p1-342 - Problem Set 1 Physics 342 Due January 14 Some...

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Problem Set 1 Physics 342 Due January 14 Some abbreviations: S - Shankar. 1 . The wave function of a particle in three dimensions is given by ψ = ( x + y + 3 z ) f ( r ) with r 2 = x 2 + y 2 + z 2 . What are the possible values of L 2 and L z which could be measured, and what is the probability of measuring each value? 2 . Three spin 1 / 2 particles have spins ~ S 1 , ~ S 2 , ~ S 3 . What are the possible eigenvalues of ~ S 2 where ~ S = ~ S 1 + ~ S 2 + ~ S 3 ? What are the multiplicities of each eigenvalue? 3 . A system of two distinguishable spin 1 / 2 particles with spin operators ~ S 1 and ~ S 2 is described by the Hamiltonian H = λ ~ S 1 · ~ S 2 where λ is a constant. (i) What are the possible eigenvalues of H ? (ii) Construct the eigenstates of H as linear combinations of the eigenstates of ~ S 2 1 , ~ S 2 2 , S 1 z and S 2 z . (iii) Now suppose that the two particles are placed in a uniform magnetic field ~ B = B ˆ z and that they have opposite gyromagnetic ratios so that the new Hamiltonian is H 0 = λ ~ S 1 · ~ S 2 + gB ( S 1 z - S 2 z ) .

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