3.12.A.1 Photon Absorption/Emission and Energy Change:

1. Photon Basics:

The basic particles of light and other kinds of electromagnetic waves are photons. They have certain properties that come directly from how they behave under different circumstances and include energy, wavelength, and frequency.

i. Energy (E):
Energy of a photon is directly proportional to its frequency. The formula for the energy of a photon is given below:

E=h⋅f

Where:
 E: energy of the photon
h: Planck’s constant

$6.62607015\times {10}^{-34}\text{J}$

f :frequency of the photon.

So high frequency photons such as ultraviolet or X-rays have more energy and correspondingly lower frequency photons like radio waves or microwaves have less energy.

ii. Frequency (f):
Frequency is the number of oscillations (or cycles) per second. It is measured in Hertz (Hz), where 1 Hz = 1 cycle per second. The frequency of a photon determines its color (for visible light) or its type (for electromagnetic radiation in general).

•  High-frequency photons, such as gamma rays, X-rays have high energy.
•  Low-frequency photons, such as radio waves, microwaves have low energy.

The frequency of a photon is inversely related to its wavelength, and so brings us directly to the next property.

iii. Wavelength (λ):

Wavelength is measured as the distance between two consecutive peaks (or troughs) of the wave. It is usually measured in meters, but for photons, it is perhaps more convenient to use nanometers for visible light or even smaller units for high-energy radiation.

The relationship between wavelength (λ) and frequency (f) is:

c=λ⋅f

Where:
c: stands for light speed in a vacuum, approximately 3×108m/s.

(λ): represents the wavelength of the photon.
f: represents the photon’s frequency.

This equation demonstrates that:
High-energy photons, such as blue light, have shorter wavelengths.

 Low-energy photons, such as red light, have longer wavelengths.

iv. Relationship Between These Properties:
Since energy, frequency, and wavelength are all related, you can use these equations to convert between them. For instance, if you know the frequency of a photon, you can calculate its energy and wavelength.

High-frequency radiation (like X-rays or gamma rays): Short wavelength and high energy.
Low-frequency radiation (like radio waves or microwaves): Long wavelength and low energy.

2. Energy Levels of Atoms/Molecules:

Energy levels in atoms and molecules are quantized, meaning that they can only exist in specific, discrete values rather than a continuous range. This concept plays a central role in understanding the behavior of light, atomic spectra, and chemical reactions.

i. Quantized Energy States:

The energy of an electron in atoms is quantized, which is imposed by quantum mechanics. An electron in an atom can only be at specific discrete energy levels. The energy levels are determined by the principal quantum number (denoted as n), which is an integer (1, 2, 3, …).

• For an atom:
  Energy levels (Eₙ) for an electron in a hydrogen atom (as an example) are given by the Bohr model equation:  E_n = – 13.6 / n^2 eV

  Where:  Eₙ  is the energy of the electron in the nth level. n  is the principal quantum number.
  The Rydberg energy of the hydrogen atom is 13.6 eV.
   Energy is more negative for smaller values of  n meaning they are more bound to the nucleus and closer together in levels. For larger values of n , it becomes less negative, or higher in magnitude, putting the electron further from the nucleus.

In molecules energy levels are much more complicated by other factors like molecular bonding, rotation, and vibration. Still, the basic principle is the same: there exist discrete energy levels that the system can occupy.

ii. Energy Transitions:

An electron or molecule can shift between energy levels by **absorbing or emitting energy**. The amount of energy necessary for this shift is the difference in energy between the initial and final states.

• Atomic Transitions:
  In atomic systems, energy transitions involve an electron being shifted from one energy level to another. The energy needed to transfer an electron from one energy level to another is:

E_transition = E_final – E_initial

Since energy levels are quantized, these transitions correspond to the absorption or emission of photons with specific energies.

Absorption occurs when an electron absorbs a photon with an energy equal to the difference between two energy levels, causing it to jump to a higher level.
E_photon = E_final – E_initial

Emission: when an electron moves down from a higher energy level to a lower energy level. In this transition, a photon having energy equal to that of the levels being separated is emitted.

The frequency of the photon emitted or absorbed depends directly on its energy by :

Ephoton​ = h⋅f

Where:

Ephoton​ is the photon’s energy, equivalent to the energy difference between levels.
h : is Planck’s constant.
f : is the frequency of the emitted or absorbed photon.

For example, when an electron in the hydrogen atom moves from  n = 3  to  n = 2 , it needs to emit a photon whose energy is equal to the difference of the two levels. This will produce a spectral line in the emission spectrum.

