Phy5645/Transformations and Symmetry Problem: Difference between revisions
SohamGhosh (talk | contribs) (New page: ===Symmetries(Problem taken from an assignment of the Department of Physics, UF)=== Consider an <math>N</math> state system with the states labeled as <math>|1\rangle , |1\rangle , ..., |N...) |
SohamGhosh (talk | contribs) No edit summary |
||
| Line 2: | Line 2: | ||
Consider an <math>N</math> state system with the states labeled as <math>|1\rangle , |1\rangle , ..., |N\rangle</math>. Let the hamiltonian for this system be | Consider an <math>N</math> state system with the states labeled as <math>|1\rangle , |1\rangle , ..., |N\rangle</math>. Let the hamiltonian for this system be | ||
<math>\sum_{n=1}^{N} (|n\rangle \langle n+1| + |n+1\rangle \langle n|</math> | <math>\sum_{n=1}^{N} (|n\rangle \langle n+1| + |n+1\rangle \langle n|</math> | ||
Notice that the Hamiltonian, in this form, is manifestly hermitian. Use periodic boundary condition, i.e, <math>|N+1\rangle = |1\rangle</math>. You can think of these states as being placed around a circle. | Notice that the Hamiltonian, in this form, is manifestly hermitian. Use periodic boundary condition, i.e, <math>|N+1\rangle = |1\rangle</math>. You can think of these states as being placed around a circle. | ||
(a) Define the translation operator, <math>T</math> as taking <math>|1\rangle \to |2\rangle, |2\rangle \to |3\rangle ,...,|N\rangle \to |1\rangle</math> .Write T in | (a) Define the translation operator, <math>T</math> as taking <math>|1\rangle \to |2\rangle, |2\rangle \to |3\rangle ,...,|N\rangle \to |1\rangle</math> .Write T in | ||
a form like <math>H</math> in | a form like <math>H</math> in the first equation and show that <math>T</math> is both unitary and commutes with <math>H</math> . It is thus a symmetry of the hamiltonian. | ||
(b) Find the eigenstates of T by using wavefunctions of the form | (b) Find the eigenstates of T by using wavefunctions of the form | ||
<math>|\psi \rangle = \sum_{n=1}^{N} e^{ikn}|n\rangle</math> | <math>|\psi \rangle = \sum_{n=1}^{N} e^{ikn}|n\rangle</math> | ||
What are the eigenvalues of these eigenstates? Do all these eigenstates have to be eigenstates of <math>H</math> as well? If not, do any of these eigenstates have to be eigenstates of <math>H</math>? Explain your answer. | What are the eigenvalues of these eigenstates? Do all these eigenstates have to be eigenstates of <math>H</math> as well? If not, do any of these eigenstates have to be eigenstates of <math>H</math>? Explain your answer. | ||
(c) Next Consider <math>F</math> which takes <math>|n\rangle \to |N+1-n\rangle.</math> Write F in a form like <math>H</math> in | (c) Next Consider <math>F</math> which takes <math>|n\rangle \to |N+1-n\rangle.</math> Write F in a form like <math>H</math> in the first equation and show that <math>F</math> both is unitary and commutes with <math>H</math>. It is thus a symmetry of the hamiltonian. | ||
unitary and commutes with <math>H</math>. It is thus a symmetry of the hamiltonian. | |||
(d) Find a complete set of eigenstates of <math>F</math> and their eigenvalues. Do all these eigenstates have to be eigenstates of <math>H</math> as well? If not, do any of these eigenstates have to be eigenstates of <math>H</math>? Explain your answer. | (d) Find a complete set of eigenstates of <math>F</math> and their eigenvalues. Do all these eigenstates have to be eigenstates of <math>H</math> as well? If not, do any of these eigenstates have to be eigenstates of <math>H</math>? Explain your answer. | ||
Revision as of 19:00, 9 December 2009
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|}
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.