Hydrogen energy levels

A glowing gas can produce a set of distinct spectral lines rather than every colour between them. Those lines are clues about allowed energy changes. The energy-level diagram is a map of possible transitions, not a set of tiny classical tracks.

We use the one-electron Coulomb model to calculate spectra while keeping the historical Bohr orbit picture separate from modern quantum states.

What you will learn to do

Explain the negative bound-energy reference and one-electron scope.
Apply Z² and principal-level scaling.
Use Bohr-model radius and energy partitions with their stated limitations.
Calculate photon gaps and spectral-series limits.
Distinguish bound-state absorption from ionization and collision excitation.
Conserve energy across cascades and interpret line-count upper bounds.

Useful before this chapterPhotons, photoelectric emission and matter waves →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Hydrogen levels, spectra and ionization
Learning goals:
- Explain the negative bound-energy reference and one-electron scope.
- Apply Z² and principal-level scaling.
- Use Bohr-model radius and energy partitions with their stated limitations.
- Calculate photon gaps and spectral-series limits.
- Distinguish bound-state absorption from ionization and collision excitation.
- Conserve energy across cascades and interpret line-count upper bounds.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

Why can a bound state have negative energy when the zero is chosen at an electron and nucleus infinitely far apart?

Compare your reasoning

Energy must be supplied to separate the bound system to that reference state. Negative energy is meaningful relative to the declared zero; it is not a negative amount of motion.

Revisit the explanation →

01 / Hydrogen levels, spectra and ionization

Negative energy describes a bound electron, not negative kinetic energy

Choose zero energy for an electron and nucleus infinitely far apart and at rest relative to one another. A bound atomic state then has negative total energy: energy must be supplied to reach the separated state.

For a hydrogen-like ion containing exactly one electron around nuclear charge +Ze, the nonrelativistic Coulomb model gives En ≈ −13.6 Z²/n² eV, with n = 1, 2, 3 and so on. The lowest n = 1 state is the ground state, and higher n values describe excited levels.

The formula neglects finite-nuclear-mass corrections, fine structure and other small effects. It is an effective level formula for the stated one-electron system, not a formula for arbitrary many-electron atoms with the same nuclear charge.

As n increases, the energies approach zero from below and the level gaps become smaller. At zero and above, the electron is no longer bound in the discrete spectrum. Negative total energy does not imply negative kinetic energy; attractive potential energy makes the sum negative.

E_n\approx-\frac{13.6Z^2}{n^2}\ \mathrm{eV},\qquad n=1,2,3,\ldots

One-electron Coulomb atom or ion in the nonrelativistic approximation, with the separated electron–nucleus state as the zero of energy.

Hydrogen level energies in the 13.6 eV approximation
n	Total energy (eV)	Energy needed to ionize from that level (eV)
1	−13.6	13.6
2	−3.40	3.40
3	−1.511…	1.511…
4	−0.850	0.850
∞ limit	0	0

WORKED EXAMPLE

Compare two one-electron systems at specified levels

Find the n = 2 level energy for hydrogen, Z = 1, and for singly ionized helium, He⁺ with Z = 2.

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Hydrogen levels, spectra and ionization
Focus: Negative energy describes a bound electron, not negative kinetic energy
Learning goals:
- Explain the negative bound-energy reference and one-electron scope.
- Apply Z² and principal-level scaling.
- Use Bohr-model radius and energy partitions with their stated limitations.
- Calculate photon gaps and spectral-series limits.
- Distinguish bound-state absorption from ionization and collision excitation.
- Conserve energy across cascades and interpret line-count upper bounds.
Equations in this section (LaTeX):
E_n\approx-\frac{13.6Z^2}{n^2}\ \mathrm{eV},\qquad n=1,2,3,\ldots
Meaning: One-electron Coulomb atom or ion in the nonrelativistic approximation, with the separated electron–nucleus state as the zero of energy.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Hydrogen levels, spectra and ionization

The Bohr construction explains the scale while modern states are not circular tracks

In the historical Bohr model, Coulomb attraction supplies circular centripetal force and angular momentum is restricted to mvr = nℏ. Combining these assumptions gives radii rn = a0n²/Z, where a0 ≈ 0.0529 nm for the usual heavy-nucleus approximation.

For an inverse-square Coulomb circular orbit, kinetic energy is half the magnitude of potential energy: K = −U/2. Therefore E = K + U = −K = U/2. This produces the same principal energy scale as the simple quantum Coulomb calculation.

The Bohr orbit is a useful historical derivation, but it is not a literal modern picture of an electron following a measured circular path. Quantum states have probability distributions and additional quantum numbers. The ground state, for example, does not have the Bohr model’s nonzero orbital angular momentum.

