In the harmonic oscillators eave functions of figure there is variation in wavelength from the middle of the extremes of the classically allowed region, most noticeable in the higher-n functions. Why does it vary as it does?
There is a variation in wavelength from the middle to the extremes because of variation in kinetic energy and hence variation in momentum.
The wavelength gets longer near the extreme edges because the kinetic energy is low there.
As the Kinetic Energy is low, so the momentum is small. Because, we know that Kinetic energy is directly proportional to momentum by the following formula
And we know that momentum is inversely proportional to wavelength by the formula
So, a low kinetic energy corresponds to a longer wavelength.
Hence, Variation of wavelength from middle to extremes is seen because of variation in kinetic energy.
In several bound systems, the quantum-mechanically allowed energies depend on a single quantum number we found in section 5.5 that the energy levels in an infinite well are given by , where and is a constant. (Actually, we known what is but it would only distract us here.) section 5.7 showed that for a harmonic oscillator, they are , where (using an with n strictly positive is equivalent to with n non negative.) finally, for a hydrogen atom, a bound system that we study in chapter 7, , where consider particles making downwards transition between the quantized energy levels, each transition producing a photon, for each of these three systems, is there a minimum photon wavelength? A maximum ? it might be helpful to make sketches of the relative heights of the energy levels in each case.
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