Complete set of commuting observables
In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose eigenvalues completely specify the state of a system.[1]
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,
.
Proofs
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
. The non-degeneracy of
implies that
and
can differ at most by a multiplicative constant. We call this constant
. So,
So,
is eigenket of the operators
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.
Discussion
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.
Discussion
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 are more than one such unitary transformations, then we can say that the set is not yet complete.
Examples
The Hydrogen Atom
- Main article: Hydrogen-like 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
.
See also
- 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)
References
- ↑ (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