1. Joined
    08 Oct '06
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    30 Mar '10 06:37
    Assume that H is an operator valid on some hilbert space, N, with nondegenerate eigenvalues, E_n, whose corresponding eigenbasis is {g_n}. If f is a vector that lies on N, then f can be expanded as a linear combination of the eigenbasis of H. Now mathematically, we can say that

    H|f> = sum(n, E_n*c_n*|g_n>😉.

    This implies that H changes the the vector, but it is still expanded in the entire eigenbasis. However, if we consider this problem from a physical perspective, |f> collapses into one of the eigenstates it is expressed in, or

    H|f> ---> E_n*c_n|g_n>

    where there is no summation. Which view is correct?
  2. Germany
    Joined
    27 Oct '08
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    30 Mar '10 10:181 edit
    H doesn't necessarily have to collapse f into one of the eigenstates, it can be a linear combination of eigenstates.

    In fact for most "measurement" processes you will not be able to find a well-defined H that does what you just described.
  3. Joined
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    30 Mar '10 16:46
    I am under the impression that an observation on a vector returns a single value. If f is expressed in the eigenbasis of H, and the eigenvalues of H are non-degenerate, wouldn't this force f to return an eigenvalue of H, and therefore collapse it into an eigenstate of H.

    For instance, assume |f> = c_1|g_1> + c_2|g_2>.

    Also assume P is an operator such that

    P|g_1> = 1
    P|g_2> = -1

    Then P|f> must return either 1 or -1, correct? Then for instance, if P|f> = 1 wouldn't |f> have collapsed into |g_1>?
  4. Germany
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    30 Mar '10 18:17
    H acting on f is not an observation. H describes how f evolves in time.
  5. Joined
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    30 Mar '10 18:20
    This is where it gets confusing for me. I understand That H|f>=E|f> describes the evolution of the wave-function. However, I was also under the impression that a hermitian operator acting on a state-vector can be seen as an observation. I am having a hard time making these two views coincide. I am very confused by all this, sorry...
  6. Germany
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    30 Mar '10 19:381 edit
    Originally posted by amolv06
    This is where it gets confusing for me. I understand That H|f>=E|f> describes the evolution of the wave-function. However, I was also under the impression that a hermitian operator acting on a state-vector can be seen as an observation. I am having a hard time making these two views coincide. I am very confused by all this, sorry...
    Umm... I'm not quite sure but I don't think you can see that as an observation. However, a hermitian operator is an observable, if you take the expectation value of H you get the energy of the system.

    And the more confused you get, the more you understand... 😉
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