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The chain of pain: Consider the tight-binding problem in the Sakurai book, but now the "chain" is finite:
- (NI
- This chain has boundary conditions so that |N + 1) = | - N) and | - N - 1) = |N). This can be used to define the action of the lattice translation operator t(a) on each (n) as t(a)(n) = (n + 1). With a small
- adjustment this problem can be mapped to the previous problem. Questions
- :a) Does the parity operator I1 commute with the discrete translation operator t(a)? Parity in this case is such that II|n) = | - n). b) Write down the eigenstates of t(a) and II as linear combinations of (n). c) Does the tight-binding Hamiltonian H commute with t(a) and II? d) Are the eigenstates of t(a) also eigenstates of I1? Justify your answer explicitly by applying II to Ik), where |k) is an eigenstate of t(a). e) Set N = 6. Can you "map" this problem to the previous problem in the homework? If your answer is yes, how would you do that?
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Question Text | The chain of pain: Consider the tight-binding problem in the Sakurai book, but now the "chain" is finite: |
Topic | All topics |
Subject | Physics |
Class | Class 11 |