Hydrogen Atom Simulator: Exploring Quantum Energy Levels and Spectral Lines

✓ Verified Content: All equations, formulas, and reference data in this simulation have been verified by the Simulations4All engineering team against authoritative sources including NIST Atomic Spectra Database, MIT OpenCourseWare, and peer-reviewed physics textbooks. See verification log

Introduction

Point a spectroscope at a hydrogen tube and you will see something beautiful: a handful of colored lines against darkness. Red, cyan, violet. Not a rainbow, not a continuous smear of color. Discrete lines, sharp as knife edges. How can glowing gas produce such specific wavelengths?

This question drove physics for a century. And hydrogen (one proton, one electron, nothing simpler exists) became the proving ground for every atomic theory we invented. Dalton's billiard balls, Thomson's plum pudding, Bohr's planetary orbits, Schrodinger's probability clouds. Each model explained something the previous one could not. Each one was eventually replaced [1].

The elegant part is that hydrogen is the only atom we can solve exactly. One electron, one nucleus, one Coulomb force. No approximations needed. When theory matches experiment to ten decimal places (as it does for hydrogen's spectral lines), you know you have found something true about nature. The characteristic red glow at 656 nm? That is an electron falling from the third orbit to the second. We can calculate that wavelength from first principles.

How to Use This Simulation

If you could slow down atomic physics enough to watch each model develop, you would see a century of scientific revolutions compressed into six buttons. This simulator lets you walk through that history, compare competing models side by side, and understand why each one was eventually replaced.

Historical Model Selection

Select any model to see its visualization and unique controls:

Model-Specific Controls

Bohr Model Controls:

Watch the energy display: E_1 = -13.6 eV, E_2 = -3.4 eV, and so on. The -1/n^2 relationship is visible in real time.

Schrodinger Model Controls (Orbital Selection):

Rutherford Model Controls (Alpha Scattering):

Set impact parameter to 0 and watch the alpha particle bounce straight back. This is what surprised Rutherford: "as if you fired a shell at tissue paper and it came back."

Hydrogen Emission Spectrum

The spectrum panel shows all major spectral series:

Balmer Series (Visible Lines):

Click any spectrum line to see the transition details and Rydberg formula calculation.

Keyboard Shortcuts

Results Panels

Quantum Numbers Display:

• n (principal): Energy level
• l (angular): Orbital shape (0=s, 1=p, 2=d)
• m_l (magnetic): Orbital orientation
• Orbital designation: Combined notation like 2p or 3d

Physical Properties:

• Energy E_n in electron volts
• Bohr radius for the current state
• Transition wavelength (when applicable)
• Electron velocity in the Bohr model

Bottom Information Tabs

Tips for Exploration

1. Watch the historical progression: Start with Dalton (1) and step through to Schrodinger (6) using number keys. Each model addresses a failure of the previous one.

2. See why Rutherford was surprised: Set impact parameter to 0.5 fm and fire an alpha particle. Compare to Thomson's model where particles would pass straight through.

3. Verify the Rydberg formula: In Bohr mode, set n=3, then transition to n=2. The spectrum shows H-alpha at 656 nm. Calculate 1/lambda = R(1/4 - 1/9) yourself.

4. Compare orbital shapes: In Schrodinger mode, switch between 2p and 3d orbitals. The increasing complexity as l increases reveals the spherical harmonic structure.

5. Find the most probable radius: In the Probability tab, look for the peak of P(r). For 1s, it is exactly at the Bohr radius (0.529 Angstroms), validating the old model within the new framework.

Historical Atomic Models

Each model here was the best understanding of its time. Each was eventually proven incomplete. Watch the progression: from solid balls to probability clouds over 120 years.

The Dalton Model (1803)

What Dalton figured out was that matter comes in discrete chunks. He imagined atoms as tiny, indestructible billiard balls with no internal structure [2]. Wrong about the details, but right about the fundamental idea: chemical reactions involve rearranging these chunks in fixed ratios.

Key Features:

• Atoms are indivisible and indestructible
• All atoms of an element are identical
• Different elements have different masses
• Chemical compounds form through combinations of atoms

Dalton could not explain spectral lines or radioactivity (both unknown in 1803), but he got chemistry onto a scientific foundation.

