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In [[quantum mechanics]], a '''complete set of commuting observables''' (CSCO) is a set of [[commutative|commuting]] [[Operator_(physics)#Operators_in_quantum_mechanics|operator]]s whose [[eigenvalue]]s completely specify the [[quantum state| state]] of a system {{harv|Gasiorowicz|1974|p=119}}.
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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]].
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</ul>


==The Compatibility Theorem==
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Let us have two observables, <math>A</math> and <math>B</math>, represented by <math>\hat{A}</math> and <math>\hat{B}</math>. Then any one of the following statements implies the other two:
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#<math>A</math> and <math>B</math> are compatible observables.
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#<math>\hat{A}</math> and <math>\hat{B}</math> have a common eigenbasis.
#The operators <math>\hat{A}</math> and <math>\hat{B}</math> are commuting, that is, <math>[\hat{A}, \hat{B}] = 0</math>.


===Proofs===
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:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
== Nike Free 3.0 V5 Review  funny ==
!Proof that compatible observables commute.


|-
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|Let <math>\{|\psi_n\rangle\}</math> be a complete set of common eigenkets of the two compatible observables <math>A</math> and <math>B</math>, corresponding to the sets <math>\{a_n\}</math> and <math>\{b_n\}</math> respectively. Then we can write
 
:<math>AB|\psi_n\rangle=Ab_n|\psi_n\rangle
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=a_nb_n|\psi_n\rangle
 
=b_na_n|\psi_n\rangle
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=BA|\psi_n\rangle</math>
 
Now, we can expand any arbitrary state ket <math>|\Psi\rangle</math> in the complete set <math>\{|\psi_n\rangle\}</math> as
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:<math>|\Psi\rangle=\sum_nc_n|\psi_n\rangle</math>
 
So, using the above result, we can see that
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:<math>(AB-BA)|\Psi\rangle=\sum_{n}c_n(AB-BA)|\psi_n\rangle=0</math>
 
This implies <math>[A,B]=0</math>, which means that the two operators commute.
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|}
 
</ul>


:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
== Nike Free Run Womens  morale problems ==
!Proof that commuting observables possess a complete set of common eigenfunctions.


|-
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|''When <math>A</math> has ''non-degenerate'' eigenvalues:''
 
----
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Let <math>\{|\psi_n\rangle\}</math> be a complete set of eigenkets of <math>A</math> corresponding to the set of eigenvalues <math>\{a_n\}</math>.
 
If the operators <math>A</math> and <math>B</math> commute, we can write
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:<math>A(B|\psi_n\rangle)=BA|\psi_n\rangle=a_n(B|\psi_n\rangle)</math>
 
So, we can say that <math>B|\psi_n\rangle</math> is an eigenket of <math>A</math> corresponding to the eigenvalue <math>a_n</math>. The non-degeneracy of <math>a_n</math> implies that <math>B|\psi_n\rangle</math> and <math>|\psi_n\rangle</math> can differ at most by a multiplicative constant. We call this constant <math>b_n</math>. So,
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:<math>B|\psi_n\rangle=b_n|\psi_n\rangle</math>
 
So, <math>|\psi_n\rangle</math> is eigenket of the operators <math>A</math> and <math>B</math> ''simultaneously''.
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----
 
''When <math>A</math> has ''degenerate'' eigenvalues:''
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----
 
We suppose <math>a_n</math> is <math>g</math> -fold degenerate. Let the corresponding ''linearly independent'' eigenkets be <math>|\psi_{nr}\rangle, (r=1,2,...g)</math>
</ul>
Since <math>[A,B]=0</math>, we reason as above to find that <math>B|\psi_{nr}\rangle</math> is an eigenket of <math>A</math> corresponding to the ''degenerate'' eigenvalue <math>a_n</math>. So, we can expand <math>B|\psi_{nr}\rangle</math> in the basis of the degenerate eigenkets of <math>a_n</math>:
:<math>B|\psi_{nr}\rangle=\sum_{s=1}^g c_{rs}|\psi_{ns}\rangle</math>
The <math>c_{rs}</math> are the expansion coefficients.
We now sum over all <math>r</math> with <math>g</math> constants <math>d_r</math>. So,
:<math>B\sum_{r=1}^gd_r|\psi_{nr}\rangle=\sum_{r=1}^g\sum_{s=1}^g d_rc_{rs}|\psi_{ns}\rangle</math>
So, <math>\sum_{r=1}^gd_r|\psi_{nr}\rangle</math> will be an eigenket of <math>B</math> with the eigenvalue <math>b_n</math> if we have
:<math>\sum_{r=1}^g d_rc_{rs}=b_nd_s, s=1,2,...g</math>
This constitutes a system of <math>g</math> linear equations for the constants <math>d_r</math>. A non-trivial solution exists if
:<math>\det[c_{rs}-b_n\delta_{rs}]=0</math>
This is an equation of order <math>g</math> in <math>b_n</math>, and has <math>g</math> roots. For each root <math>b_n=b_n^{(k)}, k=1,2,...g</math> we have a value of <math>d_r</math>, say, <math>d_r^{(k)}</math>. Now, the ket
:<math>|\phi_n^{(k)}\rangle=\sum_{r=1}^g d_r^{(k)}|\psi_{nr}\rangle</math>
is simultaneously an eigenket of <math>A</math> and <math>B</math> with eigenvalues <math>a_n</math> and <math>b_n^{(k)}</math> respectively.
|}


