Discrete Energy Levels and Atomic Spectra

Understanding why neon signs glow red, how astronomers determine the composition of distant stars, and even the principles behind laser technology all hinge on one fundamental idea: electrons in atoms can only exist at specific, separate energy levels. This concept of quantized energy revolutionised physics, providing the first successful model to explain atomic behaviour and laying the groundwork for quantum mechanics. For IB Physics, mastering this topic is crucial for explaining the interaction of light and matter and for interpreting the definitive evidence provided by atomic spectra.

The Bohr Model and Quantized Orbits

Before Niels Bohr's 1913 model, the prevailing "planetary" model of the atom faced a critical flaw: according to classical electromagnetism, an electron orbiting a nucleus should continuously lose energy as electromagnetic radiation, spiraling into the nucleus in a fraction of a second. This obviously did not happen. Bohr proposed a radical solution by introducing postulates that blended classical ideas with new quantum rules.

His first key postulate was that electrons can only orbit the nucleus in certain stable, stationary states without radiating energy. In these states, the electron's angular momentum is quantized, meaning it can only take on specific values that are integer multiples of a fundamental constant. This leads directly to the concept of discrete orbits, each with a fixed radius and, most importantly, a fixed energy. An electron in the closest permitted orbit to the nucleus is in the ground state, which is the lowest energy level ($n=1$). Orbits further out correspond to excited states ($n=2,3,\mathrm{4...}$), each with a higher, specific energy. The energy of these levels is given by the formula for a hydrogen-like atom: $$E_n=-\frac{13.6\text{ eV}}{n^2}$$ where $n$ is the principal quantum number. The negative sign indicates the electron is bound to the nucleus; zero energy represents being completely free.

Photon Emission and Absorption in Transitions

Bohr's second pivotal postulate explained how atoms emit and absorb light. An electron can "jump" from one stationary state to another. However, it cannot exist between these levels. To move to a higher energy level (excitation), the electron must absorb a precise amount of energy. Conversely, when it falls from a higher to a lower level, it must emit that exact same amount of energy in the form of a photon.

The energy of the emitted or absorbed photon is precisely equal to the absolute difference in energy between the two levels involved. This is a direct consequence of the conservation of energy. If $E_i$ is the initial energy level and $E_f$ is the final energy level, the photon energy $E_{photon}$ is: $$E_{photon}=|E_f-E_i|$$ This leads directly to the famous equations linking the photon's energy to its frequency $f$ and wavelength $\lambda$: $$E=hf=\frac{hc}{\lambda }$$ Here, $h$ is Planck's constant ($6.63\times 10^{-34}\text{ J s}$) and $c$ is the speed of light. The quantized energy levels therefore produce photons of only specific, quantized frequencies and wavelengths.

Calculation Example: The Balmer Series

Consider an electron in a hydrogen atom falling from the $n=3$ level to the $n=2$ level.

1. The energy of each level is: $E_3=-13.6/3^2=-1.51\text{ eV}$ and $E_2=-13.6/2^2=-3.40\text{ eV}$.
2. The energy of the emitted photon is: $E_{photon}=|-3.40-(-1.51)|=1.89\text{ eV}$.
3. Convert this to joules: $1.89\text{ eV}\times 1.60\times 10^{-19}\text{ J/eV}=3.02\times 10^{-19}\text{ J}$.
4. Calculate the photon's frequency: $f=E/h=(3.02\times 10^{-19})/(6.63\times 10^{-34})=4.56\times 10^{14}\text{ Hz}$.
5. Calculate its wavelength: $\lambda =c/f=(3.00\times 10^8)/(4.56\times 10^{14})=6.58\times 10^{-7}\text{ m}=658\text{ nm}$ (red light).

This transition is part of the Balmer series, which produces visible light. Different series (Lyman, Paschen) correspond to transitions ending at different final levels ($n=1$, $n=3$, etc.).

Atomic Spectra as Definitive Evidence

The phenomenon of atomic emission spectra provides the most direct proof for quantized energy levels. When a low-pressure gas is excited by heat or electricity, it emits light. When this light is passed through a diffraction grating or prism, it does not produce a continuous rainbow. Instead, it produces a line spectrum—a series of bright, coloured lines at specific wavelengths against a dark background. Each line corresponds to a specific electron transition from a higher to a lower energy level. Crucially, every element has a unique line spectrum, a "fingerprint" that allows for its identification.

