Question

Physical Chemistry problem:

How does the confinement of a particle to a finite region of space lead to the quantization of its energy? You may use any combination of words, images, or equations to support your answer.

Answer 1

The word confinement means to confine the motion of randomly moving electron to restrict its motion in specific energy levels(discreteness) and quantum reflects the atomic realm of particles.

Based on de Broglie equation : p = h/lambds

h+ = h/2*pi and vector called wave vector k=2*pi/lambda which gives

p = h/2*pi x 2*pi/lambda = hk

Energy and momentum are related as E = p^2/2*m

which gives E = h^2*k^2/2*m

The energy momentum and wave verctore corresponding to a particle are related. Any restriction on one of the quantities will automatically restrict the other two as well.

Consider an analogy

Consider a string tied at both ends. Since both ends of the string are tied is that those two end strings can only support zero displacement. In other words, the waves supported by the string must have nodes at those two end points, or the waves must be such that distance between the ends of the string must correspond to the integral multiples of half wavelengths.

Consider the distance between the walls to be a, then the wavelenght supported by the string can only be of the kind that satisfies one of the following relations

a = lambda/2 ; a = lambda ; a = 3*lambda / 2 and so on

For a string tied at both ends, not all values of wavelenghts can be supported due to the constraint that the ends of the string must be nodes. Therefore only specific values of energy corresponding to these specific values of wavelength, can be supported by the string. For every dicrete value of lambds permitted (or k permitted) there is a a corresponding value of energy, which is permitted. when a = lambda/2 corresponding energy value permitted to be displayed by the string. The next nearest value of energy that can be displayed by the string will correspond to the situation a = lambda. Between these two energy levels no other energy values can be supported by the string.

The discretization of energy values is called Quantization. Confinement of the string has directly resulted in quantization.