planck's hypothesis formula

We're 20 orders of magnitude short. The universe is 13.8 billion years old. That's about…

Twenty orders of magnitude smaller than that gives you a millisecond. If the time it took a photon to cross the diameter of a proton was slowed to the point where the photon needed the entirety of time itself to complete its task, the Planck time would still only last a thousandth of a second.

On to the Planck mass …

This one always strikes me as a let down. We're talking 22 μg. That's like a speck of dust. Compare it to an atom of uranium, the heaviest naturally occurring atom…

m  =  238 u = 3.95 × 10 −25  kg

or the heaviest known subatomic particle, the top quark…

m  =  173 GeV/c 2  = 185 u = 3.08 × 10 −25  kg

Both of these values are about 17 orders of magnitude smaller than the Planck mass. Whereas the Planck length and Planck time seem to represent some lower limit on how finely space and time can be divided, the Planck mass seems to be an upper limit on how big the small things in nature can be. No elementary particle will ever be more massive than the Planck mass. Or maybe it's a lower limit on how small big things in nature can be. No black hole will ever be smaller than the Planck mass.

With what we've got so far, we can create a whole coherent set of units for mechanics: Planck acceleration, Planck force, Planck pressure, Planck density, and so on. We'll do one more fully described calculation — the Planck temperature — and just summarize the rest in a table.

How hot is this? Nothing humans or nature has done recently comes close. The interiors of the hottest stars are close to a billion kelvin (10 9  K) — 24 orders of magnitude short. The hottest laboratory experiments take place inside large particle accelerators like the Tevatron at Fermilab near Chicago and the Large Hadron Collider (LHC) at CERN near Geneva. Here we're looking at quadrillions of kelvins (10 15  K) and we're still 18 orders of magnitude short. In contrast, the coldest temperatures ever achieved in the lab are a few hundred picokelvins (10 −10  K). The entire range of laboratory temperatures achieved so far is an astounding 25 orders of magnitude, but we're still short eight zeros. The Planck temperature is so hot as to be meaningless. As we shall soon see, that's the point.

For the next 50 years or so, Planck's notion of a natural unit system — one derived from physical laws, not accidents of human history — was considered an interesting diversion with little or no meaning. The primary reason for this was probably that quantum theory and general relativity were just too new and unfamiliar. (Relativity did not even exist at the time of Planck's publication.) The physics of the Modern era was a strange world that few understood at first.

Die Quantenmechanik ist sehr Achtung gebietend. Aber eine innere Stimme sagt mir, dass das noch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, dass der Alte nicht würfelt. Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the "old one". I, at any rate, am convinced that He is not playing dice. Albert Einstein, 1926

Denn wenn man nicht zunächst über die Quantentheorie entsetzt ist, kann man sie doch unmöglich verstanden haben. Anyone who is not shocked by quantum theory does not understand it. Niels Bohr, 1952

There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper, a lot of people understood the theory of relativity in some way or other, certainly more than twelve. On the other hand, I think I can safely say that nobody understands quantum mechanics. Richard Feynman, 1965

Eine neue wissenschaftliche Wahrheit pflegt sich nicht in der Weise durchzusetzen, daß ihre Gegner überzeugt werden und sich als belehrt erklären, sondern vielmehr dadurch, daß die Gegner allmählich aussterben und daß die heranwachsende Generation von vornherein mit der Wahrheit vertraut gemacht ist. A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. Max Planck, 1948

The last quote gives you an idea of what eventually happened. People who grew up with the theory applied it in situation after situation and found that it worked. We will end this chapter by addressing the meaning of all of this.

The Planck units have no practical application. No car odometer will be calibrated in Planck lengths, no stopwatch will tick off Planck times, and no thermometer will ever give temperatures as a teeny, tiny fraction of the Planck value. These numbers tell us the limits of physics as we currently know it and maybe even the limit of physics as it could ever be known. That's why it's an important theory.

Space and time are generally regarded as smooth and continuous. The number places between any two points is apparently infinite. We pass from one place to another with no sensation of granularity. There is no "screen resolution" to the video game of reality. There is no apparent "frame rate" either. One moment is followed by another with no perceivable jerkiness. Existence does not play itself out like a turn of the century nickelodeon movie. If the universe is some sort of computer simulation (as some have suggested), it is rendered with an apparently infinite level of detail.

