What Is h in the Wavelength Equation? Planck’s Constant

This calculator works out the energy of a photon from its wavelength, or the wavelength from its energy, using the equation E = hc/λ. You enter either the wavelength in nanometres or the energy in electronvolts, and you get the corresponding value back, with Planck’s constant (h) applied automatically.

Formula

The wavelength equation you are asking about is usually written as:

E = h × c / λ

where:

• E is the energy of the photon, in joules (J) or electronvolts (eV).
• h is Planck’s constant, 6.62607015 × 10⁻³⁴ J·s.
• c is the speed of light in a vacuum, 299,792,458 m/s.
• λ (lambda) is the wavelength of the light, in metres (m).

When you combine h and c into a single constant, you get 1.986445857 × 10⁻²⁵ J·m. If you work in electronvolts and nanometres, that becomes 1239.841984 eV·nm — the number used in the calculator above.

What Exactly Is h?

h is Planck’s constant, named after Max Planck, who introduced it in 1900 to explain the spectrum of light emitted by a hot object (blackbody radiation). It is a fundamental constant of nature, meaning it does not change with location, temperature, or any other condition. Its value is exactly 6.62607015 × 10⁻³⁴ joule-seconds, because the SI definition of the kilogram was revised in 2019 to fix h to that number.

The unit J·s (joule-seconds) looks odd at first, but it makes sense: energy (J) multiplied by time (s). That unit is the same as the unit of angular momentum, which is why h appears in quantum mechanics wherever discrete amounts of energy or spin are involved.

In the wavelength equation, h is the proportionality constant that links the energy of a single photon to its frequency (E = hf). Since frequency and wavelength are related by c = fλ, substituting f = c/λ gives you the wavelength form: E = hc/λ.

Where the Equation Comes From

The equation E = hc/λ is not a separate law — it is a combination of two relationships:

1. Planck’s relation: E = hf, where f is the frequency of the light.
2. The wave equation: c = fλ, which relates frequency, wavelength, and the speed of light.

Rearranging the wave equation to f = c/λ and substituting into Planck’s relation gives E = h(c/λ) = hc/λ. That is the entire derivation. It applies to any electromagnetic wave — radio, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays — as long as you treat the light as a stream of particles (photons), each carrying a discrete amount of energy.

Worked Example

Let’s use the default values from the calculator: a wavelength of 500 nm.

1. Convert the wavelength to metres: 500 nm = 500 × 10⁻⁹ m = 5.00 × 10⁻⁷ m.
2. Multiply h and c: (6.62607015 × 10⁻³⁴) × (2.99792458 × 10⁸) = 1.986445857 × 10⁻²⁵ J·m.
3. Divide by the wavelength: (1.986445857 × 10⁻²⁵) / (5.00 × 10⁻⁷) = 3.972891714 × 10⁻¹⁹ J.
4. Convert to electronvolts (1 eV = 1.602176634 × 10⁻¹⁹ J): (3.972891714 × 10⁻¹⁹) / (1.602176634 × 10⁻¹⁹) = 2.479 eV.

That matches the calculator’s output of 2.479 eV for 500 nm (green light). The shortcut version — 1239.841984 / 500 = 2.47968, which rounds to 2.480 eV — is what the calculator uses, and the small difference comes from rounding the combined constant.

Common Mistakes and How to Avoid Them

The most frequent error is mixing up units. If you use the wavelength in nanometres but the constant h in J·s, you will get a wildly wrong answer. Always convert the wavelength to metres when using h and c directly, or use the combined constant 1239.841984 eV·nm, which assumes nanometres and electronvolts.

Another common mistake is forgetting that the equation gives the energy of one photon. If you have a beam of light with many photons, the total energy is the photon energy multiplied by the number of photons. For example, a 1-milliwatt green laser (500 nm) emits about 2.5 × 10¹⁵ photons per second, because each photon carries only 2.48 eV ≈ 4 × 10⁻¹⁹ J.

Rounding too early also causes errors. The combined constant 1239.841984 eV·nm is accurate to six significant figures; if you round it to 1240, you introduce an error of about 0.01%, which is fine for most purposes but not for precise spectroscopy.

When to Use the Equation

You use E = hc/λ whenever you need to convert between the wavelength of light and the energy of its photons. Typical situations include:

• Calculating the energy of light emitted by an LED or laser.
• Finding the wavelength of light needed to break a chemical bond (photodissociation).
• Working out the energy of X-rays or gamma rays in medical imaging or astronomy.
• Understanding why ultraviolet light is more damaging than visible light — shorter wavelength means higher photon energy.

The equation works for any electromagnetic wave, but it does not apply to sound waves or water waves, because those are not made of photons. It also fails for very low-energy phenomena where the wave nature of light dominates, but for practical purposes (anything from radio to gamma rays) it is accurate.

Limits of the Equation

The equation E = hc/λ is a simplified, non-relativistic version. It assumes the photon is massless and travels at exactly c, which is true in a vacuum. Inside a medium like glass or water, the speed of light is slower, and the wavelength changes, but the frequency (and therefore the energy) stays the same. In that case, you should use the vacuum wavelength or the frequency, not the in-medium wavelength, to get the correct energy.

Another limitation: the equation gives the energy of a single photon. If you are dealing with a macroscopic amount of light, you need to multiply by the number of photons, which is usually not known directly. In that situation, you would measure the total power (watts) and divide by the photon energy to get the photon flux.

Finally, the equation is exact only for free photons. In some condensed-matter contexts, like excitons in semiconductors, the energy of an absorbed photon is not simply hc/λ because part of the energy goes into binding the electron-hole pair. But for most everyday and laboratory uses, E = hc/λ is the correct and standard formula.

Frequently Asked Questions

What is the value of h in the wavelength equation?

Planck’s constant h is exactly 6.62607015 × 10⁻³⁴ J·s, by definition since 2019. In the wavelength equation, you often see it combined with c as hc = 1239.841984 eV·nm, which is more convenient when working with nanometres and electronvolts.

Why is h so small?

Planck’s constant is small because it sets the scale of quantum effects, which only become noticeable at the atomic level. For everyday objects, the energy of a single photon is minuscule, so you need trillions of photons to produce even a tiny amount of light. That is why quantum behaviour is not visible in macroscopic life.

Can I use the equation for sound waves?

No. Sound waves are mechanical vibrations of matter, not photons. They have energy, but that energy is carried by the motion of atoms, not by discrete particles with a fixed energy proportional to frequency. The equation E = hf applies only to electromagnetic radiation and, in a modified form, to matter waves (de Broglie), but not to sound.

What is the difference between h and ħ (h-bar)?

ħ (h-bar) is Planck’s constant divided by 2π: ħ = h / (2π) ≈ 1.054571817 × 10⁻³⁴ J·s. It appears in quantum mechanics when dealing with angular momentum and the Schrödinger equation, because those involve cycles (2π radians). The wavelength equation uses h, not ħ.

Why do some sources use 1240 eV·nm instead of 1239.84?

1240 is a rounded value that is easy to remember and accurate to about 0.01%. For quick estimates, it is fine. For precise work, use 1239.841984 eV·nm, which comes from the exact values of h and c. The calculator above uses the precise constant.

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in wl | Wavelength | nm | 500 | 0
in en | Energy | eV | 2.48 | 0
out energy | Photon energy | eV | (1239.841984 / wl) | 3
out wavelength | Wavelength | nm | (1239.841984 / en) | 1