The condition 2πr = nλ with de Broglie λ = h/(mv) reproduces mvr = nℏ in that orbit model. It is a mnemonic connection between quantization and phase consistency, not a complete derivation of all atomic orbitals.

r_n=a_0\frac{n^2}{Z},\qquad K=-E_n,\qquad U=2E_n\quad\text{in the Bohr Coulomb construction}

These radius and orbital-motion statements are explicitly tied to the historical circular-orbit model. Modern bound-state probabilities are not circular trajectories.

WORKED EXAMPLE

A negative total energy still contains positive kinetic energy

Within the Bohr Coulomb construction, a hydrogen n = 2 state has total energy −3.40 eV. Find kinetic and potential energies and its model radius using a0 = 0.0529 nm.

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Hydrogen levels, spectra and ionization
Focus: The Bohr construction explains the scale while modern states are not circular tracks
Learning goals:
- Explain the negative bound-energy reference and one-electron scope.
- Apply Z² and principal-level scaling.
- Use Bohr-model radius and energy partitions with their stated limitations.
- Calculate photon gaps and spectral-series limits.
- Distinguish bound-state absorption from ionization and collision excitation.
- Conserve energy across cascades and interpret line-count upper bounds.
Equations in this section (LaTeX):
r_n=a_0\frac{n^2}{Z},\qquad K=-E_n,\qquad U=2E_n\quad\text{in the Bohr Coulomb construction}
Meaning: These radius and orbital-motion statements are explicitly tied to the historical circular-orbit model. Modern bound-state probabilities are not circular trajectories.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

03 / Hydrogen levels, spectra and ionization

A photon carries the difference between two levels

When an atom emits a photon while moving from a higher-energy bound level to a lower one, the photon energy equals the positive decrease in atomic energy, neglecting the small recoil correction. Thus Eγ = Ei − Ef for emission.

For an upward bound-state transition by photon absorption, the photon energy must match the level gap in the ideal narrow-line model. A photon whose energy is merely larger than one gap is not automatically absorbed into that level with the excess discarded.

Use the actual signed level energies before taking their difference. A transition from n = 4 to n = 2 has final energy more negative, so the emitted energy is positive. The photon does not carry a negative energy because the atomic levels are negative.

Line widths, selection rules and state populations determine which transitions are observed and with what strength. The principal-level energy formula predicts possible energy differences; it does not by itself guarantee every proposed spectral line has nonzero intensity.

E_\gamma=13.6Z^2\left(\frac1{n_{\mathrm{low}}^2}-\frac1{n_{\mathrm{high}}^2}\right)\ \mathrm{eV},\qquad\lambda_0=\frac{hc}{E_\gamma}

n_high > n_low. This positive gap applies to emission downward or resonant absorption upward, with recoil and small corrections neglected.

WORKED EXAMPLE

Calculate a spectral wavelength from signed energies

A hydrogen atom emits in a transition from n = 4 to n = 2. Find photon energy and vacuum wavelength using hc = 1240 eV·nm.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Hydrogen levels, spectra and ionization
Focus: A photon carries the difference between two levels
Learning goals:
- Explain the negative bound-energy reference and one-electron scope.
- Apply Z² and principal-level scaling.
- Use Bohr-model radius and energy partitions with their stated limitations.
- Calculate photon gaps and spectral-series limits.
- Distinguish bound-state absorption from ionization and collision excitation.
- Conserve energy across cascades and interpret line-count upper bounds.
Equations in this section (LaTeX):
E_\gamma=13.6Z^2\left(\frac1{n_{\mathrm{low}}^2}-\frac1{n_{\mathrm{high}}^2}\right)\ \mathrm{eV},\qquad\lambda_0=\frac{hc}{E_\gamma}
Meaning: n_high > n_low. This positive gap applies to emission downward or resonant absorption upward, with recoil and small corrections neglected.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

04 / Hydrogen levels, spectra and ionization

A spectral series fixes one endpoint and varies the other

A series of hydrogen emission lines consists of transitions from several higher n values to a fixed lower level. Transitions ending at n = 1 form the Lyman series, those ending at n = 2 the Balmer series, and those ending at n = 3 the Paschen series.

For a fixed lower level, the smallest gap comes from the nearest higher level, n_high = n_low + 1, and gives the longest wavelength in that series. The largest gap approaches the transition from n_high → ∞ and gives the short-wavelength series limit.

The wavelengths obey the Rydberg form 1/λ = RZ²(1/n_low² − 1/n_high²), with the appropriate reduced-mass constant if high precision matters. The simple 13.6 eV approximation gives a consistent approximate version.

