Phy5645/Transformations and Symmetry Problem

Symmetries(Problem taken from a quantum assignment in the Department of Physics, UF)

Problem

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

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

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 the first equation 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

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

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 the first equation 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.

Solution

$T|n\rangle =|n+1\rangle$

$\langle i|T|j\rangle =\delta _{1,j+1}.$ So

$T=\sum _{n=1}^{N}|n+1\rangle \langle n|$ with $|N+1\rangle =|1\rangle$

$T^{\dagger }=\sum _{n=1}^{N}|n\rangle \langle n+1|$ So

$TT^{\dagger }=\sum _{m=1}^{N}\sum _{n=1}^{N}(|m+1\rangle \langle m|)(|n\rangle \langle n+1|)=\sum _{m=1}^{N}\sum _{n=1}^{N}\delta _{m,n}|m+1\rangle \langle n+1|$

$TT^{\dagger }=\sum _{n=1}^{N}|n\rangle \langle n|=\left[I\right]_{NXN}$

So $T$ is unitary.

$\left[T,H\right]=\sum _{m=1}^{N}\sum _{n=1}^{N}[(|m+1\rangle \langle m|)(|n\rangle \langle n+1|+|n+1\rangle \langle n|)-(|n\rangle \langle n+1|+|n+1\rangle \langle n|)(|m+1\rangle \langle m|)]$

$=\sum _{n=1}^{N}[|n+1\rangle \langle n+1|+|n+2\rangle \langle n|-|n\rangle \langle n|-|n+1\rangle \langle n-1|]$

$=\sum _{n=2}^{N+1}|n\rangle \langle n|+\sum _{n=2}^{N+1}|n+1\rangle \langle n-1|-\sum _{n=1}^{N}|n\rangle \langle n|-\sum _{n=1}^{N}|n+1\rangle \langle n-1|$

$=\sum _{n=1}^{N}[|n+2\rangle \langle n|-|n+1\rangle \langle n-1|]$ as $|N+1\rangle =|1\rangle$