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Energy Level Diagram For Hydrogen

Key idea: Read a hydrogen energy level diagram, interpret negative energies and ionisation energy, and use ΔE = hf = hc/λ for photon emission/absorption (A Level Physics).

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Analyse atomic energy levels and emission or absorption spectra.

1. Definitions (Must Know)

A. Principal quantum number, n

n labels the energy level, with allowed values:

n = 1,2,3,…

B. Ground state and excited state

  • Ground state: the lowest energy state (n = 1).
  • Excited state: any state above the ground state (n ≥ 2).

C. Ionisation and ionisation energy

  • Ionisation means the electron is no longer bound to the atom.
  • For hydrogen, the ionisation limit corresponds to n → ∞ and is defined as E = 0 eV.

The ionisation energy from the ground state is 13.6 eV.

D. Photon energy in a transition

For a transition between two levels:

Δ E = hf = hc/λ

2. Key Ideas (What Earns Marks)

Photon absorption and emission between atomic energy levelsDiscrete atomic energy levels show an upward absorption transition and a downward emission transition, each labelled with photon energy equal to the level difference.EnergyE₁E₂E₃absorptionhf = E₃ − E₁emissionhf = E₃ − E₂
Scroll diagram horizontally to read all labels.
Absorption raises an electron only when the photon energy matches an allowed gap. Emission releases a photon whose energy equals the downward energy-level difference.
  • Energy levels in hydrogen are discrete.
  • Energies are often negative because we define E = 0 for a free electron at infinity; bound states lie below that reference.
  • Level spacing gets smaller as n increases (levels “crowd” near E = 0).
  • Emission vs absorption:
    • emission: electron drops to a lower level and emits a photon,
    • absorption: electron rises to a higher level by absorbing a photon of the exact energy gap.
Fast diagram reading

A downward arrow means emission (photon out). An upward arrow means absorption (photon in).

3. Detailed Explanations

Hydrogen energy levels n = 1 to 6 at −13.60, −3.40, −1.51, −0.85, −0.54 and −0.38 eV. They crowd toward the 0 eV ionisation limit as n increases. An arrow from n = 1 to the limit marks the 13.6 eV ground-state ionisation energy.
Hydrogen levels follow Eₙ = −13.6 eV/n² and crowd toward 0 eV as n → ∞. Removing the ground-state electron requires 13.6 eV.

A. Why energies are negative

We set the reference E = 0 as:

  • electron completely free from the atom (at infinity).

Bound states have less energy than a free electron, so their energies are negative relative to this reference.

B. Using ΔE to find photon frequency or wavelength

If the electron transitions from E_high to E_low, then the photon energy is:

Δ E = E_high-E_low

For emission, Δ E is carried away by the photon:

Δ E = hf = hc/λ

For absorption, the photon must have exactly that energy to raise the electron to the higher level.

4. Common Mistakes

  • Mixing sign: use the energy difference between the two levels (a positive photon energy).
  • Using “n = 0” as a level (not used; n = 1 is the ground state).
  • Forgetting that λ must be in metres in E = hc/λ.

5. Exam Tips

  • Convert eV to J only if needed: 1 eV = 1.60 × 10⁻¹⁹ J
  • If the question gives energies in eV, stay in eV until the last step.
  • Always show: Δ E → f or λ.

6. Worked Examples

Modelled example 1

Photon wavelength from a transition

Core

Problem

An electron drops from -1.51 eV to -3.40 eV in hydrogen. Find the emitted photon wavelength. Take hc = 1240 eV nm.
Study the worked solution

  1. Find the positive energy gap

    Method

    Δ E = 1.89 eV.

    Reason

    The photon carries the amount by which the atom’s energy decreases, so use higher minus lower level.

    Working

    Δ E = (-1.51)-(-3.40) = 1.89 eV

  2. Convert gap to wavelength

    Method

    λ = 656 nm.

    Reason

    The supplied hc value is already in compatible eV–nanometre units.

    Working

    λ = hc/(Δ E) = 1240/1.89 nm = 656 nm

Common misconception 2

Ionisation energy from n = 2

Find and correct the mistake

Learner claim

Hydrogen has E₂ = -3.40 eV. A learner says the ionisation energy from n = 2 is -3.40 eV because that is the level’s energy. Diagnose the sign and find the required energy.

Try this before viewing the solution

Unit: eV

View solution step by step

  1. Identify the final state

    Method

    The ionisation limit is 0 eV.

    Reason

    Zero is defined as a free electron at infinity.

    Working

    initial: -3.40 eV; final: 0 eV

  2. Find required energy

    Method

    The ionisation energy is + 3.40 eV.

    Reason

    Energy must be supplied to raise the bound electron to the free-electron reference.

    Working

    Δ E = 0-(-3.40) = 3.40 eV

7. Mind Stretchers

Mind stretcher 1: Crowding of levelsExtension

Explain why many spectral lines can get very close together near the ionisation limit.

Show Answer

As n increases, energy levels get closer together (smaller energy gaps).

So the photon energies for transitions between high-n states are very similar, producing closely spaced spectral lines.

8. Optional (Enrichment)

A. Spectral series names

For hydrogen, transitions ending at:

  • n = 1 are in the ultraviolet (Lyman series),
  • n = 2 are in the visible (Balmer series),
  • n = 3 are in the infrared (Paschen series).

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027