Change principal quantum number n to visualize energy levels, orbital radii, and emission spectra in real time. Explore Lyman, Balmer, and Paschen series transitions interactively.
Parameters
Results (live)
Transition
3 → 2
Series
Balmer
ΔE photon energy
1.889 eV
Wavelength λ
656.5 nm
Frequency ν
4.57e14 Hz
E₁ (init) / E₂ (final)
-1.51 / -3.40
Orbital radius r₁
0.476 nm
Orbital radius r₂
0.212 nm
Hydrogen atom — orbiting electron, transition & photon emission
The electron orbits in quantized shells → drops from n₁ to n₂ and emits a photon colored by its wavelength (absorbed if n₁<n₂).
Energy-level diagram with transition arrow
Spectral line (emitted line)
Visible range 380–700 nm. The white marker is the line just emitted; the flash marks a newly arrived photon.
Eₙ is the level energy, rₙ the orbital radius, ΔE the photon energy emitted/absorbed in a transition, and λ the wavelength. This is equivalent to the Rydberg formula 1/λ = R_H(1/n₂²−1/n₁²); Balmer-α (3→2) is 656 nm.
💬 A Conversation About the Bohr Model
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I heard the electron in a hydrogen atom can only travel in certain "orbits." Why is there such a restriction?
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In 1913 Bohr proposed that the electron's angular momentum must be an integer multiple of $\hbar$, i.e. $mvr = n\hbar$. Combine that with classical electromagnetism and you find the orbital radius can only take discrete values $r_n = n^2 a_0$. The reason behind it was later explained by quantum mechanics, but it matched the experimental hydrogen spectrum astonishingly well.
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The energy is $-13.6/n^2$ eV — why negative? What does negative energy even mean?
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It's about where you put the reference. We define the energy of an electron infinitely far from the nucleus as zero. Pulling the electron in does work, so any bound state must be negative. The $-13.6$ eV at $n=1$ means "you need 13.6 eV to strip the electron completely off the hydrogen atom (ionize it)."
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We covered the Balmer series in class. Are there others, like the Lyman or Paschen series?
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Yes — the final quantum number $n$ determines the series. Transitions to $n=1$ form the Lyman series (ultraviolet), to $n=2$ the Balmer series (visible), and to $n=3$ the Paschen series (near infrared). The dark Fraunhofer lines in the Sun's spectrum are evidence that hydrogen in the solar atmosphere absorbs specific wavelengths.
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So is the Bohr model perfect? Why do we use quantum mechanics now?
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It works for hydrogen but fails completely for multi-electron atoms (helium and beyond). It also can't explain electron spin, the fine structure of spectral lines, or behavior in a magnetic field (the Zeeman effect). Quantum mechanics via the Schrödinger equation uses a "probability density" instead of orbits and describes all of these accurately. The Bohr model remains a useful introductory picture for intuitively grasping energy levels.
Frequently Asked Questions
Increasing n expands the electron's orbital radius by a factor of n² and makes the energy level shallower at -13.6/n² eV. The wavelength of the light emitted during transitions between n values also changes, letting you observe spectral series (Lyman, Balmer, etc.) in real time.
The Balmer series involves transitions from n ≥ 3 to n = 2 and emits visible light, while the Lyman series involves transitions from n ≥ 2 to n = 1 and emits ultraviolet light. You can compare the wavelengths of each series by changing n in the simulator.
Because zero energy is referenced at infinity (the electron completely separated from the nucleus), bound-state energies are negative. The larger the absolute value, the more strongly the electron is bound, with n = 1 being the most stable ground state.
No. The Bohr model assumes classical circular orbits and ignores the electron's wave nature and the uncertainty principle. More accurate quantum-mechanical models use probability distributions (orbitals) rather than orbits, but the Bohr model is useful for a basic understanding of energy levels.
What is the physical meaning of the Bohr radius a₀ = 0.529 Å?
It is the radius of the n=1 orbit of hydrogen. $a_0 = \hbar^2 / (m_e e^2 k_e)$ is a natural constant derived from the electron's charge, mass, and Planck's constant. It sets the characteristic length scale of atoms and is widely used in quantum chemistry.
Why doesn't the electron radiate light while orbiting?
In classical electromagnetism an accelerating charge should radiate, lose energy, and spiral into the nucleus. Bohr resolved this by postulating that "stationary states do not radiate." Quantum mechanics justifies it by noting that the lowest-energy eigenstate cannot change.
Why is the hydrogen Hα line (656 nm) red?
