Phy5645/Transformations and Symmetry Problem

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

Consider an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} state system with the states labeled as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1\rangle , |1\rangle , ..., |N\rangle} . Let the hamiltonian for this system be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |N+1\rangle = |1\rangle} . You can think of these states as being placed around a circle.

(a) Define the translation operator, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} as taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1\rangle \to |2\rangle, |2\rangle \to |3\rangle ,...,|N\rangle \to |1\rangle} .Write T in a form like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} in Eq. (1) and show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} . It is thus a symmetry of the hamiltonian.

(d) Find a complete set of eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} and their eigenvalues. Do all these eigenstates have to be eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} as well? If not, do any of these eigenstates have to be eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} ? Explain your answer.