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I What does a zero-value of the Born-condition mean?

  1. Jan 3, 2018 #1

    SeM

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    Hi, I was wondering what is the physical meaning of these integrals:

    \begin{equation}
    \int_{-\infty}^{\infty} \psi \psi^* dx = 0
    \end{equation}


    \begin{equation}
    \int_{-\infty}^{\infty} \psi x \psi^* dx = 0
    \end{equation}

    \begin{equation}
    \int_{-\infty}^{\infty} \psi p \psi^* dx = NA
    \end{equation}

    ?
     
    Last edited by a moderator: Jan 3, 2018
  2. jcsd
  3. Jan 3, 2018 #2

    Demystifier

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    Where did you find this expression?
     
  4. Jan 3, 2018 #3

    SeM

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    By a function I have been calculating on
     
  5. Jan 3, 2018 #4

    Demystifier

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    What is ##t##? Time? 3-volume?
     
  6. Jan 3, 2018 #5

    SeM

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    Sorry, I should have used x. Corrected.
     
  7. Jan 3, 2018 #6

    Demystifier

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    (1) implies ##\psi=0##.
    (2) means that average position is zero.
    (3) means that the average momentum is not well defined.
     
  8. Jan 3, 2018 #7

    SeM

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    This part is not clear to me. Evidently, Psi and Psi* are hermitian counterparts, so their multiplication with one another ends up with zero.

    This is OK. But does that mean the wavefunction has not physical application?

    This is fine. Does normalization solve this?
     
  9. Jan 3, 2018 #8

    Demystifier

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    No. For properly normalized wave functions you should have ##\int \psi^*\psi dx=1##.

    No, such wave functions have a lot of applications. For instance, the ground state of harmonic oscillator has zero average position.

    Probably not. Consider, for example, the delta-function ##\delta(x)##. It has undefined average momentum and change of normalization does not help.
     
  10. Jan 3, 2018 #9

    SeM

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    Is there any chance of solving this constant by setting the integral:

    \begin{equation}
    N^2 \int_0^L \psi \psi^* dx = 1
    \end{equation}

    and solve for N?

    I tried and got complex conjugates, in fact, not only one, but two.

    Thanks for this interesting point! Do you know of a paper that reports such a wavefunction?

    What does one do with such wavefunctions without a well-defined momentum and kinetic energy?
     
  11. Jan 3, 2018 #10

    Demystifier

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    Yes, provided that N is assumed to be real. (Otherwise, you should write ##N^*N## instead of ##N^2##.)

    Take into account that N is assumed to be real.

    Any textbook on quantum mechanics (QM) should do. Which book on QM do you use?

    Such a ##\delta##-function is an idealization of a more realistic case of a very narrow Gaussian. Even though the idealization is not realistic, it is OK as long as you only consider positions and not momenta or energy.
     
  12. Jan 3, 2018 #11

    SeM

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    Tried and got two weird values.

    Let me share this strange wavefunction with the forum here:

    \begin{equation}
    \psi(x)= \frac{e^{-x(k_2-k_1)}\big(\sqrt{E-2\gamma^2}+\gamma i\big)}{2\sqrt{E-2\gamma^2}}-\frac{e^{-x(k_1+k_2)}\big(-\sqrt{E-2\gamma^2}+\gamma i\big)}{2\sqrt{E-2\gamma^2}}
    \end{equation}



    Molecular Quantum Mechanics, there they normalize it, and get nice eigenfunctions. This function however, gave two horrible Normalization constants, of the length of half a page, and cannot really give a worse starting point for generating higher energy levels.
     
  13. Jan 3, 2018 #12

    Demystifier

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    In your wave function I see only one (not two) implicit values of N, and I see nothing strange with it.
     
  14. Jan 3, 2018 #13

    SeM

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    This function is without N. I had two initial conditions for the original ODE, first y(0)=1 and y'(0)=0 , and got this function. This function follows the integrals given in the first post.

    Therefore I was looking for adding a normalization constant to the integrals.
     
  15. Jan 3, 2018 #14

    Demystifier

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    This function does not satisfy your Eq. (1).
     
  16. Jan 3, 2018 #15

    Demystifier

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    Author please!
     
  17. Jan 3, 2018 #16

    SeM

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    Atkins and Friedman
     
  18. Jan 3, 2018 #17

    SeM

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    you need k_1 and k_2, these are:


    \begin{equation}
    k_1 = {\frac{\,\sqrt{E-2\,\gamma^2}}{\hbar}}
    \end{equation}

    \begin{equation}
    k_2 = \frac{\gamma i}{\hbar}
    \end{equation}


    ##\gamma = 5^{-28}## and E is the zero point energy. If you still get that integral 1 is not satisfied, then MATLAB is crap
     
  19. Jan 3, 2018 #18

    Demystifier

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    Are you a chemist? I ask because it looks like you miss the basic foundations of QM. In any case, study first Chapters 1 and 2 of the book. For instance, in Fig. 2.27 (fourth edition) all wave functions have zero average position.
     
  20. Jan 3, 2018 #19

    SeM

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    Yes, but this model has not potential energy. and Yes I am a chemist.

    What can I do to develop this model further as it does not follow the Hamiltonian on p 55, chapter 2, where fig 2.27 is.

    Thanks!
     
  21. Jan 3, 2018 #20

    Demystifier

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    We must be using different editions of the book. Mine is the 4th edition. In any case, try to compute ##\int \psi^*\psi dx## again and show that it is a positive real number. You cannot make any progress before you do that.
     
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