1.2: Quantum hypothesis used for the blackbody radiation law (2023)

At the end of the 19th century, many physicists thought their discipline was on the way to explaining most natural phenomena. They could calculate the motions of material objects using Newton's laws of classical mechanics and describe the properties of radiant energy using mathematical relationships known asMaxwell equations, developed in 1873 by James Clerk Maxwell, a Scottish physicist. The universe appeared to be a simple, ordered place containing matter, composed of particles that had mass and whose location and motion could be precisely described, and electromagnetic radiation that was thought to be massless and whose exact position in space was not known , could be described. Matter and energy were therefore considered to be distinct and unrelated phenomena. Soon, however, scientists began to take a closer look at some inconvenient phenomena that could not be explained by the theories available at the time.

An experimental phenomenon that could not be adequately explained by classical physics was blackbody radiation (Figure 1.2.1). Attempts to explain or calculate this spectral distribution of the classical theory have failed completely. A theory developed by Rayleigh and Jeans predicted that the intensity should go to infinity at short wavelengths. Because the intensity actually drops to zero at short wavelengths, the Rayleigh-Jeans result was namedUV disaster(Figure 1.2.1 dashed line). In the ultraviolet region of the blackbody spectrum, there was no agreement between theory and experiment.

Quantification of electrons in the emitter

In 1900, the German physicist Max Planck (1858-1947) explained the UV catastrophe by proposing that the energy was electromagnetic wavesquantifiedinstead of continuous. This means that for any temperature there is a maximum intensity of radiation emitted by a blackbody, corresponding to the peaks in Figure 1.2.1, such that the intensity does not follow a smooth curve with increasing temperature. , as predicted by the classics. Physicist. Thus, energy could only be gained or lost in integer multiples of a smallest unit of energy, a quant (smallest possible unit of energy). Energy can only be gained or lost in integer multiples of a quantum.


While quantization might seem like an unfamiliar concept, we often find it in quantum mechanics (hence the name). For example, US money is an integer multiple of cents. Likewise, musical instruments like a piano or trumpet can only produce certain musical notes like C or F#. Because these instruments cannot produce a continuous range of frequencies, their frequencies are quantized. It is also similar to walking up and down a hill with discrete steps, rather than walking up and down a continuous slope. Your potential energy takes on discrete values ​​as you move from one step to the next. Even electric charge is quantized: an ion can have a charge of -1 or -2, butNot−1.33 electron charges.

1.2: Quantum hypothesis used for the blackbody radiation law (1)

The quantization of the Planck energy is described by his famous equation:

\[ E=h \nu \label{Gl1.2.1} \]

(Video) Planck's Quantum Hypothesis and Quantized Energy

where the constant of proportionality is called \(h\).Planck's constant, one of the best known natural constants in science

\[h=6,626070040(81) \times 10^{−34}\, J\cdot s \nonumber \]

For our purposes, however, its value with four significant digits is sufficient:

\[h = 6,626 \times 10^{−34} \,J\cdot s \nonumber \]

As the frequency of electromagnetic radiation increases, the magnitude of the associated quantum of radiation energy increases. Assuming that an object can only emit energy in integer multiples of \(hν\), Planck developed an equation that fits the experimental data shown in Figure 1.2.1. Qualitatively, we can understand Planck's explanation for the ultraviolet catastrophe as follows: At low temperatures, radiation is emitted with relatively low frequencies, which correspond to low-energy quanta. As an object's temperature increases, there is a greater chance of emitting radiation at higher frequencies, corresponding to higher quanta of energy. However, at any given temperature, an object is more likely to lose energy by emitting a large number of lower energy quanta than a single very high energy quantum corresponding to ultraviolet radiation. The result is a peak in the plot of emitted radiation intensity versus wavelength, as shown in Figure 1.2.1, and a shift in the position of the peak to a shorter wavelength (higher frequency). ) with increasing temperature.

When he proposed his radical hypothesis, Planck could not explain itwhyEnergies must be quantified. 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 could be developed to explain this law. As it turned out, Planck's hypothesis was the seed from which modern physics grew.

