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Unformatted text preview: Problem Set 08 Note: The following problem set is due November 07 by midnight. Please return directly to me in my office Rutherford 321. If I’m not there slip your assignment below my door. 1. A two dimensional Hilbert space is spanned by states: | ϕ 1 i and | ϕ 2 i that are eigenvec- tors of an operator H with eigenvalues E 1 and E 2 . Imagine that I add a small shift to the operator H via a matrix W : W = W 11 W 12 W * 12 W 22 ! with W 11 and W 22 real, such that the shifted operator is H ≡ H + W . Clearly for this shift, the states | ϕ 1 i and | ϕ 2 i are no longer eigenstates of the new operator H . Let us denote the new eigenstates by | ψ ± i . Determine these new eigenstates and the new eigenvalues of H . Show that you can always express these states as: | ψ + i | ψ- i = cos θ 2 e- iϕ/ 2 sin θ 2 e iϕ/ 2- sin θ 2 e- iϕ/ 2 cos θ 2 e iϕ/ 2 ! | ϕ 1 i | ϕ 2 i as long as the Hamiltonian H is hermitian. The angles θ (0 ≤ θ < π ) and ϕ are defined in the following way:...
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This note was uploaded on 12/15/2010 for the course PHYS 357 taught by Professor Keshavdasgupta during the Fall '05 term at McGill.
- Fall '05