wang
(Wang)
#1
4.5 Complete sets of commuting observables
For many simple quantum systems, the energy of a state completelyspecifies that state of
the system. This is the case for the 1-dimensional harmonic oscillator, for example. But in
more complicated systems, this is no longer the case. At the cost ofsupplying additional
mutually commuting observables, however, it is possible to characterize the state uniquely
by the simultaneous eigenvalues of this set of observables.
This concept is familiar, for example, from the description of orbitalangular momentum
states. An eigenvalueℓ(ℓ+ 1), forℓ= 0, 1 , 2 ,···, of the observableL^2 does not uniquely
specify an angular momentum state (except whenℓ= 0) but leaves a 2ℓ+1-fold degeneration.
Supplying the additional observableLz lifts this degeneracy completely as it supplements
ℓwith the eigenvaluemofLz such that−ℓ≤m≤+ℓ. The pair (ℓ,m) now completely
and uniquely specifies all quantum states. Another example, which builds on this one, is
provided by the Hydrogen atom, whose states may be completely specified by the quantum
numbers (n,ℓ,m), ignoring the spins of the nucleus and of the electron.
The above construction may be generalized to more complicated quantum systems. One
defines acomplete set of commuting observablesof a system, as a set of observables,
A 1 ,A 2 ,···,An (4.26)
(1) such that they mutually commute, [Ai,Aj] = 0 for alli,j= 1,···,n;
(2) and such that the eigenspaces common toA 1 ,A 2 ,···,Anare all one-dimensional.
The fact that the eigenspaces are all one-dimensional means precisely that there is a one-
to-one map between the simultaneous eigenvalues (a 1 ,a 2 ,···,an) and a basis of states inH.
The set of eigenvalues (a 1 ,a 2 ,···,an) which describes the states ofHuniquely is referred to
as the set ofquantum numbersof that state.
Complete sets of commuting observables can be trivial. For example,given an orthonor-
mal basis of states|n〉, forn∈N, the following is a complete set of commuting observables,
Pn=|n〉〈n| n∈N (4.27)
wherePnis the orthogonal projection operators on a single state|n〉. The possible eigenvalues
of the operators are 0 and 1, and would provide a digital numbering of all the basis vectors
inH. Describing all the states of the harmonic oscillator this way would not be very efficient.
Given an operatorA, any functionf(A) will commute withA. Also, given two mutually
commuting operatorsA 1 andA 2 , the productA 1 A 2 commutes withA 1 andA 2. Neither
f(A), norA 1 A 2 , however, produce information not already contained inA, or inA 1 and
A 2. Therefore, the most interesting complete sets of commuting operator will be the most
economical ones in which the operators are functionally independent from one another.
