Problem 2316. Spin Matrices

Created by Yaroslav

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{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The spin of a particle is a fundamental property in quantum physics. We shall inspect below matrix representations of such spin operators.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Suppose you have integer or half-integer spin of value</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>s</w:t></w:r><w:r><w:t>. The matrices</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>Sx</w:t></w:r><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>Sy</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>Sz</w:t></w:r><w:r><w:t> representing it have the following properties:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>Si</w:t></w:r><w:r><w:t> (with</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>i={x,y,z}</w:t></w:r><w:r><w:t>) are</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"en.wikipedia.org/wiki/Trace\"><w:r><w:t>traceless</w:t></w:r></w:hyperlink><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Hermitian_matrix\"><w:r><w:t>Hermitian matrices</w:t></w:r></w:hyperlink><w:r><w:t>;</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>Commutation relations (a): [</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>Si,Sj</w:t></w:r><w:r><w:t> ] = i</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>εijk Sk</w:t></w:r><w:r><w:t>, where [·,·] is the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Commutator#Ring_theory\"><w:r><w:t>commutator</w:t></w:r></w:hyperlink><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>εijk</w:t></w:r><w:r><w:t> is the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Levi-Civita_symbol\"><w:r><w:t>Levi-Civita symbol</w:t></w:r></w:hyperlink><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>Commutation relations (b): [</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>Si,S²</w:t></w:r><w:r><w:t> ] = 0, where</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>S²</w:t></w:r><w:r><w:t> =</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>Sx²+Sy²+Sz²</w:t></w:r><w:r><w:t>;</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>Eigenvalues:</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>S²</w:t></w:r><w:r><w:t> =</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>j(j+1)·I</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>Sz</w:t></w:r><w:r><w:t> = diag(</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>-j/2, -j/2+1, … ,j/2-1, j</w:t></w:r><w:r><w:t> ), where</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>I</w:t></w:r><w:r><w:t> is the identity matrix.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>See also</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Spin_operator#Mathematical_formulation_of_spin\"><w:r><w:t>this article</w:t></w:r></w:hyperlink><w:r><w:t> for more reference.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Examples</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ [Sx,Sy,Sz] = spin_matrices(1/2)\n\n Sx = \n 0 0.5\n 0.5 0\n\n Sy = \n 0 -0.5i\n 0.5i 0\n\n Sz = \n 0.5 0\n 0 0.5]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Note:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The usual cheats</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>are not</w:t></w:r><w:r><w:t> allowed!</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

The spin of a particle is a fundamental property in quantum physics. We shall inspect below matrix representations of such spin operators.Suppose you have integer or half-integer spin of value s. The matrices Sx, Sy and Sz representing it have the following properties:<ul><li>Si (with i={x,y,z}) are <a href = "en.wikipedia.org/wiki/Trace">traceless</a> <a href = "http://en.wikipedia.org/wiki/Hermitian_matrix">Hermitian matrices</a>;</li><li>Commutation relations (a): [ Si,Sj ] = i εijk Sk, where [·,·] is the <a href = "http://en.wikipedia.org/wiki/Commutator#Ring_theory">commutator</a> and εijk is the <a href = "http://en.wikipedia.org/wiki/Levi-Civita_symbol">Levi-Civita symbol</a>.</li><li>Commutation relations (b): [ Si,S² ] = 0, where S² = Sx²+Sy²+Sz²;</li><li>Eigenvalues: S² = j(j+1)·I and Sz = diag( -j/2, -j/2+1, … ,j/2-1, j ), where I is the identity matrix.</li></ul>See also <a href = "http://en.wikipedia.org/wiki/Spin_operator#Mathematical_formulation_of_spin">this article</a> for more reference.Examples<pre> [Sx,Sy,Sz] = spin_matrices(1/2)</pre><pre> Sx = 
 0 0.5
 0.5 0</pre><pre> Sy = 
 0 -0.5i
 0.5i 0</pre><pre> Sz = 
 0.5 0
 0 0.5</pre>Note:The usual cheats are not allowed!

The spin of a particle is a fundamental property in quantum physics.  We shall inspect below matrix representations of such spin operators.

Suppose you have integer or half-integer spin of value _s_. The matrices _Sx_, _Sy_ and _Sz_ representing it have the following properties:

* _Si_ (with _i={x,y,z}_) are <en.wikipedia.org/wiki/Trace traceless> <http://en.wikipedia.org/wiki/Hermitian_matrix Hermitian matrices>;
* Commutation relations (a): [ _Si,Sj_ ] = i _εijk Sk_, where [·,·] is the <http://en.wikipedia.org/wiki/Commutator#Ring_theory commutator> and _εijk_ is the <http://en.wikipedia.org/wiki/Levi-Civita_symbol Levi-Civita symbol>.
* Commutation relations (b): [ _Si,S²_ ] = 0, where _S²_ = _Sx²+Sy²+Sz²_;
* Eigenvalues: _S²_ = _j(j+1)·I_ and _Sz_ = diag( _-j/2, -j/2+1, … ,j/2-1, j_ ), where _I_ is the identity matrix.

See also <http://en.wikipedia.org/wiki/Spin_operator#Mathematical_formulation_of_spin this article> for more reference.

*Examples*

[Sx,Sy,Sz] = spin_matrices(1/2)

Sx = 
     0      0.5
     0.5    0

Sy = 
     0     -0.5i
     0.5i   0
 
 Sz = 
     0.5    0
     0      0.5

*Note:*

The usual cheats *are not* allowed!

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Last Solution submitted on Dec 27, 2021

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Jean-Marie Sainthillier on 17 May 2014

Not a problem for me...

Georges on 9 Sep 2014

Neither for me :D

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Problem 2316. Spin Matrices

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