# 1.2: Quantum Hypothesis used for Blackbody Radiation Law

Skills to Develop

- To understand how energy is quantized in black body radiation

By the late 19th century, many physicists thought their discipline was well on the way to explaining most natural phenomena. They could calculate the motions of material objects using Newton’s laws of classical mechanics, and they could describe the properties of radiant energy using mathematical relationships known as Maxwell’s equations, developed in 1873 by James Clerk Maxwell, a Scottish physicist. The universe appeared to be a simple and orderly place, containing matter, which consisted of particles that had mass and whose location and motion could be accurately described, and electromagnetic radiation, which was viewed as having no mass and whose exact position in space could not be fixed. Thus matter and energy were considered distinct and unrelated phenomena. Soon, however, scientists began to look more closely at a few inconvenient phenomena that could not be explained by the theories available at the time.

### Blackbody Radiation

One phenomenon that seemed to contradict the theories of classical physics was blackbody radiation, which is electromagnetic radiation whose wavelength and color that depends on the temperature of the object. The wavelength of energy emitted by an object depends on only its temperature, not its surface or composition. Hence an electric stove burner or the filament of a space heater glows dull red or orange when heated, whereas the much hotter tungsten wire in an incandescent light bulb gives off a yellowish light.

**Figure \(\PageIndex{1}\)***: Blackbody Radiation. When heated, all objects emit electromagnetic radiation whose wavelength (and color) depends on the temperature of the object. A relatively low-temperature object, such as a horseshoe forged by a blacksmith, appears red, whereas a higher-temperature object, such as the surface of the sun, appears yellow or white.* Images used with permission from Wikipedia.

The *intensity* of radiation is a measure of the energy emitted per unit area. A plot of the intensity of blackbody radiation as a function of wavelength for an object at various temperatures is shown in Figure \(\PageIndex{2}\). One of the major assumptions of classical physics was that energy increased or decreased in a smooth, continuous manner. For example, classical physics predicted that as wavelength decreased, the intensity of the radiation an object emits should increase in a smooth curve without limit at *all* temperatures, as shown by the broken line for 6000 K in Figure \(\PageIndex{2}\). Thus classical physics could not explain the sharp *decrease* in the intensity of radiation emitted at shorter wavelengths (primarily in the ultraviolet region of the spectrum), which was referred to as the “ultraviolet catastrophe.” In 1900, however, the German physicist Max Planck (1858–1947) explained the ultraviolet catastrophe by proposing that the energy of electromagnetic waves is *quantized* rather than continuous. This means that for each temperature, there is a maximum intensity of radiation that is emitted in a blackbody object, corresponding to the peaks in Figure Figure \(\PageIndex{2}\), so the intensity does not follow a smooth curve as the temperature increases, as predicted by classical physics. Thus energy could be gained or lost only in integral multiples of some smallest unit of energy, a quantum (the smallest possible unit of energy). Energy can be gained or lost only in integral multiples of a quantum..

* Figure \(\PageIndex{2}\)*:

**Relationship between the Temperature of an Object and the Spectrum of Blackbody Radiation It Emits. At relatively low temperatures, most radiation is emitted at wavelengths longer than 700 nm, which is in the infrared portion of the spectrum. The dull red glow of the electric stove element in Figure 1.2.1 is due to the small amount of radiation emitted at wavelengths less than 700 nm, which the eye can detect. As the temperature of the object increases, the maximum intensity shifts to shorter wavelengths, successively resulting in orange, yellow, and finally white light. At high temperatures, all wavelengths of visible light are emitted with approximately equal intensities. The white light spectrum shown for an object at 6000 K closely approximates the spectrum of light emitted by the sun (Figure 1.2.1). Note the sharp decrease in the intensity of radiation emitted at wavelengths below 400 nm, which constituted the ultraviolet catastrophe. The classical prediction fails to fit the experimental curves entirely and does not have a maximum intensity.**Although quantization may seem to be an unfamiliar concept, we encounter it frequently. For example, US money is integral multiples of pennies. Similarly, musical instruments like a piano or a trumpet can produce only certain musical notes, such as C or F sharp. Because these instruments cannot produce a continuous range of frequencies, their frequencies are quantized. Even electrical charge is quantized: an ion may have a charge of −1 or −2, but *not* −1.33 electron charges.

