KAHIBARO
Table of Contents
Change between atomic energy levels
Atomic transitions are changes of an atom from one energy state to another. In atomic physics, electrons in atoms can occupy only certain allowed energy levels. When an electron moves from one allowed level to another, the atom either absorbs energy or releases energy.
If the electron goes from a lower energy level to a higher one, the atom must gain energy. This is called excitation. If the electron goes from a higher energy level to a lower one, the atom loses energy, usually by emitting light. This is called de-excitation.
The key idea is that atomic energies are quantized. The atom cannot change its energy by just any amount. It can change only by the difference between two allowed levels.
For a transition between two atomic states,
$$\Delta E = E_f - E_i$$
where $E_i$ is the initial energy and $E_f$ is the final energy.
If the atom emits a photon,
$$E_{\gamma} = E_i - E_f \quad \text{with} \quad E_i > E_f$$
If the atom absorbs a photon,
$$E_{\gamma} = E_f - E_i \quad \text{with} \quad E_f > E_i$$
Absorption and emission
An atom can interact with electromagnetic radiation by absorbing or emitting photons. A photon carries energy
$$E_{\gamma} = hf = \frac{hc}{\lambda}$$
where $h$ is Planck's constant, $f$ is frequency, $c$ is the speed of light, and $\lambda$ is wavelength.
For absorption to happen, the photon energy must match the energy gap between two atomic levels. Then the atom can take in the photon and move to a higher state. For emission, an excited atom drops to a lower state and releases a photon with energy equal to the level difference.
This explains why atoms produce line spectra rather than continuous spectra. Each line corresponds to a particular transition.
Atomic transitions occur only when the photon energy matches an allowed energy difference:
$$hf = |E_f - E_i|$$
This is why atomic spectra contain discrete lines.
Ground state and excited states
The lowest energy state of an atom is called the ground state. Any higher allowed state is an excited state. If an atom is excited, it usually does not remain there forever. It tends to return to lower energy states, often in one step or through several intermediate transitions.
An atom may absorb energy from light, collisions with other particles, or electrical discharge. After excitation, it can emit one photon or a sequence of photons as it relaxes downward through different levels.
Energy differences and spectral lines
Every possible pair of atomic energy levels gives a possible transition energy. Since different pairs have different energy gaps, the emitted or absorbed photons have different frequencies and wavelengths.
A larger energy gap means a higher photon frequency and a shorter wavelength. A smaller energy gap means a lower photon frequency and a longer wavelength.
The relationship can be summarized clearly.
| Energy gap | Photon frequency | Photon wavelength |
|---|---|---|
| Large | High | Short |
| Small | Low | Long |
So, by measuring the wavelengths of spectral lines, we can infer the energy structure of the atom.
Spontaneous and stimulated transitions
Some transitions happen without any external photon being present. An excited atom can naturally fall to a lower energy state and emit a photon. This is called spontaneous emission.
A second possibility is that an incoming photon can trigger an excited atom to emit another photon of the same energy. This is called stimulated emission. This idea is central to lasers, but the detailed discussion belongs elsewhere.
There is also absorption, where an atom in a lower state takes in a photon and jumps upward.
| Process | Initial atomic state | Photon interaction | Result |
|---|---|---|---|
| Absorption | Lower level | Photon absorbed | Higher level |
| Spontaneous emission | Higher level | No trigger needed | Lower level plus photon |
| Stimulated emission | Higher level | Incoming photon triggers emission | Lower level plus emitted photon |
Transition paths
An excited atom does not always return directly to the ground state. It may pass through one or more intermediate levels. Each step produces a photon whose energy equals that particular energy difference.
Suppose an atom has levels $E_3$, $E_2$, and $E_1$. It could go directly from $E_3$ to $E_1$, or it could go from $E_3$ to $E_2$ and then from $E_2$ to $E_1$. These paths produce different photons.
If the atom goes in two steps, then energy conservation requires
$$E_3 - E_1 = (E_3 - E_2) + (E_2 - E_1)$$
This means the total released energy is the same, but it may be divided among several photons.
Atomic transitions and spectral identification
Each type of atom has its own set of allowed energy levels, so each type of atom has its own pattern of transition lines. This pattern acts like a fingerprint. By studying emitted or absorbed wavelengths, scientists can identify which atoms are present in a gas, a star, or a laboratory sample.
Hydrogen has one set of transitions, helium another, sodium another, and so on. Because the energy levels depend on atomic structure, the spectrum reveals information about the atom itself.
Each element has unique allowed energy levels, so each element has a unique set of spectral lines.
This is why spectroscopy can identify atoms.
Selection rules and allowed transitions
Not every energy difference necessarily appears as a strong observed spectral line. Some transitions are much more likely than others. The detailed rules depend on quantum mechanics and angular momentum, which are developed elsewhere. For beginners, it is enough to know that some transitions are called allowed and happen readily, while others are forbidden or weak and occur much less often.
So, the existence of two levels with an energy difference does not automatically guarantee a strong line in practice. Transition probability matters.
Lifetimes and line width
Excited atomic states last for a finite time. This time is called the lifetime of the state. A very short lifetime means the state decays quickly. A longer lifetime means the atom remains excited for a longer time before making a transition.
Because excited states are not infinitely long lived, spectral lines are not perfectly sharp. They have a finite width. In real measurements, additional effects such as motion of atoms and collisions can also broaden lines.
Simple example
Imagine an atom with two levels, a lower level $E_1$ and an upper level $E_2$. If
$$E_2 - E_1 = 3.0 \text{ eV}$$
then a photon of energy $3.0 \text{ eV}$ can be absorbed to excite the atom from $E_1$ to $E_2$. If the atom later returns from $E_2$ to $E_1$, it emits a photon with the same energy difference.
Using
$$E_{\gamma} = hf = \frac{hc}{\lambda}$$
the emitted wavelength can be found from that energy. This direct connection between atomic energy differences and light is the heart of atomic transitions.
What atomic transitions tell us
Atomic transitions connect atomic structure to observable light. They explain why atoms absorb only certain wavelengths and emit only certain wavelengths. They also allow us to measure atomic energy levels experimentally.
When we observe light from atoms, we are seeing evidence of electrons changing from one quantized state to another. Atomic transitions are therefore one of the clearest demonstrations that energy in atoms is discrete, not continuous.