The Thomson Model (1897)

J.J. Thomson discovered the electron by deflecting cathode rays with electric and magnetic fields. He proposed the "plum pudding" model: electrons embedded in a uniform positive charge, like raisins in pudding [3].

Key Features:

• First model to include subatomic particles
• Electrons (negative) dispersed through positive medium
• Overall atom is electrically neutral
• Mass distributed throughout the atom

This model could not explain discrete spectral lines. And then Rutherford shot alpha particles at gold foil.

The Rutherford Model (1911)

Here is what happens when you actually try this: fire alpha particles at a thin gold foil and watch where they scatter. Most go straight through. A few bounce back like they hit a brick wall. The only explanation: atoms are mostly empty space with a tiny, dense, positively charged nucleus [4].

Key Features:

• Concentrated positive charge in tiny nucleus
• Most of the atom is empty space
• Electrons orbit the nucleus
• Nuclear diameter about 10,000 times smaller than atom

Beautiful experiment, but a fatal flaw in the model: classical electromagnetism says orbiting electrons should radiate energy and spiral into the nucleus in about 10^-11 seconds. Atoms should not exist. They do. Something was missing.

The Bohr Model (1913)

Niels Bohr's fix was radical: just declare that electrons can only exist in specific orbits and do not radiate while in those orbits. Ad hoc? Yes. But it worked [5].

Key Features:

• Electrons exist only in discrete orbits
• Angular momentum is quantized: L = nh/2pi
• Energy levels: E_n = -13.6 eV/n^2
• Photon emission occurs during transitions between levels

Bohr's formula E_n = -13.6/n^2 eV matches hydrogen's spectrum exactly. But the model fails for helium. Something deeper was needed.

The de Broglie Model (1924)

Louis de Broglie asked a simple question: if light can be both wave and particle, why not electrons? His PhD thesis proposed that particles have wavelength lambda = h/mv [6].

Key Features:

• Matter exhibits wave-particle duality
• Electron wavelength: lambda = h/p
• Standing wave condition: 2pir = n*lambda
• Provides physical basis for Bohr's quantization

Suddenly Bohr's mysterious quantization made sense: only orbits where electron waves form standing waves are stable. The circumference must fit an integer number of wavelengths.

The Schrodinger Model (1926)

If electrons are waves, they need a wave equation. What Schrodinger figured out was exactly that. His equation describes the electron as a probability distribution, not a particle following a definite path [7].

Key Features:

• Electron described by wavefunction psi(r,theta,phi)
• |psi|^2 gives probability density
• Quantum numbers (n, l, m_l) define orbitals
• Explains atomic spectra, chemical bonding, and fine structure

This is where we are today. The electron is not at any particular place. It has a probability of being at each point in space. Strange? Yes. Correct? Also yes.

Key Parameters

Key Formulas

Energy Levels

Formula: E_n = -13.6 eV / n^2

Where:

• E_n = energy of level n (negative for bound states)
• n = principal quantum number (1, 2, 3, ...)
• -13.6 eV = ground state energy (ionization energy)

Used when: Calculating the energy of any hydrogen orbital or the energy released/absorbed during transitions.

Rydberg Formula

Formula: 1/lambda = R_H (1/n_1^2 - 1/n_2^2)

Where:

• lambda = wavelength of emitted photon
• R_H = 1.097 x 10^7 m^-1 (Rydberg constant)
• n_1 = lower energy level
• n_2 = upper energy level (n_2 > n_1)

Used when: Calculating spectral line wavelengths for any series (Lyman, Balmer, Paschen, etc.)

de Broglie Wavelength

Formula: lambda = h / mv

Where:

• lambda = wavelength of matter wave
• h = Planck's constant (6.626 x 10^-34 J*s)
• m = particle mass
• v = particle velocity

Used when: Determining the wave properties of electrons or other particles.

Orbital Radius

Formula: r_n = n^2 * a_0

Where:

• r_n = radius of the nth Bohr orbit
• n = principal quantum number
• a_0 = Bohr radius (0.529 Angstroms)

Used when: Finding the classical orbit radius in the Bohr model.