"Apparently" is the key word, however. The Planck length is now generally regarded as the lower limit of space. Distances less than this are meaningless. Likewise, the Planck time is the lower limit of time. No detectable change will occur in a period shorter than this. You cannot cut space and time up into infinitesimaly small parts. Eventually, you will get to the point where the notion of subdividing space and time any further becomes meaningless. Eventually there will be found an "atom" of space and an "atom" of time. Recall that atom comes from the Greek ἄ τομος ( a tomos ) meaning un-cuttable.

That matter is quantized should be evident to everyone with even the tiniest bit of education. Who doesn't know of atoms? It is less likely that the average person would know that energy was quantized, but such knowledge isn't considered exotic. Many people know of photons. Matter and energy are quantized, and as a consequence, so too is the stage on which matter and energy act. Space and time are quantized. This is perhaps the greatest meaning that one could extract from Max Planck's little excursion into units.

Electromagnetic Planck units

What about the natural units of electricity and magnetism? Planck never dealt with the subject that I know of. Your natural choice for a natural unit of electric charge might be the elementary charge…

e  =  1.602176634 × 10 −19  C

but this would not be in keeping with the spirit of Planck's work. After all, the Planck mass isn't related to the mass of an electron, proton, or any other physical thing. The Planck quantities are derived from the laws of nature. To that end some have suggested using the coulomb law constant to extend the original system since it's analogous to the universal gravitational constant G .

Including this unwieldy symbol pile gives us the following electromagnetic Planck units. The values for current and voltage look like they could be upper limits. The value for magnetic flux looks like it could be a lower limit. That's nice, I suppose. The value for resistance means… what? Resistance is a bulk property of an object. Subatomic particles or black holes — the kind of things we've been talking about in this section — don't really have a resistance. The value for charge is slightly larger than the elementary charge. Once again, I'm lost. These quantities aren't as easy to interpret as the original Planck units. I don't think anyone is really working on them as a subject of theoretical study.

6.1 Blackbody Radiation

Learning objectives.

By the end of this section, you will be able to:

• Apply Wien’s and Stefan’s laws to analyze radiation emitted by a blackbody
• Explain Planck’s hypothesis of energy quanta

All bodies emit electromagnetic radiation over a range of wavelengths. In an earlier chapter, we learned that a cooler body radiates less energy than a warmer body. We also know by observation that when a body is heated and its temperature rises, the perceived wavelength of its emitted radiation changes from infrared to red, and then from red to orange, and so forth. As its temperature rises, the body glows with the colors corresponding to ever-smaller wavelengths of the electromagnetic spectrum. This is the underlying principle of the incandescent light bulb: A hot metal filament glows red, and when heating continues, its glow eventually covers the entire visible portion of the electromagnetic spectrum. The temperature ( T ) of the object that emits radiation, or the emitter , determines the wavelength at which the radiated energy is at its maximum. For example, the Sun, whose surface temperature is in the range between 5000 K and 6000 K, radiates most strongly in a range of wavelengths about 560 nm in the visible part of the electromagnetic spectrum. Your body, when at its normal temperature of about 300 K, radiates most strongly in the infrared part of the spectrum.

Radiation that is incident on an object is partially absorbed and partially reflected. At thermodynamic equilibrium, the rate at which an object absorbs radiation is the same as the rate at which it emits it. Therefore, a good absorber of radiation (any object that absorbs radiation) is also a good emitter. A perfect absorber absorbs all electromagnetic radiation incident on it; such an object is called a blackbody .

Although the blackbody is an idealization, because no physical object absorbs 100% of incident radiation, we can construct a close realization of a blackbody in the form of a small hole in the wall of a sealed enclosure known as a cavity radiator, as shown in Figure 6.2 . The inside walls of a cavity radiator are rough and blackened so that any radiation that enters through a tiny hole in the cavity wall becomes trapped inside the cavity. At thermodynamic equilibrium (at temperature T ), the cavity walls absorb exactly as much radiation as they emit. Furthermore, inside the cavity, the radiation entering the hole is balanced by the radiation leaving it. The emission spectrum of a blackbody can be obtained by analyzing the light radiating from the hole. Electromagnetic waves emitted by a blackbody are called blackbody radiation .

