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348-problem22 - responding eigenvalues How many solutions...

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Chem. 540 Instructor: Nancy Makri PROBLEM 22 Consider a system with exactly two eigenstates (e.g., an electron with spin 1 2 ). We will denote the eigenstates of the Hamiltonian 0 ˆ H for this system as 1 φ and 2 φ and the correspond- ing eigenvalues 1 ε and 2 ε . These states are orthogonal, because the Hamiltonian in a hermitian operator, and we assume that they are also normalized to 1. Thus, the spectral expansion of 0 ˆ H is 0 1 1 1 2 2 2 ˆ H ε φ φ ε φ φ = + . Now suppose that we apply an external field to this system, such that the Hamiltonian is changed to the new Hamiltonian ˆ H , which in terms of the eigenstates of 0 ˆ H has the form ( 29 0 1 2 2 1 ˆ ˆ H H φ φ φ φ = - + h where 0 is a constant. a) Find the two normalized eigenstates 1 Ψ and 2 Ψ of this new Hamiltonian and the cor-
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Unformatted text preview: responding eigenvalues. How many solutions are there for every value of Ω ? Is there a possibility for degeneracies? b) Show explicitly that the eigenstates are orthogonal to one another and that they form a complete set; i.e., they are linearly independent, and any state in this two-dimensional space can be expressed as a linear combination of 1 Ψ and 2 Ψ . c) Write down the spectral expansion of ˆ H . d) What would happen if Ω were a complex number? (Don’t try to solve the whole problem again. Just explain what this would mean physically and how the eigenstates and eigenval-ues might be qualitatively different.)...
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