Discrete Energy Levels in Atoms

Spec mapping: OCR H556 Module 4.5 — Quantum physics (discrete energy levels of electrons in atoms; the Bohr frequency condition $hf = E_{\text{upper}} - E_{\text{lower}}$hf=Eupper​−Elower​ for emission and absorption; the use of negative energy convention with ionisation at $E = 0$E=0; the hydrogen energy-level structure and the Lyman, Balmer and Paschen series; experimental confirmation via Franck-Hertz). Refer to the official OCR H556 specification document for exact wording.

So far in this course we have seen that light is quantised into photons and that matter is quantised into de Broglie waves. In this lesson and the next, we bring the two ideas together to explain one of the most striking phenomena in physics: the discrete energy levels of atoms and the line spectra that result when electrons move between them.

The classical picture of an atom, inherited from Rutherford's 1911 experiments, is a tiny, heavy, positively charged nucleus surrounded by orbiting electrons — a miniature solar system. But this picture has a fatal problem. An electron in circular orbit is constantly accelerating (centripetal acceleration toward the nucleus). A classical accelerating charge radiates electromagnetic waves, losing energy. The orbiting electron should therefore spiral rapidly into the nucleus, emitting a continuous spectrum of radiation as it does so. The calculated time to collapse is of order 10⁻¹¹ s.

The fact that atoms exist, and are stable over astronomical timescales, is a direct refutation of this classical prediction. Something new is needed.

In 1913, Niels Bohr proposed his solution: electrons in atoms can only occupy certain specific, discrete orbits, and while in one of these stationary states they do not radiate at all. They radiate (or absorb) only when making a transition from one allowed state to another, and the energy of the emitted (or absorbed) photon is exactly equal to the difference in energy between the two states.

This was a radical and somewhat ad hoc proposal in 1913. But it worked: it correctly predicted the observed spectrum of hydrogen to extraordinary precision. And in the fully developed quantum mechanics of 1925–26, Bohr's energy levels emerged naturally as solutions of the Schrödinger equation, no longer requiring ad hoc postulates.

This lesson introduces energy levels, the language used to describe them, and the concept of quantisation in atoms. It is central to Module 4.5 of the OCR A-Level Physics A specification (H556).

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Energy Levels in a Bound System

A bound system is one in which constituents are held together by attractive forces so that they cannot escape. An electron in a hydrogen atom is bound: the Coulomb attraction between the electron and the proton holds it in place. Only by supplying sufficient energy (the ionisation energy) can one tear the electron away from the atom.

In a bound system, quantum mechanics predicts that the allowed energies form a discrete set. The system cannot have just any energy; it must have one of a definite list of values:

E_1, E_2, E_3, E_4, ...

These values are called energy levels. The lowest is called the ground state E_1; the higher ones are excited states E_2, E_3, ....

Crucially, there are no energies in between. An electron cannot have an energy half-way between E_1 and E_2; the intermediate values are simply not allowed. This is the essence of quantisation: some variables that would be continuous classically are restricted to discrete values quantum-mechanically.

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The Energy Level Diagram

The standard way of visualising energy levels is with an energy-level diagram: a vertical scale of energy, with horizontal lines drawn at each allowed value.

flowchart BT
    I["E = 0<br/>Ionised electron"]
    E5["E₅ = -0.54 eV"]
    E4["E₄ = -0.85 eV"]
    E3["E₃ = -1.51 eV"]
    E2["E₂ = -3.40 eV"]
    E1["E₁ = -13.60 eV<br/>Ground state"]
    E1 --> E2
    E2 --> E3
    E3 --> E4
    E4 --> E5
    E5 --> I

Several features are worth noting.

The energies are negative. This is a matter of convention, but a useful one. The zero of energy is chosen as the energy of an electron at infinity (i.e. a free, unbound electron). A bound electron has less energy than a free one, so its energy is negative. The ground state of hydrogen is -13.6 eV, meaning it is 13.6 eV below the free-electron level. To remove an electron from the ground state of hydrogen to infinity (to ionise the atom), you must supply +13.6 eV of energy — the hydrogen ionisation energy.

The levels converge towards zero. As n increases, the levels get closer together, piling up on the zero of energy. For hydrogen, E_n = -13.6/n² eV, so E_2 = -3.40 eV, E_3 = -1.51 eV, E_4 = -0.85 eV, and so on. Above E = 0 the electron is free, and the continuous (non-quantised) spectrum of free-electron energies begins.

The ground state is the stable state. At room temperature, the overwhelming majority of hydrogen atoms are in their ground state. Thermal energy k_B T \approx 0.025 eV is far smaller than the gap E_2 - E_1 = 10.2 eV, so thermal excitation to E_2 is extremely unlikely. Atoms have to be excited deliberately — by collisions in a discharge tube, by absorbing photons, or by other means — before they populate the excited states.