Quantum Numbers Explained

The principal quantum number n determines energy and average distance from nucleus. Angular momentum l determines orbital shape (l=0: spherical s; l=1: dumbbell p; l=2: cloverleaf d). Magnetic number m_l determines spatial orientation and becomes important in magnetic fields.

Learning Objectives

After working through this simulation, you should be able to:

1. Compare the six historical atomic models and explain what each contributed. Each one got something right and something wrong.

2. Calculate hydrogen energy levels using E_n = -13.6/n^2 eV and predict photon wavelengths for transitions using the Rydberg formula. Get comfortable with these calculations.

3. Interpret the hydrogen emission spectrum. See a red line at 656 nm? That is an electron falling from n=3 to n=2. Know this.

4. Explain quantum numbers (n, l, m_l) and describe how they determine orbital shape, size, and orientation. These labels are not arbitrary.

5. Visualize probability density distributions for s, p, and d orbitals. Understand why "where is the electron?" has no simple answer in quantum mechanics.

6. Analyze the Rutherford scattering experiment. Know why some alpha particles bounce straight back and most sail through.

Exploration Activities

Activity 1: Historical Model Comparison

Objective: Watch physics evolve from billiard balls to probability clouds.

Steps:

1. Start with the Dalton model. Featureless sphere, nothing inside.
2. Progress through each model using buttons 1-6
3. For each model, note what new feature appears
4. Pay attention to how the electron representation changes

Observe: How does our picture of the electron evolve from Thomson to Schrodinger?

Expected Result: Definite particles (Dalton, Thomson), definite orbits (Rutherford, Bohr), standing waves (de Broglie), probability clouds (Schrodinger). Each step gives up more certainty about where the electron "is."

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Activity 2: Energy Level Transitions

Objective: See the 1/n^2 law in action.

Steps:

1. Select the Bohr model
2. Set n = 1 and record the energy (-13.6 eV)
3. Increase n to 2 and record the new energy (-3.4 eV)
4. Continue for n = 3, 4, 5, 6
5. Calculate the energy difference for the 3 to 2 transition

Observe: Energy is negative because the electron is bound. Zero energy means a free electron at rest, infinitely far from the nucleus. Bound electrons have negative energy because you would have to add energy to free them.

Expected Result: E_n = -13.6/n^2 eV. The 3-to-2 transition releases 1.89 eV. Light with energy 1.89 eV has wavelength 656 nm. That is the red H-alpha line you see in hydrogen tubes.

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Activity 3: Emission Spectrum Analysis

Objective: Connect those colored lines to specific electron jumps.

Steps:

1. Observe the emission spectrum at the bottom of the visualization
2. Click on each Balmer series line (the visible ones)
3. Note the wavelength and transition displayed
4. Use the Rydberg formula to verify: 1/lambda = R(1/4 - 1/9) for H-alpha
5. Calculate what color corresponds to each energy difference

Observe: Which transitions produce visible light? Why is H-alpha the strongest?

Expected Result: Balmer series (transitions ending at n=2) produces visible light: H-alpha (656nm, red), H-beta (486nm, cyan), H-gamma (434nm, violet), H-delta (410nm, deep violet). The 3-to-2 transition is most probable because n=3 is the closest level to n=2.

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Activity 4: Orbital Shapes and Probability

Objective: See why we draw orbitals the way we do.

Steps:

1. Select the Schrodinger model
2. Start with the 1s orbital. Spherical cloud, densest near the nucleus.
3. Switch to 2s. Notice the node (zero probability shell).
4. Select 2p and observe the dumbbell shape
5. Try 3d and adjust the m_l slider. Five different orientations.
6. Go to the Probability tab to see radial distribution

Observe: How does shape depend on l? The s orbitals are always spherical. The p orbitals always have lobes. Why?

Expected Result: l=0 gives spherical orbitals. l=1 gives dumbbells. l=2 gives four-lobed cloverleaves. The most probable radius for 1s is exactly one Bohr radius (0.529 Angstroms). Bohr got the distance right even though his orbits are wrong.