• Molecular Transitions:
  For molecules, transitions can occur not only between electronic energy levels but also within vibrational and rotational energy levels. These transitions are understood in much the same way but are a bit more complex because of the additional modes of motion.
• Electronic transitions occur when electrons change between different electronic energy levels, usually by absorbing or emitting photons in the UV-visible range.
• Vibrational transitions occur when molecules absorb or emit energy that causes atoms within the molecule to vibrate in different modes (stretching, bending, etc.), typically requiring infrared radiation.
• Rotational transitions involve the rotation of the entire molecule and generally occur in the microwave range.

 iii. Selection Rules for Transitions:

Not all energy level transitions are permitted. There are selection rules governing which transitions can occur. The rules depend on the symmetries of the system and on the nature of the interactions present (such as electromagnetic interactions).

For instance:
In atoms the orbital angular momentum may change by an amount of either  +1 or  -1 .

The change in the principal quantum number, (Δn) may be any integer value.

For molecular vibrations, the change in the vibrational quantum number, Δv is generally ±1.

• Energy quantization is a phenomenon where electrons, in atoms or atoms and molecules, occupy only definite discrete values and hence exist only in quantized states.
• Transitions in energy levels are the process where electrons change their positions within these levels either by emitting energy in the form of photons or by absorbing it.
• The energy of the photon emitted or absorbed corresponds to the energy difference between the initial and final states, and this energy is related to the photon’s frequency.
• Selection rules determine what transitions are allowed, thereby affecting the spectrum of light absorbed or emitted.

3. Photon Absorption/Emission:

i. Photon Absorption:
Photon absorption is the process in which a photon whose energy is equal to the difference of two energy levels of an atom or molecule absorbs by that atom or molecule; this causes a jump of electron to a greater energy state.
Match in Energy: E photon ​= E final​−E initial​

Energy of the photon must be exact and equal to the energy space between the states initial and final.
ii. Photon Emission:
a. Spontaneous Emission:
The excited atom or molecule emits a random photon and returns to the lower energy state.
The photon travels in some arbitrary direction. It carries the difference between two levels as its energy.

b. Stimulated Emission:
In the process, an atom or molecule that is already in an excited state is excited further through interaction with an incoming photon, while at the same time emitting another photon with equal energy, phase and direction.
This is the basic process underlying all lasers.

c. Major Differences
Absorption: Photon energy is commensurate with the gap of energy between levels
Spontaneous Emission: The emission of the photon is not induced by any external interaction.
Stimulated Emission: Photon emission due to an external photon (used in lasers).

4. Energy Conservation:

The principle of energy conservation applies to both photon absorption and photon emission processes.

i. Photon Absorption:

When an atom or molecule absorbs a photon, it gains energy, and this energy is used to move an electron from a lower energy state to a higher energy state.

Change in energy: The change in energy in the system is the energy of the absorbed photon, which is precisely equal to the difference between the final and initial energy levels:

$$\Delta E = E_{\text{final}} – E_{\text{initial}} = E_{\text{photon}}$$

represents the energy difference between two energy levels, which is the energy of the photon emitted or absorbed during a transition.

The energy of the photon must equal the energy difference between the two levels.

ii.Photon Emission:
When an atom or molecule emits a photon, it is giving up energy. This energy comes out in the form of a photon. The system goes from a higher energy state to a lower energy state.

Energy change: The energy given up is equal to the energy of the emitted photon, which corresponds to the energy difference between the two energy levels:

$$\Delta E = E_{\text{final}} – E_{\text{initial}} = – E_{\text{photon}}$$

shows that the energy change of the system (which is the difference between the final and initial energy levels) is the negative of the photon energy. This happens because when an electron transitions to a lower energy state, it releases energy as a photon.

The energy transferred out of the system is the energy carried off by the emitted photon.

Example: Suppose an electron in a hydrogen atom falls from n = 2 to n = 1. The amount of energy released is the difference between the two energy states (for example, 13.6 eV).

5. Quantum Mechanics:

• Quantum Transitions: In quantum mechanics, particles (like electrons) move between discrete energy levels by absorbing or emitting photons. The energy of the photon must match the energy difference between the initial and final states.

  1. 
     $\Delta l = \pm 1$

      

     : The orbital angular momentum quantum number must change by ±1.
  2. 
     $\Delta m = 0, \pm 1$

      

     : The magnetic quantum number can change by 0, +1, or -1.
  3. 
     $\Delta s = 0$

      

     : The spin quantum number does not change for electric dipole transitions.

  These rules determine which transitions are allowed based on changes in quantum numbers (orbital angular momentum, magnetic quantum number, spin).