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Transitions Between Levels

An atom in an excited state is unstable. After a short time — typically of order 10⁻⁸ s — it makes a transition to a lower state, emitting a photon in the process. Energy conservation requires

$$hf = E_{\text{upper}} - E_{\text{lower}}$$hf=Eupper​−Elower​

where hf is the energy of the emitted photon. This is the Bohr frequency condition. It relates the photon frequencies observed experimentally to the otherwise-invisible internal energy levels of the atom.

Conversely, if an atom in the ground state absorbs a photon of exactly the right energy, it can be promoted to an excited state. Again,

$$hf = E_{\text{upper}} - E_{\text{lower}}$$hf=Eupper​−Elower​

Exactly. A photon with slightly more or less energy cannot be absorbed at all. This all-or-nothing resonance condition is the reason atomic spectra consist of sharp, discrete lines rather than a continuous smear.

flowchart TB
    A["Excited state E_2"]
    B["Ground state E_1"]
    A -- "emission: photon hf = E_2 - E_1" --> B
    B -- "absorption: photon hf = E_2 - E_1" --> A

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Experimental Evidence: Franck and Hertz

The direct experimental confirmation that atoms possess discrete energy levels came in 1914 from James Franck and Gustav Hertz. They passed electrons through mercury vapour and measured the energy lost by each electron as it traversed the vapour.

Key finding: electrons pass through unimpeded until their kinetic energy reaches about 4.9 eV; at exactly this energy, they suddenly lose most of their kinetic energy in a single collision with a mercury atom and mercury vapour begins to glow in the ultraviolet at a wavelength of about 254 nm.

The interpretation is straightforward in the Bohr picture. A mercury atom has a first excited state 4.9 eV above its ground state. An electron below this energy cannot excite the atom — there is no available state for the atom to go to — so the collision is elastic and the electron passes through. At exactly 4.9 eV, the electron can promote the mercury atom to its first excited state, giving up exactly 4.9 eV of kinetic energy. The excited mercury atom then de-excites by emitting a photon of wavelength

$$\lambda = hc/E = (1240 eV nm)/(4.9 eV) \approx 253 nm$$λ=hc/E=(1240eVnm)/(4.9eV)≈253nm

— which matches the observed glow.

This was the first direct experimental verification that atoms possess discrete energy levels, with transitions at specific, predictable energies. Franck and Hertz shared the 1925 Nobel Prize for this work. It is also a beautiful piece of confirmation that de Broglie's matter waves and Bohr's discrete levels are two sides of the same coin.

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Quantum Numbers

In Bohr's original treatment, the allowed orbits of the hydrogen atom were labelled by a single integer n = 1, 2, 3, ..., called the principal quantum number. The energy of a level depends only on n:

$$E_n = -13.6/n^{2}\,\text{eV} \quad \text{(hydrogen)}$$En​=−13.6/n2eV(hydrogen)

This formula, empirically established by Rydberg in 1888 and derived theoretically by Bohr in 1913, is one of the great successes of early quantum physics.

In the full quantum mechanics of 1926, more quantum numbers appeared: the orbital angular momentum quantum number ℓ, the magnetic quantum number m_ℓ, and (with spin, 1927) the spin quantum number m_s. For A-Level we stick with the single n-based description, which is perfectly sufficient for hydrogen.

Heavier atoms have more complex level structures, with fine structure arising from electron-electron interactions, relativistic corrections, and spin-orbit coupling. The details are not part of the A-Level curriculum, but the general principle — that every atom has a characteristic set of discrete levels — applies universally.

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The Hydrogen Energy Levels

For the hydrogen atom, the theoretical formula gives:

n	E_n (eV)
1	-13.60
2	-3.40
3	-1.51
4	-0.85
5	-0.544
6	-0.378
∞	0 (ionisation)

The gaps between successive levels:

Transition	ΔE (eV)	Photon wavelength
n = 2 → 1	10.20	121.5 nm (UV, Lyman series)
n = 3 → 1	12.09	102.5 nm (UV)
n = 3 → 2	1.89	656.1 nm (red, Balmer H-alpha)
n = 4 → 2	2.55	486.0 nm (cyan-blue, Balmer H-beta)
n = 5 → 2	2.86	434.0 nm (violet, Balmer H-gamma)
n = 4 → 3	0.66	1875 nm (IR, Paschen series)

These values match experimentally observed wavelengths of hydrogen emission lines to astonishing precision — historically, this was the clinching confirmation of Bohr's model.

Note also that for transitions to n = 1, the photons are in the Lyman series (ultraviolet). Transitions to n = 2 produce the Balmer series (visible). Transitions to n = 3 produce the Paschen series (infrared). The three series together span the ultraviolet, visible and infrared — a window into the internal structure of an atom.

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Worked Example 1: Photon from a Transition

Calculate the wavelength of the photon emitted when a hydrogen atom makes a transition from n = 3 to n = 2.

Solution.

E_3 = -1.51 eV, E_2 = -3.40 eV.

$$\Delta E = E_3 - E_2 = -1.51 - (-3.40) = 1.89 eV$$ΔE=E3​−E2​=−1.51−(−3.40)=1.89eV

(This is the photon energy — positive, as it should be. Watch the signs: E_upper - E_lower > 0.)