===Discussion===
== Nike Free 3.0 Womens  however ==
We consider the two above observables <math>A</math> and <math>B</math>. Suppose there exists a complete set of kets <math>\{|\psi_n\rangle\}</math> whose every element is simultaneously an eigenket of <math>A</math> and <math>B</math>. Then we say that <math>A</math> and <math>B</math> are ''compatible''. If we denote the eigenvalues of <math>A</math> and <math>B</math> corresponding to <math>|\psi_n\rangle</math> respectively by <math>A</math> and <math>B</math>, we can write
:<math>A|\psi_n\rangle=a_n|\psi_n\rangle</math>
:<math>B|\psi_n\rangle=b_n|\psi_n\rangle</math>


If the system happens to be in one of the eigenstates, say, <math>|\psi_n\rangle</math>, then both <math>A</math> and <math>B</math> can be ''simultaneously'' measured to any arbitrary level of precision, and we will get the results <math>a_n</math> and <math>b_n</math> respectively. This idea can be extended to more than two observables.
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===Examples of Compatible Observables===
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The Cartesian components of the position operator <math>\mathbf{r}</math> are <math>x</math>, <math>y</math> and <math>z</math>. These components are all compatible. Similarly, the Cartesian components of the momentum operator <math>\mathbf{p}</math>, that is <math>p_x</math>, <math>p_y</math> and <math>p_z</math> are also compatible.


==Formal Definition of a CSCO==
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A set of observables <math>A, B, C...</math> is called a CSCO if:
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#All the observables commute in pairs.
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#If we specify the eigenvalues of all the operators in the CSCO, we identify a unique eigenvector in the Hilbert space of the system.
 
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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.
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Let us have an operator <math>\hat{A}</math> of an observable <math>A</math>, which has all ''non-degenerate'' eigenvalues <math>\{a_n\}</math>. 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 <math>\hat{A}</math> corresponding to the eigenvalue <math>a_n</math> can be labelled as <math>|a_n\rangle</math>. Such an observable is itself a self-sufficient CSCO.
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However, if some of the eigenvalues of <math>a_n</math> 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 <math>B</math>), which is compatible with  <math>A</math>. The compatibility theorem tells us that a common basis of eigenfunctions of <math>\hat{A}</math> and <math>\hat{B}</math> can be found. Now if each pair of the eigenvalues <math>(a_n, b_n)</math> uniquely specifies a state vector of this basis, we claim to have formed a CSCO: the set <math>\{A, B\}</math>. The degeneracy in <math>\hat{A}</math> is completely removed.
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It may so happen, nonetheless, that the degeneracy is not completely lifted. That is, there exists at least one pair <math>(a_n, b_n)</math> which does not uniquely identify one eigenvector. In this case, we repeat the above process by adding another observable <math>C</math>, which is compatible with both <math>A</math> and <math>B</math>. If the basis of common eigenfunctions of <math>\hat{A}</math>, <math>\hat{B}</math> and <math>\hat{C}</math> is unique, that is, uniquely specified by the set of eigenvalues <math>(a_n, b_n, c_n)</math>, then we have formed a CSCO: <math>\{A, B, C\}</math>. If not, we add one more compatible observable and continue the process till a CSCO is obtained.
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The same vector space may have distinct complete sets of commuting operators.
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Suppose we are given a '''finite''' CSCO <math>\{A, B, C,...,\}</math>. Then we can expand any general state in the Hilbert space as
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:<math>|\psi\rangle \sum_{i,j,k,...} c_{i,j,k,...} |a_i, b_j, c_k,...\rangle</math>
where <math>|a_i, b_j, c_k,...\rangle</math> are the eigenkets of the operators <math>\hat{A}, \hat{B}, \hat{C}</math>, and form a basis space. That is,
:<math>\hat{A}|a_i, b_j, c_k,...\rangle = a_i|a_i, b_j, c_k,...\rangle</math>, etc