A visible series does not mean every line in it is visible to the human eye. For example, the Balmer series has visible low-order lines but approaches a near-ultraviolet limit. Classify series by their endpoint, then calculate the actual wavelength.

WORKED EXAMPLE

Find the shortest-wavelength Balmer limit

Find the photon energy and approximate wavelength at the hydrogen Balmer series limit, using 13.6 eV and hc = 1240 eV·nm.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Hydrogen levels, spectra and ionization
Focus: A spectral series fixes one endpoint and varies the other
Learning goals:
- Explain the negative bound-energy reference and one-electron scope.
- Apply Z² and principal-level scaling.
- Use Bohr-model radius and energy partitions with their stated limitations.
- Calculate photon gaps and spectral-series limits.
- Distinguish bound-state absorption from ionization and collision excitation.
- Conserve energy across cascades and interpret line-count upper bounds.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

05 / Hydrogen levels, spectra and ionization

Reaching the continuum has a threshold, not one discrete final gap

Ionization removes the electron from its bound state. From level n, the minimum required energy is 0 − En = |En|. This threshold is smaller for an excited atom than for the ground state.

A photon with energy above that threshold can ionize the atom, leaving surplus kinetic energy shared by the electron and recoiling ion. In a heavy-nucleus approximation, most of that surplus becomes electron kinetic energy.

This continuum process differs from bound-to-bound absorption: above threshold there is a continuous range of allowed free-particle kinetic energies. Below threshold, a photon must match an allowed bound-state transition to be resonantly absorbed in the ideal model.

An incident massive particle can transfer only part of its kinetic energy during an inelastic collision and retain the rest. Therefore a collision with energy above an excitation gap can excite that level without the projectile energy having to equal the gap exactly. Momentum and recoil still belong to a complete collision account.

WORKED EXAMPLE

An excited atom has a smaller ionization threshold

A hydrogen atom is initially in n = 2 and absorbs a 5.0 eV photon that ionizes it. Neglect nuclear recoil. Find the electron’s outgoing kinetic energy.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Hydrogen levels, spectra and ionization
Focus: Reaching the continuum has a threshold, not one discrete final gap
Learning goals:
- Explain the negative bound-energy reference and one-electron scope.
- Apply Z² and principal-level scaling.
- Use Bohr-model radius and energy partitions with their stated limitations.
- Calculate photon gaps and spectral-series limits.
- Distinguish bound-state absorption from ionization and collision excitation.
- Conserve energy across cascades and interpret line-count upper bounds.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

06 / Hydrogen levels, spectra and ionization

One atom’s path and an ensemble’s set of lines are different counts

An atom prepared in an excited level can emit directly to a lower level or pass through intermediate levels, subject to transition rules. Along any one cascade, the photon energies add to the same net initial-to-final energy difference.

If an atom starting in principal level n eventually reaches level 1 through strictly decreasing principal levels, at most n − 1 steps occur in that ideal level-only picture. A direct jump produces one photon; a chain through every lower principal level produces n − 1.

Across many atoms with populations and allowed transitions spanning levels 1 through n, there are n(n − 1)/2 possible pairs of distinct principal levels. This is a count of level-pair gaps, and an upper bound for distinct observed lines in the simplified model, not a guarantee that one atom emits that many photons.

Actual spectra depend on sublevels, selection rules, branching probabilities, populations and resolution. Keep a combinatorial upper bound separate from a physical prediction of which lines will be observed.

WORKED EXAMPLE

Check cascade energy without assuming a specific branching probability

In a simplified hydrogen level model, compare a direct n = 3 to n = 1 emission with a cascade 3 → 2 → 1. Use E1 = −13.6 eV, E2 = −3.40 eV and E3 = −13.6/9 eV.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Hydrogen levels, spectra and ionization
Focus: One atom’s path and an ensemble’s set of lines are different counts
Learning goals:
- Explain the negative bound-energy reference and one-electron scope.
- Apply Z² and principal-level scaling.
- Use Bohr-model radius and energy partitions with their stated limitations.
- Calculate photon gaps and spectral-series limits.
- Distinguish bound-state absorption from ionization and collision excitation.
- Conserve energy across cascades and interpret line-count upper bounds.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

The separated electron–nucleus state defines zero; bound total energies are negative.
The simple −13.6Z²/n² eV formula applies to one-electron systems.
Bohr circular orbits are a historical model, not literal modern electron tracks.
Photon energy is a positive level difference.
A series fixes the lower endpoint; its limit gives the largest gap and shortest wavelength.
Ionization from level n requires |En|, with excess energy going into outgoing motion.
An ensemble’s possible line pairs are not one atom’s emitted-photon count.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1