Since 20 May 2019 the SI defines the kilogram by fixing h at 6.626 070 15 × 10−34 J·s. It is a defined value, so it has no measurement uncertainty.

Quantization in physics: Planck to the atom

In physics, quantization means a quantity can take only certain discrete values. For bound systems such as an electron in an atom, energy comes in fixed levels, not a continuous range. It started with Planck's E = hf in 1900 and now underlies atomic spectra, lasers and semiconductors. This guide covers the key results, works two calculations, and separates physical from digital quantization.

Planck and Einstein: energy in packets of hf

Blackbody radiation and E = nhf

Classical physics could not explain the light spectrum of a hot object, a blackbody. The Rayleigh–Jeans law predicted intensity rising without limit at short wavelengths, the "ultraviolet catastrophe", while experiments showed the spectrum peaking and falling. In 1900 Max Planck matched the data by assuming that oscillators in the cavity walls hold energy only in whole multiples of a unit proportional to their frequency:

E = n h f (n = 1, 2, 3, …)

f is the oscillator frequency and h is Planck's constant. Since the 2019 revision of the SI, h is defined as exactly 6.626 070 15 × 10⁻³⁴ J·s, and the kilogram is defined from it.

The step hf is tiny at everyday scales. A green photon at 550 nm has f = c/λ ≈ 5.45 × 10¹⁴ Hz and energy hf ≈ 3.61 × 10⁻¹⁹ J, or about 2.25 eV. A 1 mW green laser pointer emits about 2.8 × 10¹⁵ photons per second, which is why light looks continuous. Shortcut: with hc ≈ 1,239.84 eV·nm, photon energy in eV ≈ 1,239.84 / λ in nm.

The photoelectric effect: light is quantized too

Planck quantized the oscillators. In 1905 Einstein went further and proposed that light itself arrives in quanta, now called photons, each carrying energy hf. That explained the photoelectric effect, where light ejects electrons from a metal surface:

K_max = h f − φ

K_max is the maximum kinetic energy of an ejected electron; φ is the work function, the minimum energy needed to free an electron from that metal.

None of the key observations fits a classical wave picture. Below a cutoff frequency f_c = φ/h no electrons leave, however bright the light. Brighter light gives more electrons, not faster ones. Emission starts with no measurable delay even in dim light, because one photon delivers all its energy to one electron.

Worked example: sodium under violet light

Sodium's work function is about 2.46 eV, so its cutoff wavelength is 1,239.84 / 2.46 ≈ 504 nm. Violet light at 400 nm carries 1,239.84 / 400 ≈ 3.10 eV per photon, so the fastest electrons leave with K_max ≈ 3.10 − 2.46 = 0.64 eV. Red light at 650 nm (1.91 eV) ejects nothing at any intensity.

Hydrogen energy levels and the Bohr model

Hydrogen gas does not glow with a continuous rainbow; it emits sharp lines at specific wavelengths. In 1885 Johann Balmer found an empirical formula for the visible lines. In 1913 Niels Bohr explained them by quantizing the electron's orbital angular momentum to L = nħ, where ħ = h/2π. Combined with Coulomb attraction, that condition gives a ladder of allowed energies:

E_n = −13.6 eV / n² (n = 1, 2, 3, …)

The constant is the Rydberg energy, 13.605 693 eV in CODATA 2022, computed for an infinitely heavy nucleus. Accounting for the proton's finite mass (the reduced-mass correction) gives 13.598 eV, the measured ionization energy of hydrogen. The negative sign means the electron is bound; E = 0 is the threshold for escaping the atom.

The levels crowd together as n grows: E₁ = −13.61 eV, E₂ = −3.40 eV, E₃ = −1.51 eV, E₄ = −0.85 eV. Above zero the electron is free and its energy is continuous: quantization comes from confinement.

Transitions and the Rydberg formula

When an electron drops from level n_i to a lower level n_f, it emits a photon carrying the energy difference. Dividing by hc gives the Rydberg formula:

1/λ = R (1/n_f² − 1/n_i²)

R_∞ = 10,973,731.568 m⁻¹ (CODATA 2022). Transitions ending at n_f = 1 form the ultraviolet Lyman series; n_f = 2 is the Balmer series, whose first lines are visible; n_f = 3 is the infrared Paschen series.