Wavelength:

$$\lambda = 1240/1.89 \approx 656 nm$$λ=1240/1.89≈656nm

Answer: 656 nm — the red H-alpha line of the Balmer series, familiar from photographs of hydrogen-rich astronomical objects.

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Worked Example 2: Ionisation from an Excited State

What is the minimum photon energy (and corresponding maximum wavelength) required to ionise a hydrogen atom that is initially in its first excited state (n = 2)?

Solution.

From n = 2 (at -3.40 eV) to the ionisation level (E = 0), the energy change is +3.40 eV. So the minimum photon energy is 3.40 eV.

Corresponding wavelength:

$$\lambda _max = 1240/3.40 \approx 365 nm$$λm​ax=1240/3.40≈365nm

Answer: 365 nm, in the near-ultraviolet.

Note that this is much easier than ionising from the ground state (which requires 13.6 eV, corresponding to λ ≈ 91 nm in the extreme UV). Atoms in excited states are "soft targets" for ionisation — relevant for understanding photoionisation rates in stars and interstellar gas.

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Worked Example 3: Excitation from the Ground State

Can a photon of wavelength 500 nm excite a ground-state hydrogen atom to the n = 2 state? What about a photon of 121 nm?

Solution.

Photon 1: E = 1240/500 = 2.48 eV. Energy needed for 1 → 2: 10.20 eV. Insufficient — no absorption.

Photon 2: E = 1240/121 = 10.25 eV. Energy needed: 10.20 eV. The photon energy is close to the transition energy but not exact. In fact, the hydrogen Lyman-alpha line is at exactly 121.57 nm (E = 10.20 eV). A photon of 121.0 nm would not be absorbed — its energy is slightly too high, and there is no corresponding level, so it passes straight through.

Answer: No to both. Atomic absorption is a sharp resonance, not a "threshold" as in the photoelectric effect. Only photons of exactly the right energy can excite the transition.

Exam Tip: Atomic absorption is a resonance. Photon energy must match the transition energy exactly (not merely exceed it). This is in marked contrast to the photoelectric effect, where photon energies above the threshold are all absorbed, with the excess becoming electron kinetic energy. Make sure you understand the difference: in the photoelectric effect the electron ends up free in a continuum of final states; in atomic excitation the electron ends up in a specific discrete state.

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Worked Example 4: Transitions to Multiple Lower States

A hydrogen atom is in the n = 3 state. List all the possible photons it can emit as it returns to the ground state.

Solution. From n = 3, the atom can make a direct transition 3 \to 1, or a two-step transition 3 \to 2 \to 1.

• 3 → 1: ΔE = -1.51 - (-13.60) = 12.09 eV, λ = 1240/12.09 ≈ 102.5 nm (UV)
• 3 → 2: ΔE = -1.51 - (-3.40) = 1.89 eV, λ = 1240/1.89 ≈ 656 nm (red)
• 2 → 1: ΔE = -3.40 - (-13.60) = 10.20 eV, λ = 1240/10.20 ≈ 121.5 nm (UV)

So the possible emissions are: 102.5 nm, 121.5 nm, and 656 nm.

Total emitted energy in either pathway: 12.09 eV (agreeing, as it must, with energy conservation).

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Hydrogen Energy-Level Diagram — Visual Reference

The following SVG shows the first five hydrogen energy levels with three labelled transitions (Lyman, Balmer, Paschen series). Vertical axis represents energy on the conventional negative scale (with $E = 0$E=0 at ionisation).

The vertical lines indicate downward transitions (photon emission). Lyman series ends on $n=1$n=1 and produces UV photons; Balmer ends on $n=2$n=2 producing visible photons; Paschen ends on $n=3$n=3 producing infrared photons. The same three series, reversed (upward), are the corresponding absorption lines.

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Synoptic Links

• Photon model (Module 4.5, lesson 1) — $E = hf$E=hf gives the energy of the photon emitted or absorbed in a transition. Bohr's frequency condition $hf = \Delta E$hf=ΔE is the atomic counterpart of the photoelectric equation $hf = \phi + KE$hf=ϕ+KE.
• de Broglie matter waves (Module 4.5, lesson 6) — discrete energy levels arise from imposing standing-wave boundary conditions on the matter wave of a confined electron. The "Bohr postulate" of allowed orbits, in modern terms, is the requirement that an integer number of de Broglie wavelengths fit around the orbit ($n\lambda = 2\pi r$nλ=2πr). This is the conceptual bridge from lesson 6 to lesson 9.
• Standing waves on a string (Module 4.4) — boundary conditions $\sin(k L) = 0$sin(kL)=0 give discrete allowed wavenumbers $k_n = n\pi/L$kn​=nπ/L and hence discrete frequencies. The Schrödinger equation in a binding potential gives the same structure for an electron, with energies discrete.
• Line spectra (next lesson) — discrete levels generate discrete photon frequencies, observable as bright/dark lines in atomic spectra. This is the experimental face of the energy-level structure.

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Specimen question modelled on the OCR H556 paper format