Question

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?

Short answer

Answer:

There is a variation in wavelength from the middle to the extremes because of variation in kinetic energy and hence variation in momentum.

Step-by-step solution

Wavelength

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

$p=\sqrt{2m(KE)}$

Variations in Wavelengths

And we know that momentum is inversely proportional to wavelength by the formula

$\left(\lambda =\frac{h}{p}\right)$

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.