Phy5645/Transformations and Symmetry Problem

Symmetries(Problem taken from an assignment of the Department of Physics, UF)

Consider an $N$ state system with the states labeled as $|1\rangle ,|1\rangle ,...,|N\rangle$. Let the hamiltonian for this system be

$\sum _{n=1}^{N}(|n\rangle \langle n+1|+|n+1\rangle \langle n|$ (1)

Notice that the Hamiltonian, in this form, is manifestly hermitian. Use periodic boundary condition, i.e, $|N+1\rangle =|1\rangle$. You can think of these states as being placed around a circle.

(a) Define the translation operator, $T$ as taking $|1\rangle \to |2\rangle ,|2\rangle \to |3\rangle ,...,|N\rangle \to |1\rangle$ .Write T in a form like $H$ in Eq. (1) and show that $T$ is both unitary and commutes with $H$ . It is thus a symmetry of the hamiltonian.

(b) Find the eigenstates of T by using wavefunctions of the form

$|\psi \rangle =\sum _{n=1}^{N}e^{ikn}|n\rangle$ (2)

What are the eigenvalues of these eigenstates? Do all these eigenstates have to be eigenstates of $H$ as well? If not, do any of these eigenstates have to be eigenstates of $H$? Explain your answer.

(c) Next Consider $F$ which takes $|n\rangle \to |N+1-n\rangle .$ Write F in a form like $H$ in Eq.(1) and show that $F$ both is unitary and commutes with $H$. It is thus a symmetry of the hamiltonian.

(d) Find a complete set of eigenstates of $F$ and their eigenvalues. Do all these eigenstates have to be eigenstates of $H$ as well? If not, do any of these eigenstates have to be eigenstates of $H$? Explain your answer.