Spring 2021 Homework Assignments
Suppose you have two different angular momenta corresponding to total angular momentum values ℓ1 = 3/2 and ℓ2 = 1 (i.e. they are the states that make up the 3/2 x 1 portion of the Clebsch-Gordan table.) (a) List all outer-product states that are possible. (b) List all total angular momentum states that are possible. (c) Choose a non-trivial case (i.e. the expansion should have more than one term) and give an expansion of any one of the outer product states in terms of the total angular momentum states. (d) Choose a non-trivial case and give an expansion of any one of the total angular momentum states in terms of the outer product states.
Suppose you are given two electrons in the singlet state ( |↑z> |↓z> - |↓z> |↑z> ) / √2 . Find an expansion of this state in terms of |↑n> and |↓n>, the eigenstates of the spin operators in an arbitrary direction n,
Consider the problem of the two particles in the ground state of a quantum box of size L with the unperturbed wavefunctions Ψnm(0)(x1,x2) = 2/L sin(nπx1/L) sin(mπx2/L) for 0 < x1,x2 < L and zero otherwise. A perturbation of the form H1 = α δ( x1 - x2 ) with α = ℏ2/(2mL) is also present (this is the problem for which we found the first order shift in the ground state energy). (a) Find the first order correction to the ground state wavefunction. (b) Keeping a few of the terms with the smallest denominators in the series expansion, plot |Ψ11(0) + Ψ11(1)|2 as a function of x1 and x2.
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