The intensity I ( λ , T ) I ( λ , T ) of blackbody radiation depends on the wavelength λ λ of the emitted radiation and on the temperature T of the blackbody ( Figure 6.3 ). The function I ( λ , T ) I ( λ , T ) is the power intensity that is radiated per unit wavelength; in other words, it is the power radiated per unit area of the hole in a cavity radiator per unit wavelength. According to this definition, I ( λ , T ) d λ I ( λ , T ) d λ is the power per unit area that is emitted in the wavelength interval from λ λ to λ + d λ . λ + d λ . The intensity distribution among wavelengths of radiation emitted by cavities was studied experimentally at the end of the nineteenth century. Generally, radiation emitted by materials only approximately follows the blackbody radiation curve ( Figure 6.4 ); however, spectra of common stars do follow the blackbody radiation curve very closely.

Two important laws summarize the experimental findings of blackbody radiation: Wien’s displacement law and Stefan’s law . Wien’s displacement law is illustrated in Figure 6.3 by the curve connecting the maxima on the intensity curves. In these curves, we see that the hotter the body, the shorter the wavelength corresponding to the emission peak in the radiation curve. Quantitatively, Wien’s law reads

where λ max λ max is the position of the maximum in the radiation curve. In other words, λ max λ max is the wavelength at which a blackbody radiates most strongly at a given temperature T . Note that in Equation 6.1 , the temperature is in kelvins. Wien’s displacement law allows us to estimate the temperatures of distant stars by measuring the wavelength of radiation they emit.

Example 6.1

Temperatures of distant stars.

When simplified, Equation 6.2 gives

Therefore, Betelgeuse is cooler than Rigel.

Significance

Check your understanding 6.1.

The flame of a peach-scented candle has a yellowish color and the flame of a Bunsen’s burner in a chemistry lab has a bluish color. Which flame has a higher temperature?

The second experimental relation is Stefan’s law , which concerns the total power of blackbody radiation emitted across the entire spectrum of wavelengths at a given temperature. In Figure 6.3 , this total power is represented by the area under the blackbody radiation curve for a given T . As the temperature of a blackbody increases, the total emitted power also increases. Quantitatively, Stefan’s law expresses this relation as

where A A is the surface area of a blackbody, T is its temperature (in kelvins), and σ σ is the Stefan–Boltzmann constant , σ = 5.670 × 10 −8 W / ( m 2 · K 4 ) . σ = 5.670 × 10 −8 W / ( m 2 · K 4 ) . Stefan’s law enables us to estimate how much energy a star is radiating by remotely measuring its temperature.

Example 6.2

Power radiated by stars.

The power emitted per unit area by a white dwarf is about 5000 times that the power emitted by a red giant. Denoting this ratio by a = 4.8 × 10 3 , a = 4.8 × 10 3 , Equation 6.5 gives

We see that the total power emitted by a white dwarf is a tiny fraction of the total power emitted by a red giant. Despite its relatively lower temperature, the overall power radiated by a red giant far exceeds that of the white dwarf because the red giant has a much larger surface area. To estimate the absolute value of the emitted power per unit area, we again use Stefan’s law. For the white dwarf, we obtain

The analogous result for the red giant is obtained by scaling the result for a white dwarf:

Check Your Understanding 6.2

An iron poker is being heated. As its temperature rises, the poker begins to glow—first dull red, then bright red, then orange, and then yellow. Use either the blackbody radiation curve or Wien’s law to explain these changes in the color of the glow.

Check Your Understanding 6.3

Suppose that two stars, α α and β , β , radiate exactly the same total power. If the radius of star α α is three times that of star β , β , what is the ratio of the surface temperatures of these stars? Which one is hotter?

The term “blackbody” was coined by Gustav R. Kirchhoff in 1862. The blackbody radiation curve was known experimentally, but its shape eluded physical explanation until the year 1900. The physical model of a blackbody at temperature T is that of the electromagnetic waves enclosed in a cavity (see Figure 6.2 ) and at thermodynamic equilibrium with the cavity walls. The waves can exchange energy with the walls. The objective here is to find the energy density distribution among various modes of vibration at various wavelengths (or frequencies). In other words, we want to know how much energy is carried by a single wavelength or a band of wavelengths. Once we know the energy distribution, we can use standard statistical methods (similar to those studied in a previous chapter) to obtain the blackbody radiation curve, Stefan’s law, and Wien’s displacement law. When the physical model is correct, the theoretical predictions should be the same as the experimental curves.