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Activity 5: Alpha Particle Scattering

Objective: Experience Rutherford's experiment yourself.

Steps:

1. Select the Rutherford model
2. Set impact parameter to 5 fm (grazing trajectory)
3. Fire an alpha particle. Small deflection.
4. Decrease impact parameter to 1 fm. Fire again. Bigger angle.
5. Try impact parameter near 0. Watch it bounce back.
6. Increase particle energy and observe how deflection changes

Observe: Most particles sail through. A few bounce back. What does that tell you about atomic structure?

Expected Result: The nucleus is tiny. Most of the atom is empty space. Only direct hits on the nucleus produce large-angle scattering. This is exactly what Rutherford found in 1911, and it demolished the plum pudding model.

Real-World Applications

1. Stellar Spectroscopy: The Balmer series lines in stellar spectra identify hydrogen and determine stellar temperature. Hot stars show strong hydrogen absorption; the strength pattern indicates surface temperature [8].

2. MRI Technology: Magnetic resonance imaging uses hydrogen nuclei (protons) in water molecules. Radio waves flip proton spins, and the re-alignment signal creates images. Understanding hydrogen's quantum properties is essential for MRI physics.

3. Atomic Clocks: The most precise atomic clocks use the hyperfine transition in hydrogen (21 cm line) or similar transitions in cesium. These quantum mechanical transitions provide the definition of the second.

4. Laser Development: Hydrogen's discrete energy levels enable specific wavelength laser emission. The 656 nm H-alpha transition is used in tunable dye lasers for spectroscopy applications.

5. Semiconductor Physics: The hydrogen atom model extends to electron-hole pairs in semiconductors. Excitons (bound electron-hole pairs) behave like hydrogen atoms, with modified Bohr radii depending on the material.

6. Fusion Energy Research: Understanding hydrogen isotope quantum mechanics is crucial for fusion reactor design. The deuterium-tritium reaction requires precise knowledge of nuclear wavefunctions.

Reference Data

Hydrogen Spectral Series

Physical Constants for Hydrogen

Challenge Questions

1. Conceptual: Why does the Bohr model work well for hydrogen but fail for helium and heavier atoms? What physics is missing?

2. Calculation: Calculate the wavelength of light emitted when an electron transitions from n=5 to n=2. What color is this light? (Use R_H = 1.097 x 10^7 m^-1)

3. Analysis: In the 3d orbital, there are five possible m_l values (-2, -1, 0, +1, +2). How many different spatial orientations does this represent? What happens to these orbitals in a magnetic field?

4. Application: An astronomer observes the H-alpha line from a distant galaxy at 670 nm instead of 656 nm. What does this tell you about the galaxy? Calculate its recession velocity.

5. Design: You need to design a gas discharge tube that produces the most visible light. Which energy levels should you excite electrons to? Explain your reasoning based on the Balmer series.

Common Mistakes to Avoid

These errors appear constantly. Avoid them.

1. Confusing orbit and orbital: Bohr orbits are definite circular paths. Schrodinger orbitals are probability distributions. The electron in a hydrogen atom does not orbit the nucleus like a planet. It has no trajectory at all.

2. Misapplying the Rydberg formula: The formula gives 1/lambda, not lambda directly. Do the algebra carefully. Also, n_2 must be greater than n_1 for emission (energy leaving the atom). Swap them for absorption.

3. Forgetting energy is negative: Bound electrons have negative energy. Zero energy means the electron has escaped to infinity with no kinetic energy left over. E = -13.6 eV means you need to add 13.6 eV to free the electron.

4. Misinterpreting probability density: |psi|^2 gives probability per unit volume. The probability of finding the electron at exactly one point is zero (a point has zero volume). You integrate |psi|^2 over a region to get meaningful probabilities.

5. Confusing l and m_l quantum numbers: l determines orbital shape (0=s, 1=p, 2=d). m_l determines orientation. There are 2l+1 orientations for each l. They are not interchangeable.

Frequently Asked Questions

Q1: Why are hydrogen spectral lines discrete rather than continuous?