Worked example: the n = 3 → 2 photon (H-alpha)

ΔE = 13.6057 eV × (1/2² − 1/3²) = 13.6057 × (1/4 − 1/9) = 13.6057 × 5/36 = 1.890 eV
λ = hc / ΔE = 1,239.84 eV·nm / 1.8897 eV ≈ 656.1 nm
Same result from the Rydberg formula: 1/λ = 10,973,731.6 × 5/36 = 1,524,129 m⁻¹, so λ ≈ 656.1 nm.

This is the red Balmer-alpha (H-alpha) line. The reduced-mass correction moves it to about 656.47 nm, close to the measured 656.46 nm in vacuum (656.28 nm in air). The shortest Balmer line (n_i → ∞) is at about 364.6 nm.

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Particle in a box: quantization from confinement

The simplest fully quantum example is a particle trapped between two impenetrable walls a distance L apart. The wavefunction must vanish at both walls, so only standing waves with a whole number of half-wavelengths fit, like the modes of a guitar string:

E_n = n² h² / (8 m L²) = n² π² ħ² / (2 m L²) (n = 1, 2, 3, …)

Two lessons follow. The lowest energy is not zero; n = 0 would mean no wave at all, so a confined particle always keeps some zero-point motion. And the spacing scales as 1/(mL²): quantization is dramatic for light particles in small spaces and negligible for anything large.

Worked example: an electron in a 1 nm box

h² = (6.626 070 15 × 10⁻³⁴)² = 4.3905 × 10⁻⁶⁷ J²·s²
8mL² = 8 × 9.109 384 × 10⁻³¹ kg × (1 × 10⁻⁹ m)² = 7.2875 × 10⁻⁴⁸ kg·m²
E₁ = 4.3905 × 10⁻⁶⁷ / 7.2875 × 10⁻⁴⁸ = 6.025 × 10⁻²⁰ J
In electronvolts: 6.025 × 10⁻²⁰ / 1.602 177 × 10⁻¹⁹ ≈ 0.376 eV
E₂ = 4E₁ ≈ 1.504 eV and E₃ = 9E₁ ≈ 3.384 eV

A 2 → 1 transition releases 3E₁ ≈ 1.128 eV, a near-infrared photon at about 1,099 nm. Thermal energy at room temperature is k_BT ≈ 0.026 eV, so these levels are well resolved. In a 1 cm box E₁ falls by 10¹⁴, to about 4 × 10⁻¹⁵ eV. Quantum dots exploit this scaling: changing a nanocrystal's size changes its colour.

Try it: Energy levels calculator

Hydrogen transitions with photon wavelength and colour, plus particle-in-a-box energies.

Open the tool

Quantized angular momentum and quantum numbers

Solving the Schrödinger equation for hydrogen in three dimensions gives the same energies as Bohr's model. It also shows that several quantities are quantized together, each labelled by a quantum number:

Quantum number	Allowed values	What it fixes
Principal, n	1, 2, 3, …	Energy level (in hydrogen, energy depends on n alone)
Orbital, l	0, 1, …, n − 1	Orbital angular momentum magnitude, L = √(l(l+1)) ħ
Magnetic, m_l	−l, …, 0, …, +l	Component along a chosen axis, L_z = m_l ħ
Spin, m_s	+½ or −½	Electron spin component, S_z = m_s ħ

That gives n² orbital states per level, doubled by spin to 2n²: 2, 8, 18, 32. With the Pauli exclusion principle (one electron per complete set of quantum numbers), this counting underlies the periodic table. Angular momentum is quantized twice over: its magnitude takes only the values √(l(l+1))ħ, and its projection on any axis only whole multiples of ħ.

The quantum harmonic oscillator

A mass on an ideal spring, or an atom vibrating in a molecule, is a harmonic oscillator. Quantum mechanics gives it evenly spaced levels:

E_n = (n + ½) ħω (n = 0, 1, 2, …)

ω = 2πf is the angular frequency. Adjacent levels differ by exactly ħω = hf; the ground state keeps a zero-point energy of ħω/2.