The Hα line is emitted in the n=3→n=2 transition. ΔE = -1.51 - (-3.40) = 1.89 eV, giving λ = 1240/1.89 ≈ 656 nm, which falls in the red part of the visible spectrum. The red glow of stars and nebulae (HII regions) comes from this Hα line.
How does the Bohr model differ from modern quantum mechanics?
The Bohr model is a semi-classical picture that adds a quantization condition to circular orbits. The Schrödinger equation describes the electron with a wavefunction, representing it as a "probability cloud." The energy-level values are the same for hydrogen, but only the Schrödinger equation correctly describes orbital shapes (s/p/d/f) and the spin quantum number.
What is the Bohr Hydrogen Atom Model?
In the Bohr hydrogen atom model, the electron is assumed to occupy only specific quantized circular orbits. This condition arises because the electron's angular momentum is restricted to integer multiples of $\hbar = h/2\pi$, and the orbit is fixed by the principal quantum number $n$. The energy levels are given by $E_n = -\frac{13.6\,\mathrm{eV}}{n^2}$, lowest at the ground state $n=1$ and approaching zero as $n$ increases. The orbital radius is $r_n = n^2 a_0$ (where $a_0$ is the Bohr radius), growing as the square of $n$. When an electron transitions from a higher level $n_i$ to a lower level $n_f$, a photon equal to the energy difference is emitted, and its wavelength $\lambda$ follows the Rydberg formula $\frac{1}{\lambda} = R\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$ — letting you observe spectral lines such as the Balmer series ($n_f=2$) in real time.
Real-World Applications
Industrial use cases: In the semiconductor industry, Balmer-series analysis informs the emission-wavelength design of excimer lasers (ArF: 193 nm, KrF: 248 nm) used in lithography. In hydrogen fuel-cell catalyst development, estimating shifts in hydrogen-atom energy levels on platinum surfaces helps screen efficient catalyst materials.
Education and research: University quantum-mechanics lectures use the visualization of the Lyman series (e.g. n=1 to n=∞ transitions) to convey the limits of the Bohr model and the bridge to quantum theory. In astrophysics, identifying spectral lines emitted by hydrogen in interstellar clouds (such as the Paschen series) relies on the same physics.
CAE integration and practical role: This tool serves as a pre-stage to first-principles (DFT) calculations, providing quick estimates of atomic-level energy levels. In practice it is built into the calibration of plasma optical-emission spectrometers and the validation of initial conditions for hydrogen-plasma simulations, helping reduce the computational load of large-scale CAE analyses.
Common Misconceptions and Points of Caution
People often assume "the larger n is, the larger the orbital radius grows in simple proportion," but it actually grows as n² times the Bohr radius — 4× at n=2, 9× at n=3 — increasing rapidly. Another common misconception is that "the spacing between energy levels widens as n grows"; in fact the spacing narrows as n increases, converging toward the continuum limit (ionization). Finally, many believe "only the visible Balmer lines make up the hydrogen emission spectrum," but other series such as Lyman (ultraviolet) and Paschen (infrared) also exist — switching the initial and final states in the tool lets you confirm transitions across all series.
Set the initial quantum number n1 (ground state) using the n1Slider, typically starting at n=1 for hydrogen transitions from the ground state.
Adjust n2Slider to define the excited state quantum number; higher values (n=2,3,4...) reveal transitions in Lyman (UV), Balmer (visible), and Paschen (infrared) series.
Use nmaxSlider to set the maximum principal quantum number displayed in the energy level diagram, enabling visualization of multiple orbital radii and corresponding transition wavelengths.
Worked Example
For a Balmer-series transition in hydrogen: set n1=2 (lower level) and n2=4 (upper level). The simulator calculates the energy difference ΔE = 13.6 eV × (1/4 − 1/16) = 2.55 eV, yielding a photon wavelength of 487 nm (hydrogen-beta line, cyan emission). The orbital radius increases from a₀=0.053 nm at n=2 to r₄=0.212 nm at n=4, demonstrating the n² scaling relationship governing Bohr orbits.
Practical Notes
Lyman series (n1=1, n2≥2) produces UV photons above 91.2 nm; critical for stellar spectroscopy and plasma diagnostics in fusion reactors.
Balmer series (n1=2, n2≥3) generates visible wavelengths 656–365 nm; historically used for identifying hydrogen in nebulae and laboratory spectral analysis.
Paschen series (n1=3, n2≥4) emits infrared radiation (1875+ nm); relevant in thermal imaging and astrophysical observations of cool stellar atmospheres.
Larger nmax values reveal fine structure splitting and quantum defects neglected by the classical Bohr model, bridging toward quantum mechanical treatments.