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=== The Hydrogenic Atom ===
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:''Main article'': [[hydrogen-like atom| Hydrogen-like Atom]].
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Two components of the angular momentum operator <math>\mathbf{L}</math> do not commute, but satisfy the commutation relations:
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:<math>[L_i, L_j]=i\hbar\epsilon _{ijk} L_k</math>
So, any CSCO cannot involve more than one component of <math>\mathbf{L}</math>. It can be shown that the square of the angular momentum operator, <math>L^2</math>, commutes with <math>\mathbf{L}</math>.
:<math>[L_x, L^2] = 0, [L_y, L^2] = 0, [L_z, L^2] = 0</math>
Also, the [[Hamiltonian]] <math>\hat{H} = -\frac{\hbar^2}{2\mu}\nabla^2 - \frac{Ze^2}{r} </math> is a function of <math>r</math> only and has rotational invariance, where <math>\mu</math> is the reduced mass of the system. Since the components of <math>\mathbf{L}</math> are generators of rotation, it can be shown that
:<math>[\mathbf{L}, H] = 0, [L^2, H] = 0</math>
Therefore a commuting set consists of <math>L^2</math>, one component of <math>\mathbf{L}</math> (which is taken to be <math>L_z</math>) and <math>H</math>. The solution of the problem tells us that disregarding spin of the electrons, the set <math>\{H, L^2, L_z\}</math> forms a CSCO. Let <math>|E_n, l, m\rangle</math> be any basis state in the Hilbert space of the hydrogenic atom. Then
:<math>H|E_n, l, m\rangle=E_n|E_n, l, m\rangle</math>
:
:<math>L^2|E_n, l, m\rangle=l(l+1)\hbar^2|E_n, l, m\rangle</math>
:
:<math>L_z|E_n, l, m\rangle=m\hbar|E_n, l, m\rangle</math>
That is, the set of eigenvalues <math>\{E_n, l, m\}</math> or more simply, <math>\{n, l, m\}</math> completely specifies a unique eigenstate of the Hydrogenic atom.


=== The Free Particle ===
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:''Main article'': [[Free_particle#Non-Relativistic_Quantum_Free_Particle|Quantum Free Particle]]
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For a [[free particle]], the Hamiltonian is <math>H = -\frac{\hbar^2}{2m} \nabla^2 </math> is invariant under translations. Translation commutes with the Hamiltonian: <math>[H,\mathbf{\hat T}]=0</math>. However, if we express the Hamiltonian in the basis of the translation operator, we will find that <math>H</math> has doubly degenerate eigenvalues. It can be shown that to make the CSCO in this case, we need another operator called the [[parity (physics)|parity]] operator <math>\Pi</math>, such that <math>[H,\Pi]=0</math>.<math>\{H,\Pi\}</math> forms a CSCO.
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Again, let <math>|k\rangle</math> and <math>|-k\rangle</math> be the '''degenerate''' eigenstates of <math>H</math>corresponding the eigenvalue <math>H_k=\frac{{\hbar^2}{k^2}}{2m}</math>, i.e.
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:<math>H|k\rangle=\frac{{\hbar^2}{k^2}}{2m}|k\rangle</math>
:<math>H|-k\rangle=\frac{{\hbar^2}{k^2}}{2m}|-k\rangle</math>
The degeneracy in <math>H</math> is removed by the momentum operator <math>\mathbf{\hat p}</math>.
:<math>\mathbf{\hat p}|k\rangle=k|k\rangle</math>
:<math>\mathbf{\hat p}|-k\rangle=-k|-k\rangle</math>
So, <math>\{\mathbf{\hat p}, H\}</math> forms a CSCO.


===Addition of Angular Momenta===
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We consider the case of two systems, 1 and 2, with respective angular momentum operators <math>\mathbf{J_1}</math> and <math>\mathbf{J_2}</math>. We can write the eigenstates of  <math>J_1^2</math> and <math> J_{1z}</math> as <math>|j_1m_1\rangle</math> and of <math>J_2^2</math> and <math> J_{2z}</math> as <math>|j_2m_2\rangle</math>.
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:<math>J_1^2|j_1m_1\rangle=j_1(j_1+1)\hbar^2|j_1m_1\rangle</math>
:<math>J_{1z}|j_1m_1\rangle=m_1\hbar|j_1m_1\rangle</math>
:<math>J_2^2|j_2m_2\rangle=j_2(j_2+1)\hbar^2|j_2m_2\rangle</math>
:<math>J_{2z}|j_2m_2\rangle=m_2\hbar|j_2m_2\rangle</math>
Then the basis states of the complete system are <math>|j_1m_1;j_2m_2\rangle</math> given by
:<math>|j_1m_1;j_2m_2\rangle=|j_1m_1\rangle \otimes |j_2m_2\rangle</math>
Therefore, for the complete system, the set of eigenvalues <math>\{j_1,m_1,j_2,m_2\}</math> completely specifies a unique basis state, and <math>\{J_1^2, J_{1z}, J_2^2, J_{2z}\}</math> forms a CSCO.
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#[[Mathematical_formulation_of_quantum_mechanics#Mathematical_structure_of_quantum_mechanics|Mathematical structure of quantum mechanics]]
 
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*{{Citation | last1=Gasiorowicz | first1=Stephen | title=Quantum Physics | publisher=[[John Wiley & Sons]] | location=New York | isbn=978-0-471-29281-4 | year=1974}}.
*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


[[Category:Quantum mechanics]]
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There is no denying that many individuals in the UK have been, and continue to be, involved in various efforts back in India; whether it is fundraising, community development, school education or direct clinical provision. Additionally, there are various organisations like BIDA, IMA and BAPIO who've potentially the infrastructure albeit their focus has largely been on activities within the UK. The united Uggs Cheapest kingdom Government has recognised its responsibility in supporting international efforts included in the recent Health is Fake Oakleys Ebay Global Strategy (6), which provides another timely opportunity.

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