Phy5645/Transformations and Symmetry Problem

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

Problem

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} . Consider 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} to be even. 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|}

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 the first equation 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 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.

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

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 |\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 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.

(c) Next Consider 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} which takes 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\rangle \to |N+1-n\rangle.} Write F 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 the first equation 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 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.

(a)

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|n\rangle = |n+1\rangle}

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 \langle i|T|j\rangle = \delta_{1,j+1}.} So

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 = \sum_{n=1}^{N} |n+1\rangle\langle n|} 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 |N+1\rangle = |1\rangle}

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^{\dagger} = \sum_{n=1}^{N} |n\rangle\langle n+1|} So

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 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|}

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

So 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 unitary.

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 \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|)]}

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+1\rangle\langle n+1| + |n+2\rangle\langle n| - |n\rangle\langle n| - |n+1\rangle\langle n-1|]}

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=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|}

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+2\rangle\langle n| - |n+1\rangle\langle n-1|]} 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 |N+1\rangle = |1\rangle}

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 = 0 } So 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} commutes with the Hamiltonian.

(b)

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 \psi \rangle = \sum_{n=1}^{N} e^{ikn}|n\rangle}

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\psi \rangle = \sum_{m=1}^{N}\sum_{n=1}^{N}|m+1\rangle\langle m| e^{ikn}|n\rangle}

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}\sum_{m=1}^{N}e^{ikn}|m+1\rangle \delta{mn}}

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} e^{ikn}|n+1\rangle}

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=2}^{N+1} e^{ik(n-1)}|n\rangle}

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 e^{-ik}[\sum_{n=2}^{N} e^{ikn}|n\rangle + e^{ik(13)}|1\rangle}

Now the state 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 \to |1\rangle} but the coefficient remains 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 e^{13ik}} We need that to 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 e^{ik(1)}|n\rangle}

So 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 e^{12ik} = e^{2\pi in}}

So 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 k = n\pi/6 , n = 1,2,....,11}

Hence 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 \psi\rangle} is an eigenstate 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 T} with eigenvalue 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 e^{-ik}}

Since $H,T$ commute, any eigenstate of $T$ is an eigenstate of $H$

explicit proof:

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

$|\sum _{n=1}^{N}\sum _{m=1}^{N}e^{ikm}|n\rangle \langle n+1|n\rangle +\sum _{n=1}^{N}\sum _{m=1}^{N}e^{ikm}|n+1\rangle \langle n|n\rangle$

$=2\cos(k)|\psi \rangle$

So they are eigenstates of $H$ also.

(c)

$F|j\rangle =|n+1-j\rangle$

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

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

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

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

$=1$

So $F$ is unitary.

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

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

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

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

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

Hence they commute.

(d)

It has already been proved that F is both unitary and hermitian. Thus the eigenvalues of F are given by

$\pm 1$

Since $F|n\rangle =|N+1-n\rangle$ by intuition let us construct states given by

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 |\psi\pm\rangle = |n\rangle \pm |N+1-n\rangle n = 1,2,3,….N/2. }

$F|\psi \pm \rangle =|N+1-n\rangle \pm |N-N+n\rangle$

$=\pm (|n\rangle \pm |N+1-n\rangle )$

$=\pm |\psi \pm \rangle$

Thus $|\psi \pm \rangle$ thus constructed are eigenstates of F. It is noted that we have $N/2$ symmetric eigenstates given by

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 + |N\rangle ; |2\rangle + |N-1\rangle ; |3\rangle +|N-2\rangle … } and $N/2$ anti-symmetric eigenstates given by

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 - |N\rangle ; |2\rangle - |N-1\rangle ; |3\rangle -|N-2\rangle ; … }

$H(|n\rangle \pm |N+1-n\rangle )$

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

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

$=|n+1\rangle +|n-1\rangle \pm |N+2-n\rangle \pm |N-n\rangle$

Thus we find that the eigenstate of F may not necessarily be eigenstates oh H. This is because F is degenerate. However there may exist linear superposition of eigenstates of F which is also an eigenstates of H and vice versa.