Complete set of commuting observables

The Compatibility Theorem

Let us have two observables, A and B , represented by A ^ and B ^ . Then any one of the following statements implies the other two:

1. A and B are compatible observables.
2. A ^ and B ^ have a common eigenbasis.
3. The operators A ^ and B ^ are commuting, that is, [ A ^ , B ^ ] = 0 .

Discussion

We consider the two above observables A and B . Suppose there exists a complete set of kets { | ψ n ⟩ } whose every element is simultaneously an eigenket of A and B . Then we say that A and B are compatible. If we denote the eigenvalues of A and B corresponding to | ψ n ⟩ respectively by a n and b n , we can write

If the system happens to be in one of the eigenstates, say, | ψ n ⟩ , then both A and B can be simultaneously measured to any arbitrary level of precision, and we will get the results a n and b n respectively. This idea can be extended to more than two observables.

Examples of Compatible Observables

The Cartesian components of the position operator r are x , y and z . These components are all compatible. Similarly, the Cartesian components of the momentum operator p , that is p x , p y and p z are also compatible.

Formal Definition of a CSCO (Complete Set of Commuting Observables)

A set of observables A , B , C . . . is called a CSCO if:

1. All the observables commute in pairs.
2. If we specify the eigenvalues of all the operators in the CSCO, we identify a unique eigenvector in the Hilbert space of the system.

If we are given a CSCO, we can choose a basis for the space of states made of common eigenvectors of the corresponding operators. We can uniquely identify each eigenvector by the set of eigenvalues it corresponds to.

Discussion

Let us have an operator A ^ of an observable A , which has all non-degenerate eigenvalues { a n } . As a result, there is one unique eigenstate corresponding to each eigenvalue, allowing us to label these by their respective eigenvalues. For example, the eigenstate of A ^ corresponding to the eigenvalue a n can be labelled as | a n ⟩ . Such an observable is itself a self-sufficient CSCO.

However, if some of the eigenvalues of a n are degenerate, then the above result no longer holds. In such a case, we need to distinguish between the eigenfunctions corresponding to the same eigenvalue. To do this, a second observable is introduced (let us call that B ), which is compatible with A . The compatibility theorem tells us that a common basis of eigenfunctions of A ^ and B ^ can be found. Now if each pair of the eigenvalues ( a n , b n ) uniquely specifies a state vector of this basis, we claim to have formed a CSCO: the set { A , B } . The degeneracy in A ^ is completely removed.

It may so happen, nonetheless, that the degeneracy is not completely lifted. That is, there exists at least one pair ( a n , b n ) which does not uniquely identify one eigenvector. In this case, we repeat the above process by adding another observable C , which is compatible with both A and B . If the basis of common eigenfunctions of A ^ , B ^ and C ^ is unique, that is, uniquely specified by the set of eigenvalues ( a n , b n , c n ) , then we have formed a CSCO: { A , B , C } . If not, we add one more compatible observable and continue the process till a CSCO is obtained.

The same vector space may have distinct complete sets of commuting operators.

Suppose we are given a finite CSCO { A , B , C , . . . , } . Then we can expand any general state in the Hilbert space as

where | a i , b j , c k , . . . ⟩ are the eigenkets of the operators A ^ , B ^ , C ^ , and form a basis space. That is,

If we measure A , B , C , . . . in the state | ψ ⟩ then the probability that we simultaneously measure a i , b j , c k , . . . is given by | c i , j , k , . . . | 2 .

For a complete set of commuting operators, we can find a unique unitary transformation which will simultaneously diagonalize all of them. If there are more than one such unitary transformations, then we can say that the set is not yet complete.

The Hydrogen Atom

Two components of the angular momentum operator L do not commute, but satisfy the commutation relations:

So, any CSCO cannot involve more than one component of L . It can be shown that the square of the angular momentum operator, L 2 , commutes with L .

Also, the Hamiltonian H ^ = − ℏ 2 2 μ ∇ 2 − Z e 2 r is a function of r only and has rotational invariance, where μ is the reduced mass of the system. Since the components of L are generators of rotation, it can be shown that

Therefore a commuting set consists of L 2 , one component of L (which is taken to be L z ) and H . The solution of the problem tells us that disregarding spin of the electrons, the set { H , L 2 , L z } forms a CSCO. Let | E n , l , m ⟩ be any basis state in the Hilbert space of the hydrogenic atom. Then

That is, the set of eigenvalues { E n , l , m } or more simply, { n , l , m } completely specifies a unique eigenstate of the Hydrogenic atom.

The Free Particle

For a free particle, the Hamiltonian is H = − ℏ 2 2 m ∇ 2 is invariant under translations. Translation commutes with the Hamiltonian: [ H , T ^ ] = 0 . However, if we express the Hamiltonian in the basis of the translation operator, we will find that H has doubly degenerate eigenvalues. It can be shown that to make the CSCO in this case, we need another operator called the parity operator Π , such that [ H , Π ] = 0 . { H , Π } forms a CSCO.

Again, let | k ⟩ and | − k ⟩ be the degenerate eigenstates of H corresponding the eigenvalue H k = ℏ 2 k 2 2 m , i.e.

The degeneracy in H is removed by the momentum operator p ^ .

So, { p ^ , H } forms a CSCO.

Addition of Angular Momenta

We consider the case of two systems, 1 and 2, with respective angular momentum operators J 1 and J 2 . We can write the eigenstates of J 1 2 and J 1 z as | j 1 m 1 ⟩ and of J 2 2 and J 2 z as | j 2 m 2 ⟩ .

Then the basis states of the complete system are | j 1 m 1 ; j 2 m 2 ⟩ given by

Therefore, for the complete system, the set of eigenvalues { j 1 , m 1 , j 2 , m 2 } completely specifies a unique basis state, and { J 1 2 , J 1 z , J 2 2 , J 2 z } forms a CSCO. Equivalently, there exists another set of basis states for the system, in terms of the total angular momentum operator J = J 1 + J 2 . The eigenvalues of J 2 are j ( j + 1 ) ℏ 2 where j takes on the values j 1 + j 2 , j 1 + j 2 − 1 , . . . , | j 1 − j 2 | , and those of J z are m where m = − j , − j + 1 , . . . j − 1 , j . The basis states of the operators J 2 and J z are | j 1 j 2 ; j m ⟩ . Thus we may also specify a unique basis state in the Hilbert space of the complete system by the set of eigenvalues { j 1 , j 2 , j , m } , and the corresponding CSCO is { J 1 2 , J 2 2 , J 2 , J z } .