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...) |
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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 state system with the states labeled as . Let the hamiltonian for this system be
Notice that the Hamiltonian, in this form, is manifestly hermitian. Use periodic boundary condition, i.e, . 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 .Write T in a form like in the first equation and show that is both unitary and commutes with . It is thus a symmetry of the hamiltonian.
(b) Find the eigenstates of T by using wavefunctions of the form
What are the eigenvalues of these eigenstates? Do all these eigenstates have to be eigenstates of as well? If not, do any of these eigenstates have to be eigenstates of ? Explain your answer.
(c) Next Consider which takes Write F in a form like 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 . It is thus a symmetry of the hamiltonian.
(d) Find a complete set of eigenstates of and their eigenvalues. Do all these eigenstates have to be eigenstates of 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.