Planck postulated that the energy of a particular quantum of radiant energy could be described Max Planck (1858–1947) by the equation

\[ E=h \nu \label{Eq1.2.1}\]

where the proportionality constant *h* is called Planck’s constant, one of the most accurately known fundamental constants in science. For our purposes, its value to four significant figures is generally sufficient:

\[h = 6.626 \times 10^{−34} J\cdot s\]

As the frequency of electromagnetic radiation increases, the magnitude of the associated quantum of radiant energy increases. By assuming that energy can be emitted by an object only in integral multiples of *h*ν, Planck devised an equation that fit the experimental data shown in Figure \(\PageIndex{2}\). We can understand Planck’s explanation of the ultraviolet catastrophe qualitatively as follows: At low temperatures, radiation with only relatively low frequencies is emitted, corresponding to low-energy quanta. As the temperature of an object increases, there is an increased probability of emitting radiation with higher frequencies, corresponding to higher-energy quanta. At any temperature, however, it is simply more probable for an object to lose energy by emitting a large number of lower-energy quanta than a single very high-energy quantum that corresponds to ultraviolet radiation. The result is a maximum in the plot of intensity of emitted radiation versus wavelength, as shown in Figure \(\PageIndex{2}\), and a shift in the position of the maximum to lower wavelength (higher frequency) with increasing temperature.

At the time he proposed his radical hypothesis, Planck could not explain *why* energies should be quantized. Initially, his hypothesis explained only one set of experimental data—blackbody radiation. If quantization were observed for a large number of different phenomena, then quantization would become a law. In time, a theory might be developed to explain that law. As things turned out, Planck’s hypothesis was the seed from which modern physics grew.

### The “Ultraviolet Catastrophe”

One experimental phenomenon that could not be adequately explained by classical physics was black-body radiation. The exact spectrum depends upon properties of the material and the temperature. It was found that the observed intensity of black-body radiation as a function of wavelength varies with temperature. Attempts to explain or calculate this spectral distribution from classical theory were complete failures. A theory developed by Rayleigh and Jeans predicted that the intensity should go to infinity at short wavelengths. Since the intensity actually drops to zero at short wavelengths, the Rayleigh-Jeans result was called the “ultraviolet catastrophe.” There was no agreement between theory and experiment in the ultraviolet region of the black-body spectrum (Figure \(\PageIndex{3}\)).

**Figure \(\PageIndex{3}\)**: Experimental (blue) and Rayleigh-Jeans (red) radiation distribution curves show the radiation density for each model at 4000 K as a function of wavelength \(\lambda\).

### Quantizing Electrons in the Radiator

Max Planck explain the spectral distribution of black-body radiation as result from oscillations of electrons. Similarly, oscillations of electrons in an antenna produce radio waves. Max Planck concentrated on modeling the oscillating charges that must exist in the oven walls, radiating heat inwards and—in thermodynamic equilibrium—themselves being driven by the radiation field. He found he could account for the observed curve if he required these oscillators not to radiate energy continuously, as the classical theory would demand, but they could only lose or gain energy in chunks, called **quanta**, of size \(h\nu\), for an oscillator of frequency \(\nu\) (Equation \(\ref{Eq1.2.1} \)).

With that assumption, Planck calculated the following formula for the radiation energy density inside the oven:

\[d\rho(\nu,T) = \rho_\nu (T) d\nu = \dfrac {8\pi h}{c^2} \cdot \dfrac {\nu^3 d\nu}{e^{\frac {h\nu}{k_B T}}-1} \label{Eq2a}\]

using:

- \(\pi = 3.14159\)
- \(h\) = \(6.626 \times 10^{-34} J\cdot s\)
- \(c\) = \(3.00 \times 10^{8} \dfrac {M}{s}\)
- \(\nu\) = \(\frac {1}{s}\)
- \(k_B\) = \(1.38 \times 10^{-23} \dfrac {J}{K}\)
- \(T\) is absolute temperature (in Kelvin)

