Quantization of Energy

Energy Quantization Definition

First, let's define what exactly is energy quantization!

These discrete energy packets are known as quanta. To explain the photoelectric effect, the energy absorbed by the metal surface from the incident ray is modeled as consisting of quanta. The photoelectric effect is therefore a demonstration of the quantization of energy.

Planck defines energy quantization in terms of the quantization of electromagnetic waves.

Energy Quantization Formula

Quantum mechanics describes electromagnetic waves as a collection of small energy packets known as photons. The energy of each of these photons in terms of the frequency of the wave is \[E=h\nu,\]

where \(h=6.626\times 10^{-34}\,\mathrm{J\,s}\,\left(\mathrm{joule\,second}\right)\) is Planck's constant, and \(\nu\) is the frequency of the wave.

The speed of an electromagnetic wave in a vacuum is \(c=3\times10^{8}\,\mathrm{m\,s^{-1}}\). The relation between the speed \(\left(c\right)\), wavelength \(\left(\lambda\right)\), and frequency \(\left(\nu\right)\) of the ray is

\[c=\nu\lambda.\]

Using this relation, the energy of each quantum in terms of the wavelength is

\[E=h\frac{c}{\lambda}.\]

The above equation describes the energy of each photon, but electromagnetic waves consist of many photons. The amount of energy emitted or absorbed by the body due to the incidence of electromagnetic waves on the body is,

\[E=nh\nu,\]

where \(n\) is the number of the incident photons (integers) i.e. \(n=\pm1,\pm2,\pm3,...\)

Some examples of energy quantization are the vibrations on the harp (only specific musical notes are possible), speed boat waves (waves formed are discrete instead of in a continuum), etc.

Cause of Energy Quantization

The cause of energy quantization is the wave nature of the matter. With the dual nature of matter by De Broglie, Bohr explained the electron revolution around the nucleus in the form of its wave nature. Bohr postulated that the electrons only revolve in the orbit where their waves interfere constructively.

Fig. 2 - The figure shows the wave nature of electrons while revolving around the nucleus in the Bohr model of an atom.

Before explaining the quantization of energy with the Bohr model, we need to know about the postulates of the Bohr model of atoms.

1. Electrons revolve around the positively charged nucleus in a particular circular orbit.
2. Each circular orbit in which the electron is revolving has a fixed energy value.
3. Quantum number \(n=1,2,3,...\) denotes the circular orbit where \(n=1\) is the quantum number for the lower energy orbit (closer to the nucleus). A higher value of the quantum number represents higher energy orbits (away from the nucleus).
4. An electron in its lower energy state revolves around the nucleus without losing energy. In other words, the electron is stable while revolving around the nucleus in a lower energy state.
5. With the gain of required energy, the electrons can move from the lower energy orbit to a higher energy orbit. But due to the instability, it returns to its initial state with the emission of the same amount of gained energy.

Fig. 3 - The figure shows the transition of an electron from a higher to a lower energy orbit with the emission of quantized energy in the Bohr model of the atom.

In the Figure above, let \(n_\mathrm{i}\) and \(n_\mathrm{f}\) be an initial and a final energy orbit of the electron. Then the energy of the electromagnetic wave emitted is \(\Delta E=\left(n_{\mathrm{f}}-n_{\mathrm{i}}\right)h\nu\). Substituting the known values of the quantum number \(n_\mathrm{f}\) and \(n_\mathrm{i}\) in the equation, gives\[\begin{align*}\Delta E&=\left(3-2\right)h\nu \\&=h\nu.\end{align*}\]

This shows that the amount of energy emitted from the atom is quantized. Thus, the Bohr model proves energy quantization.

Energy Quantization Examples

An important modern example of energy quantization is the solar cell based on a well-known phenomenon, i.e. the photoelectric effect. So, let's start with the photoelectric effect.

After five years of the discovery of energy quantization by Planck, Albert Einstein explained the mysterious interaction of light and matter called the photoelectric phenomena discovered by Heinrich Rudolf Hertz. According to Albert Einstein, an incident ray of light on a metal surface behaves like a collection of small energy packets called photons. The energy of each photon is equivalent to the energy of each quantum, i.e., \(E=h\nu\), where \(\nu\) is the frequency of the incident ray.

Fig. 4 - The figure shows the photoelectric effect in which the emission of electrons from the metal's surface occurs when the photons of sufficient energy are incident on the metal surface.

Einstein postulated that each electron is bound to the metal with a characteristic energy associated with this binding called the work function \(\left(E_\circ=h\nu_\circ\right)\). Therefore, to remove the electron from the surface of the metal, the frequency of the incident ray must be equal to or greater than the threshold frequency \(\left(\nu_\circ\right)\). The maximum kinetic energy of the emitted electron in terms of the frequency of the incident ray is

\[E_\mathrm{max}=h\nu-h\nu_\circ.\]

These emitted electrons are known as photoelectrons.

So, how is this phenomenon used in solar cells? First, let's think about what we can do with the emitted electron from the metal surface. If we provide a potential difference in the setup such that all the emitted electrons start moving from one point to another, this can generate an electric current. Well, scientists use a similar process in solar cells to generate electricity from solar rays.

Importance of Energy Quantization

Energy quantization explains many phenomena that classical mechanics cannot explain. Some of these examples are listed below.

1. Bohr explained the discrete lines in the spectrum of a hydrogen atom as being caused by the transition of electrons between fixed energy orbits due to energy quantization. The energy emitted by electrons in the hydrogen atom is quantized. Thus, the spectrum of hydrogen consists of discrete lines instead of a continuum.

2. Energy quantization explains the photoelectric phenomenon, which cannot be explained by classical physics. Also, it helps in determining the maximum kinetic energy of the emitted electrons from the surface of the metal.

3. Energy quantization provides evidence for wave-particle duality; like the incident ray as a collection of photons or the wave nature of electrons revolving around the nucleus.

In conclusion, energy quantization throws light on several physical phenomena which couldn't be previously explained using classic physics theories.