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I Symmetric, self-adjoint operators and the spectral theorem

  1. Jan 20, 2017 #1
    Hi Guys,

    at the moment I got a bit confused about the notation in some QM textbooks. Some say the operators should be symmetric, some say they should be self-adjoint (or in many cases hermitian what maybe means symmetric or maybe self-adjoint). Which condition do we need for our observables (cause they are not the same in the case of an infinite-dimensional Hilbertspace)?

    If symmetric is enough why can we find an othonormal basis of eigenvectors (since the spectral theorem holds only for self-adjoint operators)?

    Thanks for your help
     
  2. jcsd
  3. Jan 20, 2017 #2

    dextercioby

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    The observables should be self-adjoint operators, but essential self-adjointness would do. There's no guarrantee for a purely real spectrum for a symmetric operator.
     
  4. Jan 21, 2017 #3

    vanhees71

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  5. Jan 21, 2017 #4
    Thanks for your answers. The fact that the operator should be self-adjoint makes sense but there is one problem left.

    If we assume that all the operators are self-adjoint and not defined everywhere (since they are unbounded) how can we make sure that the products of operators are well-definied, i.e. why is the image of the first in the domain of the second and so on?
     
  6. Jan 22, 2017 #5

    dextercioby

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    This is a very good question. Self-adjoint operators won't typically have the same domain, but it may happen that the maximal common domain of them is an essential self-adjointness domain and moreover this domain is also invariant for the polynomial algebra.
    Example: the Schwartz test function space in R is a common dense everywhere invariant domain for the x, p and p^2 (this is the free particle Hamiltonian) operators. All 3 of them are esa when restricted to this space.
     
  7. Jan 22, 2017 #6

    A. Neumaier

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    In general this is not the case and the product need not exist. However, in most quantum theories of interest, there is an algebra of operators of interest with a fixed common dense domain (for ##N##-particle QM, the Schwartz space in ##3N## dimensions).
     
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