This is Planck's picture made exact: the oscillator exchanges energy only in units of hf, plus a zero-point term he did not include. Equal steps mean a vibrating molecule absorbs one characteristic infrared frequency per mode; real molecules deviate at higher n. Each mode of the electromagnetic field is also an oscillator, which leads to the last section.

Physical, digital and "second" quantization

The word quantization is used in three different senses that are easy to confuse.

Physical quantization is a property of nature. A bound system has only certain allowed values of energy or angular momentum. Measuring the system finds one of those values, and nothing has been rounded or lost.
Digital quantization is an engineering choice. A converter or a compressed AI model rounds a continuous value to the nearest of a finite set of levels and accepts a known error, ±½ step, as the price. The levels are chosen by people, not by nature. See ADC quantization error and what is quantization.
Canonical and second quantization are procedures for building quantum theories. Canonical quantization turns classical position and momentum into operators that obey [x̂, p̂] = iħ. Second quantization applies the same idea to fields, describing states by how many particles occupy each mode, with creation and annihilation operators. The name is a historical misnomer: nothing is quantized twice.

The common thread is discreteness; the cause differs. Physical levels come from boundary conditions on waves, digital levels from finite storage.

Key takeaways

Planck's E = nhf (1900) introduced energy quanta; h = 6.626 070 15 × 10⁻³⁴ J·s exactly since 2019.
The photoelectric effect, K_max = hf − φ, shows that light itself comes in photons of energy hf.
Hydrogen levels follow E_n = −13.6057 eV / n²; the 3 → 2 transition emits H-alpha at about 656 nm.
A particle in a box has E_n = n²h²/(8mL²): an electron in a 1 nm box has E₁ ≈ 0.376 eV.
Quantum numbers n, l, m_l and m_s label the allowed states, 2n² per hydrogen level.
The harmonic oscillator has equal steps of ħω and a zero-point energy of ħω/2.
Physical quantization is a property of nature; digital quantization is a rounding choice made by engineers.

Frequently asked questions

Is all energy quantized?

No. Only bound or confined systems have discrete levels. A free electron, or one with enough energy to leave an atom, can have any kinetic energy. That is why hydrogen's spectrum has sharp lines plus a continuum beyond the ionization limit.

Why is the hydrogen ground state −13.6 eV and not −13.6057 eV?

Both are roundings. The Rydberg energy for an infinitely heavy nucleus is 13.6057 eV; including the proton's finite mass gives 13.598 eV, which matches the measured ionization energy. Textbooks round either to 13.6 eV.

Why can't a particle in a box have zero energy?

Zero energy would require n = 0, which makes the wavefunction zero everywhere, meaning no particle. Equivalently, confining a particle to a region of size L gives it a momentum spread of order h/L, so it always has some kinetic energy.

Why is h now exact?

Since 20 May 2019 the SI defines the kilogram by fixing h at 6.626 070 15 × 10⁻³⁴ J·s. It is a defined value, so it has no measurement uncertainty.

Sources and further reading

BIPM: SI defining constants — exact value of the Planck constant in the revised SI
NIST: CODATA fundamental physical constants — 2022 values for the Rydberg constant, electron mass and hc
NIST: SI redefinition — the revised SI and its effective date
OpenStax University Physics Vol. 3, 6.1 Blackbody radiation — ultraviolet catastrophe and Planck's hypothesis
OpenStax, 6.2 Photoelectric effect — Einstein's equation and work function table
OpenStax, 6.4 Bohr's model of the hydrogen atom — energy levels, Rydberg formula and series
OpenStax, 7.4 The quantum particle in a box — energy levels and zero-point energy
OpenStax, 7.5 The quantum harmonic oscillator — equally spaced levels and ħω/2
OpenStax, 8.1 The hydrogen atom — quantum numbers and angular momentum quantization
NIST Handbook of Basic Atomic Spectroscopic Data: hydrogen — measured ionization energy and strong lines

About this guide. Written and fact-checked by the Quantization.com editorial team against the primary sources listed above. Spotted an error or an outdated command? Send a correction — we fix verified issues within a few days. See our editorial policy.