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Quantized energy in infinite potential well

  1. Sep 14, 2009 #1
    How does energy become quantized in an infinite potential well??
     
  2. jcsd
  3. Sep 14, 2009 #2

    alxm

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    How do the harmonics of a finite string become quantized?
     
  4. Sep 14, 2009 #3
    Honestly from a mathematical standpoint it comes about due to the boundary conditions of the Schrodinger equation for an infinite potential well.
     
  5. Sep 14, 2009 #4
    The proper functions (eigenfunctions) of the Hamiltonian are sin(πnz) where z is the dimensionless length z=x/L. The proper values (eigenvalues) are proportional to (πn)2.

    Any, I repeat, any wave inside the well can be decomposed in a sum of proper waves with some amplitudes. In general case the wave energy is not certain but dispersed. Only in the eigenstates the energy is certain.
     
  6. Sep 14, 2009 #5
    As the frequencies of a string in a guitar [tex]E_n = h v_n = n h v [/tex]


    620px-Harmonic_partials_on_strings.svg.png
     
  7. Sep 14, 2009 #6
    That is through (periodic) boundary conditions.
     
  8. Sep 14, 2009 #7
    For this aspect, Pythagoras was the first to study a problem of QM mechanics.

    http://img523.imageshack.us/img523/5874/pitagoradagafuriotheorixk5.jpg [Broken]
     
    Last edited by a moderator: May 4, 2017
  9. Sep 14, 2009 #8
    But what if the solution we assume of Schrodinger equation be in exponential form??
     
  10. Sep 15, 2009 #9
    it is completely equivalent.
     
  11. Sep 15, 2009 #10
    In order to satisfy the boundary conditions the two complex exponentials have to have certain coefficients that make their sum to be sin(pi*n*x/L).
     
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