In a classical approach to the blackbody radiation problem, in which radiation is treated as waves (as you have studied in previous chapters), the modes of electromagnetic waves trapped in the cavity are in equilibrium and continually exchange their energies with the cavity walls. There is no physical reason why a wave should do otherwise: Any amount of energy can be exchanged, either by being transferred from the wave to the material in the wall or by being received by the wave from the material in the wall. This classical picture is the basis of the model developed by Lord Rayleigh and, independently, by Sir James Jeans. The result of this classical model for blackbody radiation curves is known as the Rayleigh–Jeans law . However, as shown in Figure 6.6 , the Rayleigh–Jeans law fails to correctly reproduce experimental results. In the limit of short wavelengths, the Rayleigh–Jeans law predicts infinite radiation intensity, which is inconsistent with the experimental results in which radiation intensity has finite values in the ultraviolet region of the spectrum. This divergence between the results of classical theory and experiments, which came to be called the ultraviolet catastrophe , shows how classical physics fails to explain the mechanism of blackbody radiation.

The blackbody radiation problem was solved in 1900 by Max Planck . Planck used the same idea as the Rayleigh–Jeans model in the sense that he treated the electromagnetic waves between the walls inside the cavity classically, and assumed that the radiation is in equilibrium with the cavity walls. The innovative idea that Planck introduced in his model is the assumption that the cavity radiation originates from atomic oscillations inside the cavity walls, and that these oscillations can have only discrete values of energy. Therefore, the radiation trapped inside the cavity walls can exchange energy with the walls only in discrete amounts. Planck’s hypothesis of discrete energy values, which he called quanta , assumes that the oscillators inside the cavity walls have quantized energies . This was a brand new idea that went beyond the classical physics of the nineteenth century because, as you learned in a previous chapter, in the classical picture, the energy of an oscillator can take on any continuous value. Planck assumed that the energy of an oscillator ( E n E n ) can have only discrete, or quantized, values:

In Equation 6.9 , f is the frequency of Planck’s oscillator. The natural number n that enumerates these discrete energies is called a quantum number . The physical constant h is called Planck’s constant :

Each discrete energy value corresponds to a quantum state of a Planck oscillator . Quantum states are enumerated by quantum numbers. For example, when Planck’s oscillator is in its first n = 1 n = 1 quantum state, its energy is E 1 = h f ; E 1 = h f ; when it is in the n = 2 n = 2 quantum state, its energy is E 2 = 2 h f ; E 2 = 2 h f ; when it is in the n = 3 n = 3 quantum state, E 3 = 3 h f ; E 3 = 3 h f ; and so on.

Note that Equation 6.9 shows that there are infinitely many quantum states, which can be represented as a sequence { hf , 2 hf , 3 hf ,…, ( n – 1) hf , nhf , ( n + 1) hf ,…}. Each two consecutive quantum states in this sequence are separated by an energy jump, Δ E = h f . Δ E = h f . An oscillator in the wall can receive energy from the radiation in the cavity (absorption), or it can give away energy to the radiation in the cavity (emission). The absorption process sends the oscillator to a higher quantum state, and the emission process sends the oscillator to a lower quantum state. Whichever way this exchange of energy goes, the smallest amount of energy that can be exchanged is hf . There is no upper limit to how much energy can be exchanged, but whatever is exchanged must be an integer multiple of hf . If the energy packet does not have this exact amount, it is neither absorbed nor emitted at the wall of the blackbody.

