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Electron Spin Statistics and Pauli Matrices

John Deeble Jonathan Hurowitz Friday June 25, 2010. Electron Spin Statistics and Pauli Matrices. Electron Spin. Electron Spin is measured by the direction of an electron after being exposed to a magnet. Electron Spin can only be measured from one perspective (or a vector) at a time.

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Electron Spin Statistics and Pauli Matrices

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  1. John Deeble Jonathan Hurowitz Friday June 25, 2010 Electron Spin Statistics and Pauli Matrices

  2. Electron Spin • Electron Spin is measured by the direction of an electron after being exposed to a magnet. • Electron Spin can only be measured from one perspective (or a vector) at a time. • From this vector, the electron can only be seen as having spin up, or spin down.

  3. Experimental Observation: • Given unit vectors V and W that are Θ degrees apart, what is the probability that vector W will have the same direction of spin as vector V? • Experimentally the answer is cos2(Θ/2)

  4. Examples • Case I: Θ = 0, the same direction • cos^2(Θ/2)= 1 • Case II: Θ = 90, the perpendicular direction • cos^2(Θ/2)= 1/2 • Case III: Θ=180, the opposite direction • cos^2(Θ/2)= 0 • But how do achieve that value mathematically?

  5. Functions of a Matrix • Trace [Tr] : a + d • Determinant [Det] : ad-bc

  6. What Defines a Pauli Matrix? • A Pauli Matrix is a matrix with these properties: • Dimensions: 2x2 • Trace: 0 • Determinant: -1

  7. Pauli Matrices (and Identity)

  8. Finding a Matrix • Matrix Formula: ½*[I+ σ1v1 +σ2v2 + σ3v3 ], v1, v2, and v3 are x, y, and z coordinates of vector V on the unit sphere. • Matrix: [(1+v3)/2 (v1-iv2)/2] [(v1+iv2)/2 (1-v3)/2] • For any given vector W on the unit sphere: • Replace v1, v2, and v3 with w1, w2, and w3 • Vector W Matrix: [(1+w3)/2 (w1-iw2)/2] [(w1+iw2)/2 (1-w3)/2]

  9. What Now? • To find the probability, take Tr(Pv*Pw) • The trace of the matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. • Physics Background: The eigenvalues of all the Pauli Matrices are 1 and -1, for up and down respectively.

  10. The dot product: • (v1,v2,v3) · (w1,w2,w3) = v1w1 + v2w2 + v3w3 • (v1,v2,v3)=vector V • (w1,w2,w3)=vector W • V*W=||V||*||W||*cos(Θ), • ||V|| denotes the length, or magnitude, of V.

  11. The Answer: • cos2(Θ/2)

  12. Algebraic Property of Pauli Matrices • Question: Can any complex 2x2 matrix be constructed by only adding, subtracting, and multiplying Pauli matrices? (this includes multiplying by real scalars) • Short Answer: YES!!! • But how can we do this?

  13. Approach • Break into individual matrices • Add them

  14. Acknowledgements • Dr. Ambar Sengupta • Qingxia Li • Xinyao Yang • Rick Barnard • Alex Frieden • Lee Windsperger • Susan Abernathy

  15. Bibliography • http://static.howstuffworks.com/gif/quantum-computer-2.jpg • https://www.etap.org/demo/algebra2/Image59.gif • http://demonstrations.wolfram.com/PauliSpinMatrices/HTMLImages/index.en/popup_16.jpg

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