2 Working with spin ½ angular momentum for a 2 particle system There are

2 working with spin ½ angular momentum for a 2

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2. Working with spin ½ angular momentum for a 2-particle system. There are several ways to find the eigenvalues and eigenvectors of a system with two spins. Here you will find the eigenvalues and eigenvectors by constructing the appropriate matrices. In question 1 above, you worked with the 2 × 2 Pauli matrices that describe spin ½ in the basis α and β . Now it is time to construct the 4 × 4 matrices which describe the spin of 2 electrons in the basis formed by αα , βα , αβ , ββ (often called the tensor product basis because it is the made from the product of the two functions for spin 1 with the two functions for spin 2) . (a) Use the 2 × 2 Pauli matrices to evaluate the 4 × 4 matrix describing ˆ S z = ˆ s z 1 + ˆ s z 2 . What are its eigenvalues and their degeneracies? (b) Use the 2 × 2 Pauli matrices to evaluate the 4 × 4 matrix describing ˆ S x = ˆ s x 1 + ˆ s x 2 (c) Use the 2 × 2 Pauli matrices to evaluate the 4 × 4 matrix describing ˆ S y = ˆ s y 1 + ˆ s y 2 (d) Use your answers to the above to make the 4 × 4 matrix describing ˆ S 2 = ˆ S ˆ S
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(e) Find the eigenvalues of ˆ S 2 = ˆ S ˆ S and the common eigenvectors of ˆ S 2 = ˆ S ˆ S and ˆ S z = ˆ s z 1 + ˆ s z 2 . You should find a singlet and the three components of a triplet, as claimed in lecture!
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  • Summer '16
  • Alistair Sinclair
  • Matrices, Eigenvalue, eigenvector and eigenspace, Use python, HFCO

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