Planck’s Quantum Hypothesis

Planck’s hypothesis of energy quanta states that the amount of energy emitted by the oscillator is carried by the quantum of radiation, Δ E : Δ E :

Recall that the frequency of electromagnetic radiation is related to its wavelength and to the speed of light by the fundamental relation f λ = c . f λ = c . This means that we can express [link] equivalently in terms of wavelength λ . λ . When included in the computation of the energy density of a blackbody, Planck’s hypothesis gives the following theoretical expression for the power intensity of emitted radiation per unit wavelength:

where c is the speed of light in vacuum and k B k B is Boltzmann’s constant, k B = 1.380 × 10 −23 J/K . k B = 1.380 × 10 −23 J/K . The theoretical formula expressed in Equation 6.11 is called Planck’s blackbody radiation law . This law is in agreement with the experimental blackbody radiation curve (see Figure 6.7 ). In addition, Wien’s displacement law and Stefan’s law can both be derived from Equation 6.11 . To derive Wien’s displacement law, we use differential calculus to find the maximum of the radiation intensity curve I ( λ , T ) . I ( λ , T ) . To derive Stefan’s law and find the value of the Stefan–Boltzmann constant, we use integral calculus and integrate I ( λ , T ) I ( λ , T ) to find the total power radiated by a blackbody at one temperature in the entire spectrum of wavelengths from λ = 0 λ = 0 to λ = ∞ . λ = ∞ . This derivation is left as an exercise later in this chapter.

Example 6.3

Planck’s quantum oscillator, check your understanding 6.4.

A molecule is vibrating at a frequency of 5.0 × 10 14 Hz . 5.0 × 10 14 Hz . What is the smallest spacing between its vibrational energy levels?

Example 6.4

Quantum theory applied to a classical oscillator.

The energy quantum that corresponds to this frequency is

When vibrations have amplitude A = 0.10 m , A = 0.10 m , the energy of oscillations is

Check Your Understanding 6.5

Would the result in Example 6.4 be different if the mass were not 1.0 kg but a tiny mass of 1.0 µ g, and the amplitude of vibrations were 0.10 µ m?

When Planck first published his result, the hypothesis of energy quanta was not taken seriously by the physics community because it did not follow from any established physics theory at that time. It was perceived, even by Planck himself, as a useful mathematical trick that led to a good theoretical “fit” to the experimental curve. This perception was changed in 1905 when Einstein published his explanation of the photoelectric effect, in which he gave Planck’s energy quantum a new meaning: that of a particle of light.

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Blackbody radiation: from rayleigh–jeans to planck and vice versa.

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This Demonstration shows the result of an unwilling revolutionary's revolution: Max Planck's law for the spectral density of radiation emitted by a blackbody. By a heuristic approach, we derive Planck's formula from the classical Rayleigh–Jeans formula [1, 2].

Use the "temperature (K)" slider to change the temperature. Higher values lead to a shift of the maximum, according to Wien's displacement law.

Contributed by: A. Ratti , D. Meliga , L. Lavagnino and S. Z. Lavagnino   (August 2022) Additional contribution by: G. Follo Open content licensed under CC BY-NC-SA

Snapshot 1: quantum mechanics: Planck's law is applied to the energy quantization; the resulting curve follows the blackbody radiation prediction

Snapshot 2: quantum mechanics: increasing the temperature leads to a shift of the maximum toward the visible spectrum frequencies, according to Wien's law

A blackbody is an idealized object absorbing all incident electromagnetic radiation, regardless of frequency or angle of incidence. Consequently, it absorbs all colors of light as well and thus appears completely black.

A blackbody in thermal equilibrium (at a constant temperature) emits electromagnetic blackbody radiation. The first theoretical analysis resulted in the Rayleigh–Jeans formula, based on the classical hypothesis that the energies of the photons inside the blackbody could have any values from an energy continuum.

We begin with the Boltzmann distribution for thermal equilibrium with noninteracting particles distributed over energy states:

and thus, for the blackbody spectral radiance:

which is, as noted, the Rayleigh–Jeans formula.

This equation predicts a spectral radiance that diverges for high frequencies, leading to the so-called "ultraviolet catastrophe."

In this way, as shown, we get a new average photon energy value:

The average energy value becomes

[1] R. A. Serway, C. A. Moses and C. J. Moyer, Modern Physics , 2nd ed., Philadelphia: Saunders College Publishing, 1997.

[2] SpectralCalc.com. "The Planck Blackbody Formula in Units of Frequency." (Jul 15, 2021) www.spectralcalc.com/blackbody/planck_blackbody.html .