Planck's radiation energy density (Equation \(\ref{Eq2a}\)) can also be expressed in terms of wavelength \(\lambda\).

\[\rho (\lambda, T) d \lambda = \frac {8 \pi hc^2}{\lambda ^5} \frac {d \lambda}{ e^{\frac {hc}{\lambda k_B T}} - 1} \label{Eq2b}\]

He then was able to derive an equation (Equation \(\ref{Eq2b}\)) that gave excellent agreement with the experimental observations for all temperatures provided that the value of \(6.62618 \times 10^{-34}\) Joule.sec was used for h. This new fundamental constant, which is an essential component of Quantum Mechanics, now is called Planck’s constant. The Boltzmann constant, \(k_B\), and the speed of light (c), also appear in the equation.

**Figure \(\PageIndex{2}\): **The Sun is an excellent approximation of a black body. Its effective temperature is ~5777 K. Image used with permission (CC-SA-BY 3.0; Sch).

Max Planck (1858–1947)

In addition to being a physicist, Planck was a gifted pianist, who at one time considered music as a career. During the 1930s, Planck felt it was his duty to remain in Germany, despite his open opposition to the policies of the Nazi government.

*The German physicist Max Planck had a major influence on the early development of quantum mechanics, being the first to recognize that energy is sometimes quantized. Planck also made important contributions to special relativity and classical physics. (credit: Library of Congress, Prints and Photographs Division via Wikimedia Commons)*

One of his sons was executed in 1944 for his part in an unsuccessful attempt to assassinate Hitler, and bombing during the last weeks of World War II destroyed Planck’s home. After WWII, the major German scientific research organization was renamed the Max Planck Society.

Exercise \(\PageIndex{1}\)

Use Equation \(\ref{Eq2b}\) to show that the units of ρ(λ,T)dλ are \(J/m^3\) as expected for an energy density.

The perfect agreement of this formula with precise experiments, and the consequent necessity of energy quantization, was the most important advance in physics in the century. His black body curve was completely accepted as the correct one: more and more accurate experiments confirmed it time and again, yet the radical nature of the quantum assumption didn’t sink in. Planck wasn’t too upset—he didn’t believe it either, he saw it as a technical fix that (he hoped) would eventually prove unnecessary.

Part of the problem was that Planck’s route to the formula was long, difficult and implausible—he even made contradictory assumptions at different stages, as Einstein pointed out later. However, the result was correct anyway!

The mathematics implied that the energy given off by a blackbody was not continuous, but given off at certain specific wavelengths, in regular increments. If Planck assumed that the energy of blackbody radiation was in the form

\[E = nh \nu\]

where \(n\) is an integer (now called a quantum number), then he could explain what the mathematics represented. This was indeed difficult for Planck to accept, because at the time, there was no reason to presume that the energy should only be radiated at specific frequencies. Nothing in Maxwell’s laws suggested such a thing. It was as if the vibrations of a mass on the end of a spring could only occur at specific energies. Imagine the mass slowly coming to rest due to friction, but not in a continuous manner. Instead, the mass jumps from one fixed quantity of energy to another without passing through the intermediate energies.

To use a different analogy, it is as if what we had always imagined as smooth inclined planes were, in fact, a series of closely spaced steps that only presented the illusion of continuity.

Nature, it seemed, was quantized (non-continuous, or discrete). If this was so, how could Maxwell’s equations correctly predict this result? Planck spent a good deal of time attempting to reconcile the behavior of electromagnetic waves with the discrete nature of the blackbody radiation, to no avail. It was not until 1905, with yet another paper published by Albert Einstein, that the wave nature of light was expanded to include the particle interpretation of light which adequately explained Planck’s equation. To learn more about blackbody radiation, follow the link below.

### Summary

The agreement between Planck’s theory and the experimental observation provided strong evidence that the energy of electron motion in matter is quantized. In the next two sections, we will see that the energy carried by light also is quantized in units of h \(\bar {\nu}\). These packets of energy are called “photons.”

### Contributors

Michael Fowler (Beams Professor, Department of Physics, University of Virginia)

Adapted from "Quantum States of Atoms and Molecules" by David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski

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