Quantum Physics Tutorial 11 - Questions PDF

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Questions from the eleventh tutorial covering the quantum physics part of the course....

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PH352 Quantum Physics. Tutorial 11: Spin ½ and Pauli matrices Planck’s constant / 2π = ħ = 1.055x10-34 Js Electronic charge e = 1.6 10-19 C Mass of the electron = M = 9.1 10-31 Kg Permittivity = ε0 = 8.854x10-12 Fm-1 Consider the Pauli spin ½ matrices: 𝑆! =

ℏ !

0 1

1 ℏ 0 ,!!!!!𝑆! = ! 0 𝑖

−𝑖 ℏ 1 ,!!!!!𝑆! = ! 0 0

0 !ℏ! 1 ,!!!!!!𝑆 ! = ! 0 −1

0 1

1) Determine the eigenvalues and eigenvectors of the matrices 𝑆! and 𝑆! . Normalise the eigenvectors. 2) Can you express the eigenvectors of 𝑆! and 𝑆! as a function of the 1 0 eigenvectors (known as eigenspinors) |+>= of the and |−>= 0 1 matrices 𝑆 ! !and!𝑆! ? 3) Explain the physical meaning of eigenvalues and eigenvectors of each Pauli matrix. 4) What are the commutators 𝑆 ! , 𝑆! ,!!!!! 𝑆 ! , 𝑆! ,!!!!! 𝑆 ! , 𝑆! ? 5) What are the commutators 𝑆! , 𝑆! ,!!!!! 𝑆! , 𝑆! ,!!!!! 𝑆! , 𝑆! ? 6) From the answers of questions 4) and 5) what can you say about the (unknown) operators represented by the matrices 𝑆 ! !and!𝑆! ?...