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Quantum Mechanics 3, Spring 2012 CMI Problem set 2 Due by beginning of class on Monday Jan 16, 2012 Adiabatic approximation & Spin 1. Show that the geometric phase angle occurring in the adiabatic theorem Z t ∂ψn (t0 ) G θn (t) = γ = hψn (t0 ) | i i dt0 . ∂t0 0 (1) is real, so that we are justified in calling it a phase angle. Here ψn (t) are orthonormal eigenstates of the hamiltonians H(t) for each t with eigenvalues En (t). 2. With the same notation as above, show that Ėn = hψn |Ḣ|ψn i. (2) 3. Find a matrix representation of the component of spin S~ · n̂ in the direction of the unit vector n̂ = (n x , ny , nz ), for a spin half particle. 4. Find the eigenvalues of the component of spin S~ · n̂ in any direction n̂ for a spin-half particle by evaluating the square of this operator and its trace. 5. Find the corresponding normalized eigenvectors of the component of spin S~ · n̂ where n̂ is a unit vector with spherical polar coordinates (1, θ, φ). 1