Complete set of commuting observables
Since each pair of observables in the set commutes, the observables are all compatible so that the measurement of one observable has no effect on the result of measuring another observable in the set. It is therefore not necessary to specify the order in which the different observables are measured. Measurement of the complete set of observables constitutes a complete measurement, in the sense that it projects the quantum state of the system onto a unique and known vector in the basis defined by the set of operators. That is, to prepare the completely specified state, we have to take any state arbitrarily, and then perform a succession of measurements corresponding to all the observables in the set, until it becomes a uniquely specified vector in the Hilbert space.
The Compatibility Theorem
Let us have two observables, and , represented by and . Then any one of the following statements implies the other two:
- and are compatible observables.
- and have a common eigenbasis.
- The operators and are commuting, that is, .
Proof that compatible observables commute. Let be a complete set of common eigenkets of the two compatible observables and , corresponding to the sets and respectively. Then we can write
Now, we can expand any arbitrary state ket in the complete set as
So, using the above result, we can see that
This implies , which means that the two operators commute.
Proof that commuting observables possess a complete set of common eigenfunctions.
When has non-degenerate eigenvalues:
Let be a complete set of eigenkets of corresponding to the set of eigenvalues . If the operators and commute, we can write
So, we can say that is an eigenket of corresponding to the eigenvalue . Since both and are eigenkets associated with the same non-degenerate eigenvalue , they can differ at most by a multiplicative constant. We call this constant . So,
which means is an eigenket of , and thus of and simultaneously.
When has degenerate eigenvalues:
We suppose is -fold degenerate. Let the corresponding linearly independent eigenkets be Since , we reason as above to find that is an eigenket of corresponding to the degenerate eigenvalue . So, we can expand in the basis of the degenerate eigenkets of :
The are the expansion coefficients. We now sum over all with constants . So,
So, will be an eigenket of with the eigenvalue if we have
This constitutes a system of linear equations for the constants . A non-trivial solution exists if
This is an equation of order in , and has roots. For each root we have a value of , say, . Now, the ket
is simultaneously an eigenket of and with eigenvalues and respectively.
We consider the two above observables and . Suppose there exists a complete set of kets whose every element is simultaneously an eigenket of and . Then we say that and are compatible. If we denote the eigenvalues of and corresponding to respectively by and , we can write
If the system happens to be in one of the eigenstates, say, , then both and can be simultaneously measured to any arbitrary level of precision, and we will get the results and respectively. This idea can be extended to more than two observables.
Examples of Compatible Observables
The Cartesian components of the position operator are , and . These components are all compatible. Similarly, the Cartesian components of the momentum operator , that is , and are also compatible.
Formal Definition of a CSCO (Complete Set of Commuting Observables)
A set of observables is called a CSCO if:
- All the observables commute in pairs.
- 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.
Let us have an operator of an observable , which has all non-degenerate eigenvalues . 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 corresponding to the eigenvalue can be labelled as . Such an observable is itself a self-sufficient CSCO.
However, if some of the eigenvalues of 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 ), which is compatible with . The compatibility theorem tells us that a common basis of eigenfunctions of and can be found. Now if each pair of the eigenvalues uniquely specifies a state vector of this basis, we claim to have formed a CSCO: the set . The degeneracy in is completely removed.
It may so happen, nonetheless, that the degeneracy is not completely lifted. That is, there exists at least one pair which does not uniquely identify one eigenvector. In this case, we repeat the above process by adding another observable , which is compatible with both and . If the basis of common eigenfunctions of , and is unique, that is, uniquely specified by the set of eigenvalues , then we have formed a CSCO: . 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 . Then we can expand any general state in the Hilbert space as
where are the eigenkets of the operators , and form a basis space. That is,
- , etc
If we measure in the state then the probability that we simultaneously measure is given by .
For a complete set of commuting operators, we can find a unique unitary transformation which will simultaneously diagonalize all of them. If there is more than one such unitary transformation, then we can say that the set is not yet complete.
The Hydrogen Atom
Two components of the angular momentum operator do not commute, but satisfy the commutation relations:
So, any CSCO cannot involve more than one component of . It can be shown that the square of the angular momentum operator, , commutes with .
Also, the Hamiltonian is a function of only and has rotational invariance, where is the reduced mass of the system. Since the components of are generators of rotation, it can be shown that
Therefore, a commuting set consists of , one component of (which is taken to be ) and . The solution of the problem tells us that disregarding spin of the electrons, the set forms a CSCO. Let be any basis state in the Hilbert space of the hydrogenic atom. Then
That is, the set of eigenvalues or more simply, completely specifies a unique eigenstate of the Hydrogenic atom.
The Free Particle
For a free particle, the Hamiltonian is is invariant under translations. Translation commutes with the Hamiltonian: . However, if we express the Hamiltonian in the basis of the translation operator, we will find that has doubly degenerate eigenvalues. It can be shown that to make the CSCO in this case, we need another operator called the parity operator , such that . forms a CSCO.
Again, let and be the degenerate eigenstates of corresponding the eigenvalue , i.e.
The degeneracy in is removed by the momentum operator .
So, forms a CSCO.
Addition of Angular Momenta
We consider the case of two systems, 1 and 2, with respective angular momentum operators and . We can write the eigenstates of and as and of and as .
Then the basis states of the complete system are given by
Therefore, for the complete system, the set of eigenvalues completely specifies a unique basis state, and forms a CSCO. Equivalently, there exists another set of basis states for the system, in terms of the total angular momentum operator . The eigenvalues of are where takes on the values , and those of are where . The basis states of the operators and are . Thus we may also specify a unique basis state in the Hilbert space of the complete system by the set of eigenvalues , and the corresponding CSCO is .
- Mathematical structure of quantum mechanics
- Operators in Quantum Mechanics
- Canonical commutation relation
- Measurement in quantum mechanics
- Degenerate energy levels
- Good quantum number
- Collapse of the wavefunction
- Angular Momentum (Quantum Mechanics)
- (Gasiorowicz 1974, p. 119)
- Gasiorowicz, Stephen (1974), Quantum Physics, New York: John Wiley & Sons, ISBN 978-0-471-29281-4.
- Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe, Quantum Mechanics, John Wiley & Sons (1977).
- P.A.M. Dirac: The Principles of Quantum Mechanics, Oxford University Press, 1958
- R.P. Feynman, R.B. Leighton and M. Sands: The Feynman Lectures on Physics, Addison-Wesley, 1965
- R Shankar, Principles of Quantum Mechanics, Second Edition, Springer (1994).
- J J Sakurai, Modern Quantum Mechanics, Revised Edition, Pearson (1994).
- B. H. Bransden and C. J. Joachain, Quantum Mechanics, Second Edition, Pearson Education Limited, 2000.
- For a discussion on the Compatibility Theorem, Lecture Notes of School of Physics and Astronomy of The University of Edinburgh. http://www2.ph.ed.ac.uk/~ldeldebb/docs/QM/lect2.pdf.
- A slide on CSCO in the lecture notes of Prof. S Gupta, Tata Institute of Fundamental Research, Mumbai. http://theory.tifr.res.in/~sgupta/courses/qm2013/hand3.pdf
- A section on the Free Particle in the lecture notes of Prof. S Gupta, Tata Institute of Fundamental Research, Mumbai. http://theory.tifr.res.in/~sgupta/courses/qm2013/hand6.pdf