Quantized energy in infinite potential well

rozan977

How does energy become quantized in an infinite potential well??

alxm

How do the harmonics of a finite string become quantized?

Feldoh

Honestly from a mathematical standpoint it comes about due to the boundary conditions of the Schrodinger equation for an infinite potential well.

Bob_for_short

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

As the frequencies of a string in a guitar [tex]E_n = h v_n = n h v [/tex]

Halcyon-on

That is through (periodic) boundary conditions.

Halcyon-on

For this aspect, Pythagoras was the first to study a problem of QM mechanics.

rozan977

But what if the solution we assume of Schrodinger equation be in exponential form??

Halcyon-on

it is completely equivalent.

Bob_for_short

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).