Conversely, atomic absorption spectra occur when white light (containing all wavelengths) passes through a cooler gas. Atoms in the gas absorb photons whose energies exactly match possible transitions from their ground state to excited states. This results in a continuous spectrum with specific dark lines at precisely the same wavelengths as the element's emission lines. The absorption spectrum directly shows which photons have the correct energy to promote electrons, again confirming the existence of discrete levels.

Limitations of the Bohr Model

While the Bohr model was a monumental leap forward and perfectly explains the spectrum of hydrogen and other single-electron ions (like ${\text{He}}^{+}$), it has significant shortcomings that highlight its status as a semi-classical hybrid, not a full quantum theory.

• Failure with Multi-Electron Atoms: The model cannot accurately calculate the energy levels or spectra for atoms with two or more electrons. It ignores the repulsive forces between electrons, which drastically alter the energy level structure.
• No Explanation for Line Intensity: The model predicts the wavelengths of spectral lines but cannot explain why some lines are brighter (more intense) than others, which relates to the probability of specific transitions occurring.
• Violates the Uncertainty Principle: The model assumes electrons have a well-defined orbit and path, which is forbidden by Heisenberg's Uncertainty Principle—a cornerstone of modern quantum mechanics established after Bohr.
• Limited to Circular Orbits: It initially considered only circular orbits, though later modifications included elliptical ones. Modern quantum mechanics describes electrons in terms of probability clouds (orbitals), not defined paths.

Despite these limitations, the Bohr model remains a powerful pedagogical tool. It correctly introduces the core concepts of quantization, stationary states, and photon emission/absorption via transitions, creating a vital bridge between classical physics and the more abstract, but more complete, quantum mechanical model.

Common Pitfalls

1. Confusing Emission and Absorption Spectra: A common mistake is to think absorption spectra show coloured lines on a dark background. Remember: Emission = bright lines on dark; Absorption = dark lines on a continuous coloured background. A good memory aid is that the cooler gas in absorption experiments "steals" specific colours from the white light, leaving dark gaps.
2. Misapplying the Energy Formula: The formula $E_n=-13.6/n^2\text{ eV}$ is only valid for hydrogen and hydrogen-like ions (atoms stripped of all but one electron). Using it for helium or lithium atoms will yield incorrect answers. For multi-electron atoms, the energy levels are different and not given by this simple formula.
3. Sign Errors in Energy Calculations: The energy levels are negative. When calculating the photon energy for a transition from $n=2$ to $n=1$, the correct calculation is $E_{photon}=E_1-E_2=(-13.6)-(-3.40)=-10.2\text{ eV}$. The photon energy is the absolute value, $10.2\text{ eV}$. Forgetting the signs can lead to a negative photon energy, which is physically nonsensical.
4. Assuming Electrons "Decide" to Fall: Avoid classical thinking that an excited electron "wants" to fall down. In quantum mechanics, the transition is a probabilistic event. An electron in an excited state has a certain probability per unit time to spontaneously emit a photon and drop to a lower level. It does not happen immediately or by a conscious decision.

Summary

• Electrons in atoms occupy discrete, quantized energy levels, as first successfully modelled by Bohr for hydrogen. The lowest level is the ground state; higher levels are excited states.
• Atoms emit or absorb light when electrons make transitions between these levels. The energy of the involved photon is exactly equal to the difference in energy between the two levels: $E_{photon}=|\Delta E|=hf=hc/\lambda$.
• Atomic emission spectra (bright lines on dark) and absorption spectra (dark lines on a continuous spectrum) provide direct, experimental evidence for quantized energy levels, as each element has a unique spectral fingerprint.
• While groundbreaking, the Bohr model has limitations: it fails for multi-electron atoms, cannot explain line intensities, and relies on classical concepts of orbits incompatible with full quantum mechanics.
• Mastering calculations of photon wavelength/frequency from energy level differences is a core IB skill, requiring careful attention to units (often converting between electronvolts and joules) and the sign conventions for bound energy levels.