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quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, a complete set of commuting observables (CSCO) is a set of
commuting Commuting is periodically recurring travel between one's place of residence and place of work or study, where the traveler, referred to as a commuter, leaves the boundary of their home community. By extension, it can sometimes be any regul ...
operators whose common eigenvectors can be used as a basis to express any quantum
state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * ''Our S ...
. In the case of operators with discrete spectra, a CSCO is a set of commuting observables whose simultaneous eigenspaces span the Hilbert space, so that the eigenvectors are uniquely specified by the corresponding sets of eigenvalues. 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 In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
(up to a phase).


The compatibility theorem

Let us have two observables, A and B, represented by \hat and \hat. Then the following statements are equivalent: #A and B are compatible observables. #\hat and \hat have a common eigenbasis. #The operators \hat and \hat are
commuting Commuting is periodically recurring travel between one's place of residence and place of work or study, where the traveler, referred to as a commuter, leaves the boundary of their home community. By extension, it can sometimes be any regul ...
, that is, hat, \hat= 0.


Proofs


Discussion

We consider the two above observables A and B. Suppose there exists a complete set of kets \ 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 , \psi_n\rangle respectively by a_n and b_n, we can write :A, \psi_n\rangle=a_n, \psi_n\rangle :B, \psi_n\rangle=b_n, \psi_n\rangle If the system happens to be in one of the eigenstates, say, , \psi_n\rangle, 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 \mathbf are x, y and z. These components are all compatible. Similarly, the Cartesian components of the momentum operator \mathbf, that is p_x, p_y and p_z are also compatible.


Formal definition

A set of observables A, B, C... 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 (up to a phase) 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 (up to a phase) by the set of eigenvalues it corresponds to.


Discussion

Let us have an operator \hat of an observable A, 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 \hat corresponding to the eigenvalue a_n can be labelled as , a_n\rangle. Such an observable is itself a self-sufficient CSCO. However, if some of the eigenvalues of a_n are ''degenerate'' (such as having
degenerate energy levels In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give th ...
), 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 \hat and \hat 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 \. The degeneracy in \hat 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 \hat, \hat and \hat is unique, that is, uniquely specified by the set of eigenvalues (a_n, b_n, c_n), 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 :, \psi\rangle = \sum_ \mathcal_ , a_i, b_j, c_k,...\rangle where , a_i, b_j, c_k,...\rangle are the eigenkets of the operators \hat, \hat, \hat, and form a basis space. That is, :\hat, a_i, b_j, c_k,...\rangle = a_i, a_i, b_j, c_k,...\rangle, etc If we measure A, B, C,... in the state , \psi\rangle then the probability that we simultaneously measure a_i, b_j, c_k,... is given by , \mathcal_, ^2. For a complete set of commuting operators, we can find a unitary transformation which will simultaneously diagonalize all of them.


Examples


The hydrogen atom without electron or proton spin

Two components of the angular momentum operator \mathbf do not commute, but satisfy the commutation relations: : _i, L_ji\hbar\epsilon _ L_k So, any CSCO cannot involve more than one component of \mathbf. It can be shown that the square of the angular momentum operator, L^2, commutes with \mathbf. : _x, L^2= 0, _y, L^2= 0, _z, L^2= 0 Also, the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
\hat = -\frac\nabla^2 - \frac is a function of r only and has rotational invariance, where \mu is the reduced mass of the system. Since the components of \mathbf are generators of rotation, it can be shown that : mathbf, H= 0, ^2, H= 0 Therefore, a commuting set consists of L^2, one component of \mathbf (which is taken to be L_z) and H. The solution of the problem tells us that disregarding spin of the electrons, the set \ forms a CSCO. Let , E_n, l, m\rangle be any basis state in the Hilbert space of the hydrogenic atom. Then :H, E_n, l, m\rangle=E_n, E_n, l, m\rangle : :L^2, E_n, l, m\rangle=l(l+1)\hbar^2, E_n, l, m\rangle : :L_z, E_n, l, m\rangle=m\hbar, E_n, l, m\rangle 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 In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
, the Hamiltonian H = -\frac \nabla^2 is invariant under translations. Translation commutes with the Hamiltonian: ,\mathbf0. 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 \Pi, such that ,\Pi0.\ forms a CSCO. Again, let , k\rangle and , -k\rangle be the degenerate eigenstates of Hcorresponding the eigenvalue H_k=\frac, i.e. :H, k\rangle=\frac, k\rangle :H, -k\rangle=\frac, -k\rangle The degeneracy in H is removed by the momentum operator \mathbf. :\mathbf, k\rangle=k, k\rangle :\mathbf, -k\rangle=-k, -k\rangle So, \ forms a CSCO.


Addition of angular momenta

We consider the case of two systems, 1 and 2, with respective angular momentum operators \mathbf and \mathbf. We can write the eigenstates of J_1^2 and J_ as , j_1m_1\rangle and of J_2^2 and J_ as , j_2m_2\rangle. :J_1^2, j_1m_1\rangle=j_1(j_1+1)\hbar^2, j_1m_1\rangle :J_, j_1m_1\rangle=m_1\hbar, j_1m_1\rangle :J_2^2, j_2m_2\rangle=j_2(j_2+1)\hbar^2, j_2m_2\rangle :J_, j_2m_2\rangle=m_2\hbar, j_2m_2\rangle Then the basis states of the complete system are , j_1m_1;j_2m_2\rangle given by :, j_1m_1;j_2m_2\rangle=, j_1m_1\rangle \otimes , j_2m_2\rangle 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 \mathbf = \mathbf+\mathbf. The eigenvalues of J^2 are j(j+1)\hbar^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_1j_2;jm\rangle. 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

*
Quantum number In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
**
Good quantum number In quantum mechanics, given a particular Hamiltonian H and an operator O with corresponding eigenvalues and eigenvectors given by O, q_j\rangle=q_j, q_j\rangle, the q_j are said to be good quantum numbers if every eigenvector , q_j\rangle remain ...
*
Degenerate energy levels In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give th ...
* Mathematical structure of quantum mechanics * Operators in Quantum Mechanics *
Canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p_ ...
*
Measurement in quantum mechanics In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what m ...
*
Collapse of the wavefunction In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an ''observat ...
* Angular Momentum (Quantum Mechanics)


References

*. * * *{{cite book, first=P.A.M. , last=Dirac, author-link=Paul Dirac, title= The Principles of Quantum Mechanics, publisher=Clarendon Press , publication-place=Oxford , year=1958 , isbn=978-0-19-851208-0 , oclc=534829 *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 ''Modern Quantum Mechanics'', often called ''Sakurai'' or ''Sakurai and Napolitano'', is a standard graduate-level quantum mechanics textbook written originally by J. J. Sakurai and edited by San Fu Tuan in 1985, with later editions coauthored ...
'', 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 Quantum mechanics