Max Planck explains the spectral distribution of blackbody radiation by means of electron oscillations. Similarly, vibrations of electrons in an antenna generate radio waves. Max Planck focused on modeling the oscillating charges that must be present on the furnace walls, radiating heat inwards and, in thermodynamic equilibrium, are themselves driven by the radiation field. He found that he could explain the observed curve by postulating that these oscillators did not continuously radiate energy, as classical theory would require, but couldjustlose or gain energy in chunks, calledhow many, of magnitude \(h\nu\), for an oscillator of frequency \(\nu\) (equation \(\ref{Eq1.2.1} \)).

(Video) Planck's Constant and BlackBody Radiation

With this assumption, Planck calculated the following formula for the radiant energy density inside the furnace:

\[ \begin{align} d\rho(\nu,T) &= \rho_\nu (T) d\nu \\[4pt] &= \dfrac {2 h \nu^3}{c^2} \cdot \dfrac {1 }{\exp \left( \dfrac {h\nu}{k_B T}\right)-1} d\nu \label{Eq2a} \end{align} \]


  • \(\pi = 3,14159\)
  • \(h\) = \(6,626 \times 10^{-34} J\cdot s\)
  • \(c\) = \(3,00 \times 10^{8}\, m/s\)
  • \(\nu\) = \(1/s\)
  • \(k_B\) = \(1,38 \times 10^{-23} J/K\)
  • \(T\) is the absolute temperature (in Kelvin)

The energy density of the Planck radiation (equation \(\ref{Gl2a}\)) can also be expressed by the wavelength \(\lambda\).

\[\rho(\lambda, T) d \lambda = \dfrac {2 hc^2}{\lambda ^5} \dfrac {1}{ \exp \left(\dfrac {hc}{\lambda k_B T} \right) - 1} d \lambda \label{Eq2b} \]

Planck's equation (Equation \(\ref{Eq2b}\)) showed excellent agreement with experimental observations for all temperatures (Figure 1.2.2).

1.2: Quantum hypothesis used for the blackbody radiation law (2)
Max-Planck (1858-1947)
(Video) Quantum Chemistry 1.1 - Blackbody Radiation

Planck made many significant contributions to theoretical physics, but his fame as a physicist stems mainly from his role as the founder of quantum theory. In addition to being a physicist, Planck was also a gifted pianist who viewed music as a career. In the 1930s, Planck felt it his duty to remain in Germany, despite his outspoken opposition to Nazi government policies.

1.2: Quantum hypothesis used for the blackbody radiation law (3)
1.2: Quantum hypothesis used for the blackbody radiation law (4)

One of his sons was executed in 1944 for his part in a failed assassination attempt on Hitler, and a bombing raid in the final weeks of World War II destroyed Planck's home. After the Second World War, the most important German scientific research organization was renamed the Max Planck Society.

Exercise 1.2.1

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

The almost perfect agreement of this formula with precise experiments (e.g. Figure 1.2.3) and the consequent need for energy quantization was the most important advance in physics of the century. His black body curve was fully accepted as correct: Increasingly precise experiments confirmed it again and again, but the radical nature of the quantum assumption did not prevail. Planck wasn't too upset, nor did he believe it, seeing it as a technical solution that (he hoped) would eventually become unnecessary.

Part of the problem was that Planck's road to the formula was long, difficult, and implausible; he even made contradictory assumptions at different points in time, as Einstein later pointed out. But the result was still correct!

(Video) Black Bodies and Planck Explained

The math implied that the energy emitted by a black body was not continuous, but emitted in regular steps at certain specific wavelengths. If Planck assumed the black body radiant energy was in the form

\[E = nh \nu \not number\]

where \(n\) is an integer, so could you explain what the math represents. This was actually difficult for Planck to accept, because at the time there was no reason to assume that energy should only be radiated at certain frequencies. Nothing in Maxwell's laws suggested such a thing. It was as if the oscillations of a mass at the end of a spring could only occur at certain energies. Imagine that the mass stops slowly but not continuously due to friction. Instead, the mass jumps from one fixed amount of energy to another without going through the energies in between.

To use another analogy, it's as if what we've always thought of as gently sloping planes are actually a series of widely spaced steps that present only the illusion of continuity.


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

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(Video) Development of Quantum Mechanics: Blackbody Radiation & UV Catastrophe


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