The physical consequence of this is that not all three components of L are
measurable simultaneously. Instead one normally considers the length square
of L
L2 =L2 +L2+L2. (4.65)
x y z
This quantity does commute with all components of L as can be shown
by simply carrying out the commutator algebra, using (4.64). Therefore, in
order to specify simultaneously measurable quantities for a rotating system,
one normally chooses L2 and one component of L, the standard choice being
the z-component L .
z
Thisisallveryabstract,andinaconcretecase,asforexamplewhentreating
the electron in a hydrogen atom, these operators are represented as differential
operatorsactingontheconfigurationspacewave-functionoftheelectron. Sucha
concreteanalysisshowsthattheangularmomentumstatescanbecharacterized
8A goodreference treating the historyof atomic, nuclear andelementary particle physics
is[34].
89
bytwoquantumnumbersl andmrelatedtotheeigenvaluesoftheoperatorsL2
and L . This involves quite a lot of long-winded calculations, and we will not
z
performthemhere. Theycanbefoundinanytextbookonquantummechanics.
The type ofcalculationsaresimilarto thosethat wereviewedinthe caseofthe
particle in a potential box, essentially solving the Scro¨dinger equation in three
dimensions of space.
Instead we will work in a more abstract way. The constant ¯h setting the
scaleofquantumphenomena,playsnosignificantroleinthefollowingalgebraic
treatmentofangularmomentum,sowewillrescaleitto1,ortoputitdifferently,
we will choose units of measurement where ¯h=1.9
Theangularmomentumeigenstateswillbedenotedby l,m wheremisthe
| i
eigenvalue corresponding to the z-component of L, i.e.
L l,m =ml,m . (4.66)
z
| i | i
Sinceangularmomentumisdescribedbytwocommutingoperators,itmakes
sense to label the eigenstates with two labels. As said, l is a quantum number
in some way related to the length of the angular momentum vector L, but the
exact correspondence is left open at the moment. We will work with a fixed
value for l.
The states are orthonormal
l,ml,n =δ . (4.67)
mn
h | i
Some further properties of this representation will now be derived. It will
become apparentthat this can be done in rather close analogy with the way in
which the harmonic oscillator was treated. In that case the operators x and p
wasreplacedby the creationandannihilationoperators. The creationoperator
increases the quantum number n (acting like a kind of successor) whereas the
annihilationoperator,decreases the quantum number (like a predecessor). The
firststep willbe to introduce new operatorsL andL with properties similar
+
−
to creation and annihilation operators with the difference that now we will get
a finite spectrum.
L =L +iL (4.68)
+ x y
L =L IL . (4.69)
x y
− −
Next, the commutation relations (4.64), are rewritten in terms of these op-
erators and L
z
[L ,L ]=L , [L ,L ]= L , [L ,L ]=2L . (4.70)