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A. Ratti , D. Meliga , L. Lavagnino and S. Z. Lavagnino "Blackbody Radiation: from Rayleigh–Jeans to Planck and Vice Versa" http://demonstrations.wolfram.com/BlackbodyRadiationFromRayleighJeansToPlanckAndViceVersa/ Wolfram Demonstrations Project Published: August 15, 2022

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Book contents

• Frontmatter
• Preface and Acknowledgements
• 1 Introduction
• Case Study I The Origins of Newton’s Laws of Motion and of Gravity
• Case Study II Maxwell’s Equations
• Case Study III Mechanics and Dynamics: Linear and Non-linear
• Case Study IV Thermodynamics and Statistical Physics
• Case Study V The Origins of the Concepts of Quantisation and Quanta
• 13 Black-Body Radiation up to 1895
• 14 1895–1900: Planck and the Spectrum of Black-Body Radiation
• 15 Planck’s Theory of Black-Body Radiation
• 16 Einstein and the Quantisation of Light
• 17 The Triumph of the Light Quantum Hypothesis
• Case Study VI Special and General Relativity
• Case Study VII Cosmology and Physics
• Author Index
• Subject Index

15 - Planck’s Theory of Black-Body Radiation

from Case Study V - The Origins of the Concepts of Quantisation and Quanta

Published online by Cambridge University Press:  27 March 2020

Planck immediately set about attempting to understand the significance of his formula for black-body radiation. He began by using Boltzmann's procedure in statistical mechanics, an approach he had previously rejected, but then adopted empirically a definition of the entropy of the oscillators which introduced the concept of quantisation. HIs derivation was not understood by his contemporaries, including Einstein, because of the lack of a theoretical motivation for the definition of entropy. Despite a major effort to understand his formula, Planck found no classical solution to the meaning of h , Planck's constant.

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• Planck’s Theory of Black-Body Radiation
• Malcolm S. Longair , University of Cambridge
• Book: Theoretical Concepts in Physics
• Online publication: 27 March 2020
• Chapter DOI: https://doi.org/10.1017/9781108613927.021

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Planck’s Hypothesis

Assumptions used:

1. The particles/oscillators near the surface of the blackbody which emits the blackbody radiation can only have discrete values of energy, E n : E n =nhf, where n is a positive interger, f is the frequency of the oscillating particle, h is the Planck’s constant.

• Particles can only have discrete values of energy. Each energy value corresponds to a quantum state of the particle, represented by the corresponding quantum number.

2. In order for a particle to transit from one quantum state to another, the particle has to absorb or emit energy. The difference in the energy between the initial and the final state in the transition must be absorbed or emitted as a SINGLE quantum of energy. Smallest amount of energy involved is when the transition occurs between two adjacent states: E = hf.

• Energy is quantized in steps of hf
• One quanta of energy or radiation is known as a photon

Planck’s Law Of Blackbody Radiation:

$$I \left( v, T \right) dv = \left( \frac{2hv^{3}}{c^{2}} \right) \frac{1}{e^{\frac{hv}{kt}} – 1} dv$$

, where I(ν,T) dν is the amount of energy per unit surface area per unit time per unit solid angle emitted in the frequency range between ν and ν + dν by a black body at temperature T; h is the Planck constant; c is the speed of light in a vacuum; k is the Boltzmann constant; ν is frequency of electromagnetic radiation; and T is the temperature in kelvins. Deriving the momentum of a photon: Using E = pc and E = hf,

$$\begin{eqnarray*} E &=& hf \\ E &=& h \left( \frac{c}{\lambda} \right), \, \text{since} \, f = \frac{c}{\lambda} \end{eqnarray*}$$

Substituting E = pc into $E = h \left(\frac{c}{λ} \right)$:

$$\begin{eqnarray*} h \left( \frac{v}{\lambda} \right) &=& pv \\ p &=& \left( \frac{h}{\lambda} \right) \end{eqnarray*}$$

Back To Quantum Theory Of Light

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• Planck’s Quantum Theory

Most natural phenomena could be made clear by Newton’s Laws of classical mechanics or classical theory by the end of the 19th century, thanks to the work of scientists. Until this point, matter and energy were seen as separate entities with no connection to one another. In 1873, Scottish physicist James Clerk Maxwell published his Maxwell equations, which enable scientists to characterize the qualities of radiant radiation . The photoelectric effect and other phenomena like black body radiation could not be explained by classical theory or mechanics until the twentieth century. German scientist Max Planck developed his quantized energy theory of electromagnetic waves during this period. Let us now discuss planck’s quantum theory in detail.

