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I Physical interpretation of a Hamiltonian with a constraint

  1. Dec 14, 2017 #1
    Dear physics forums,

    What is the physical interpretation of imposing the following constrain on a Hamiltonian:
    [tex] Tr(\hat H^2)=2\omega ^2 [/tex]
    where [itex]\omega[/itex] is a given constant. I am not very familiar with why is the trace of the hamiltonian there.

    Thanks in advance,
    Last edited: Dec 14, 2017
  2. jcsd
  3. Dec 15, 2017 #2


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    The physical interpretation requires a physical context, and you didn't explain the physical context. At the very least, you should provide the reference in which this constraint is introduced.
  4. Dec 15, 2017 #3


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    Without the context, I would guess that the Hamiltonian describes a two-state system (hence the factor 2) both of which have the same energy ##\omega##.
  5. Dec 15, 2017 #4


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    In case the OP didn't know this already---for any operator, the trace is equal to the sum of the eigenvalues. So if the trace of [itex]H^2[/itex] is [itex]2\omega^2[/itex], then it means that the energy eigenvalues are such that [itex]E_1^2 + E_2^2 + ... = 2\omega^2[/itex]. Demystifier's example is one of the simplest: [itex]E_1 = E_2 = \omega[/itex]. Or it could be [itex]E_1 = \omega, E_2 = -\omega[/itex]. Or it could be [itex]E_1 = \omega, E_2 = \omega/\sqrt{2}, E_3 = \omega/\sqrt{4}, E_4 = \omega/\sqrt{8} ...[/itex]

    The sum of the eigenvalues doesn't seem to me to be a "constraint" on the Hamiltonian, it's just a fact about it.
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