rozan977
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How does energy become quantized in an infinite potential well??
Quantized energy in infinite potential well
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How do the harmonics of a finite string become quantized?
Feldoh
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Honestly from a mathematical standpoint it comes about due to the boundary conditions of the Schrödinger equation for an infinite potential well.
Bob_for_short
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rozan977 said:
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.
Halcyon-on
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rozan977 said:
As the frequencies of a string in a guitar [tex]E_n = h v_n = n h v[/tex]
Halcyon-on
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That is through (periodic) boundary conditions.
Halcyon-on
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For this aspect, Pythagoras was the first to study a problem of QM mechanics.
http://img523.imageshack.us/img523/5874/pitagoradagafuriotheorixk5.jpg
http://img523.imageshack.us/img523/5874/pitagoradagafuriotheorixk5.jpg
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rozan977
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Feldoh said:
But what if the solution we assume of Schrödinger equation be in exponential form??
Halcyon-on
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rozan977 said:
it is completely equivalent.
Bob_for_short
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rozan977 said:
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).