Since the Hamiltonian for atoms shows rotational invariance, the angular momentum is a conserved
property. In addition the spin is directly related to angular momentum; it is a form of angular
momentum without a classical equivalent. Thus the quantum mechanical description of the angular
momentum will be discussed here as one of the first examples, although it is mathematically quite
Classically the angular momentum is defined as
Following the correspondence principle we therefore get
for the quantum mechanical operator
the operators for position and momentum.
One can easily show (see exercises) that the following relation holds
the same commutator relations hold for the cyclic permutation of
We introduce now a general operator with this commutator relations
This property of a vectorial observable may be summarized as
The 3 components , and are scalar observables (i.e., square matrices with Hermitian symmetry). Let’s introduce another scalar observable:
is just a square
matrix. It would be classically associated with the square of the length of a classical 3-vector associated with
(if there’s one). We
will show now, that
commutates with :
Proof: Since the commutator is clearly zero, we have:
Each of those two terms can be evaluated using the above commutation relations:
Therefore, those two terms add up to zero and we obtain:
The above definition of ensures that is nonnegative for any ket (HINT: this is the sum of 3 real squares).
Therefore, this operator can only have nonnegative Eigenvalues, which (for the sake of future simplicity) we may as well put in the following form, for some non negative number .
The punch line will be that
is restricted to integer or half-integer values. For now however, we may just accept
this expression because it spans all non negative values once and only once when
from zero to infinity.
So, we may use as an index to denote each Eigenvalue of . Similarly, we may use another index to identify the Eigenvalue of . For now, nothing special is assumed about (we’ll show later that is an integer).
Since those two observables commute, there is an orthonormal Hilbertian basis consisting entirely of Eigenvectors common to both of them. We may specify it by introducing a third index (needed to distinguish between kets having identical Eigenvalues for each of our two observables). Those conventions are summarized by the following relations, which clarify the notation used for base kets:
To determine the restrictions that and must obey, we introduce the following two non-Hermitian operators, which are conjugate of each other. They are collectively known as ladder operators; and are respectively called lowering operator (or anihilation operator) and raising operator (or creation operator) because it turns out that each transforms an Eigenvector into another Eigenvector corresponding to a lesser or greater Eigenvalue, respectively.
Both commute with (because and do). The following holds:
As the non negative square bracket is equal to
we see that
. We would find
that cannot exceed
by performing the
same computation for .
Therefore, all told:
Note that the above also proves that the ket
only when .
nonzero unless .
Except in the aforementioned cases where they vanish, such kets are Eigenvectors of associated with the Eigenvalue of index . Let’s prove that:
So, if is an Eigenvector of associated with the value , then:
Thus, the ket is either
zero or an Eigenvector of
associated with the value .
The same is true of
Since we know that is between and , we see that both and must be integers (or else iterating one of the two constructions above would yield a nonzero Eigenvector with a value of outside of the allowed range). Thus, and must be integers (they are the sum and the difference of the integers and ). If is an integer, so is . If is an half-integer, so is (by definition, an ”half-integer” is half the value of an odd integer).
The above demonstration is quite remarkable: It shows how a 3-component observable is quantized whenever it obeys the same commutation relation as an orbital angular momentum. Although half-integer values of the numbers l and m are allowed, those do not correspond to an orbital momentum but to a quantum mechanical spin. Only orbital momenta can lead to whole numbers of and (which we will not proof here).
Here the adjoined Legendre polynomials are defined as
and the scaling factor
The first spherical harmonic functions are
© J. Carstensen (Quantum Mech.)