Same reason standing waves on a string come in specific frequencies. Energy levels are quantized. Electrons can only occupy specific states [1]. When they jump between states, they emit or absorb photons with precisely defined energies. Energy determines wavelength. Discrete energy differences mean discrete wavelengths.

Q2: What was the key evidence that disproved the Thomson model?

Rutherford's backscattering. Fire alpha particles at atoms and some bounce straight back [4]. Thomson's model (positive charge spread uniformly) could only produce small deflections. Large-angle scattering requires hitting something small and dense. That something is the nucleus.

Q3: Why can hydrogen's Schrodinger equation be solved exactly?

One electron, one proton, one Coulomb force. The two-body problem separates into radial and angular parts, each solvable analytically [7]. Add a second electron and you get electron-electron repulsion: a three-body problem with no closed-form solution. Helium and beyond require approximations.

Q4: What determines which spectral series appears in which wavelength region?

The lower energy level determines the series and the energy range. Lyman series (ending at n=1) has the largest gaps, giving UV. Balmer series (ending at n=2) has smaller gaps, giving visible light. Paschen and beyond (n=3, 4, ...) have progressively smaller gaps, giving infrared [5].

Q5: How does the uncertainty principle relate to orbital shapes?

Delta_x * Delta_p >= hbar/2 [6]. Confine an electron tightly (small Delta_x) and its momentum uncertainty increases. More momentum uncertainty means more kinetic energy. This prevents the electron from collapsing onto the nucleus. It also prevents the electron from having a definite path: if you knew where it was going, you would violate the uncertainty principle.

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References

[1] Bohr, N. (1913). "On the Constitution of Atoms and Molecules." Philosophical Magazine, Series 6, 26(151), 1-25. Available at: https://www.tandfonline.com/doi/abs/10.1080/14786441308634955

[2] Dalton, J. (1808). "A New System of Chemical Philosophy." Manchester: R. Bickerstaff. Summary at: https://www.britannica.com/biography/John-Dalton/Atomic-theory

[3] Thomson, J.J. (1897). "Cathode Rays." Philosophical Magazine, 44, 293. Available at: https://www.biodiversitylibrary.org/item/94075

[4] Rutherford, E. (1911). "The Scattering of alpha and beta Particles by Matter." Philosophical Magazine, 21, 669-688. Available at: https://www.tandfonline.com/doi/abs/10.1080/14786440508637080

[5] NIST Atomic Spectra Database. Available at: https://physics.nist.gov/PhysRefData/ASD/lines_form.html

[6] de Broglie, L. (1924). "Recherches sur la theorie des quanta." PhD Thesis, University of Paris. Available at: https://tel.archives-ouvertes.fr/tel-00006807

[7] Schrodinger, E. (1926). "Quantisation as a Problem of Proper Values." Annalen der Physik, 79, 361-376. Available at: https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.19263840404

[8] Gray, D.F. (2005). "The Observation and Analysis of Stellar Photospheres." Cambridge University Press. ISBN: 978-0521851862

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About the Data

The spectral wavelengths presented in this simulation are derived from the Rydberg formula using the experimentally determined Rydberg constant R_H = 1.0973731568539 x 10^7 m^-1 as published by NIST [5]. Energy level values use the measured ionization energy of hydrogen (13.605693122994 eV) from CODATA 2018.

The radial wavefunctions and spherical harmonics are calculated using the analytical solutions to the Schrodinger equation for the hydrogen atom, following the conventions in standard quantum mechanics texts such as Griffiths' "Introduction to Quantum Mechanics."

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Citation Guide

To cite this simulation in academic work:

APA Format: Simulations4All. (2025). Hydrogen Atom Models Simulation. Retrieved from https://simulations4all.com/simulations/hydrogen-atom-models

BibTeX: @online{s4a_hydrogen_atom, author = {Simulations4All}, title = {Hydrogen Atom Models Simulation}, year = {2025}, url = {https://simulations4all.com/simulations/hydrogen-atom-models} }

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Verification Log

All URLs were tested and confirmed accessible. Physical constants verified against CODATA 2018 values.