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Ladder Operator/hermiticity

  1. Jun 13, 2008 #1
    Hello everybody, I have few queries in Quantum.Can anybody tell me:-
    1.) Ladder operators are very well defined as L+/- = LX (+/-) iLY.
    But what is its significance and from where it found its origination.
    2.) Condition for any operator being HERMITIAN, plz don’t give mathematical response of this i.e. adjoint(operator) = operator, i know that. Please tell me in sense of physically , by giving example if possible.
    With regards
     
  2. jcsd
  3. Jun 13, 2008 #2
    1) Ladder operators originate from the specific derivation you are doing, e.g. in to solution of the harmonic oscillator. The significance is that if that operator acts on your system you are raised/lowered to the next eigenstate.

    2) It looks like there's math, but this is a physical response....

    By definition, a an observable Q is something that you can directly measure, therefore it must be real. Mathematically: [tex]Q = Q*[/tex]
    But we also know that the expectation of Q is: [tex]<\psi|Q\psi>[/tex]
    But by definition: [tex]<\psi|Q\psi>=<Q\psi|\psi>[/tex]
    Which tells us that Q equals its adjoint.
     
  4. Jun 13, 2008 #3

    FNL

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    1- I think that the existance of shift operators is a simple result posed by the method of solution of some kinds of problems in QM theory
     
  5. Jun 13, 2008 #4
    Whenever you do a measurement of an observable corresponding to some operator, your measurement yields one of its eigenvalues (a postulate of QM). Hermitian operators are operators with real eigenvalues, since things we measure are real numbers. Therefore a Hermitian operator has potential to correspond to a measurable observable.
     
  6. Jun 14, 2008 #5

    malawi_glenn

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  7. Jun 14, 2008 #6
    1.) Physical states are vectors in certain representation of the symmetry group. In the case of symmetry group SO(3)(or SU(2), they share same algebra), the way to construct the irreducible representation is given by Cartan. We first choose a standard state, then consecutively operating on the standard state by certain well-designed operator. It turns out that from the Lie algebra of SO(3) group, we can construct the so-called ladder operators which operate on the standard state would generate all basis of certain irreducible representation.

    Similarly, in another physical case, the SHO, from the commutators of position and momentum operators, we can construct the similar ladder operators which transform the state to another state belonging to different energy level.

    (2) One seeming reasonable sense is that, the eigenvalues(the quantities we measured in the laboratory) of Hermitian operators are real. This is consistent with the postulates of QM. Moreover, since the generators of a unitary symmetry can be Hermitian. Hence, the symmetry would give us main origin of physical observables.
    BTW, precisely speaking, we should say the observables in QM must be represented as Hermitian operators. Not all operators in QM should be Hermitian.

    -----
    Everyone is welcome to correct my concept.
    Ismaili
     
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