                                                                              Planck’s Quantum Theory

Planck ’ s Quantum Theory Presuppositions

When it comes to radiation, Planck’s quantum theory is the best way to go. Planck’s quantum theory has the following presuppositions:

• Discrete packets or bundles of energy are emitted or absorbed intermittently by matter.
• Quantum is the name given to the smallest bundle or packet of energy. Using light as an example, a photon is a quantum of light.
• The absorb quantum energy is proportional to the radiation frequency.

Quantization of Energy

In quantum physics, energy is construed as occurring in discrete “packets,” or photons, at the subatomic level. Photons appear in a variety of denominations, much like paper money.

Photons are energy bundles in quantum physics, and they correspond to various hues in the visible spectrum and other forms of electromagnetic radiation (radio waves, microwaves, X -rays, etc).

The energy value of a red photon is distinct from that of a blue photon. Red and blue photons, like dollar note denominations, are quantized in the same way. Each photon carries a discrete quantity of energy that is all it is own.

It is similar to Plank’s constant, which describes “how quantum” energy may be, that energy is unique or “quantized.”

Electromagnetic Radiation

The phenomena of electromagnetic waves travelling across space do refer to as electromagnetic radiation. A medium isn’t necessary for the propagation of electromagnetic waves . When the electric and magnetic components of a wave interact with one another, the radiation is prolonged. The planes of both components are perpendicular.

Electromagnetic waves all share a set of characteristics. The frequency and wavelength of electromagnetic radiation may be used to classify it. There are three types of waves that are higher in frequency than visible light, which are x-rays, gamma rays, and UV rays. Infrared rays, radio waves, and microwaves are examples of waves with a lower frequency than visible light.

Black Body Radiation

All the radiation that falls on it is lost by the dark body. All frequencies of radiation can be caught and exhaled by a perfect black body. Temperature affects the emission of electromagnetic radiation from a black substance. Using Planck’s law, a black body’s emitted radiation may be termed by its frequency variation. Planck’s radiation, a form of thermal radiation, is another name for this type of radiation. The greater the temperature of the body, the more radiation of all wavelengths is emitted by it.

Relation of black body relation with Planck’s law

E = Energy of the radiation

h = Planck’s constant (6.626×10–34 J.s)

ν= Frequency of radiation

As a side note, Planck also found that these processes constituted just a small part of the whole. Nothing to do with the radiation’s physical actuality. Albert Einstein, a well-known German scientist, reworked Planck’s theory later in 1905 to explain the photoelectric effect in more detail. He believed that if a light source was done at certain materials, electrons may eject from the substance. As a result of Planck’s study, Einstein was able to determine that light existed as quanta of energy, or photons.

Postulated Planck’s Quantum Theory

• Continuous radiation or emission of energy is not possible. Quanta, the term for the discrete packets of energy it emits, are the smallest units of its radiation.
• A photon is the unit of measurement for a particle of radiation when it is in the form of light. In the case of light, photons are tiny energy particles.
• The frequency of the radiation is exactly proportional to the energy of a photon or quantum of energy. Where h denotes Planck’s constant and v denotes radiation frequency, E = Hv.
• There are a lot of ways to express the total energy of radiation, including h, 2h, and so on.

Evidence of Planck’s Quantum Theory

Planck’s quantum theory was the subject of several experiments. Quantum theory was backed up by a wealth of data from a wide range of experiments. Electron motion in the matter is quantized, according to all the evidence. Light may be split into wavelengths using a prism. Prisms produce rainbows only when light behaves as a wave. In addition, this lends credence to Max Planck’s quantum mechanics. Planck’s quantum theory of radiation is verified using the nitrogen gas emission spectra, as well.

FAQs on P lanck’s Quantum Theory

Question 1: State some applications of Planck’s Quantum Theory.

Answer: In optical communications, lasers are done. The quantum nature of light causes the photons that make up the laser beam to be released into the atmosphere in distinct packets or quanta. Large volumes of data may be sent at rapid rates with this method.

Quantum computing is a further application of quantum theory. In a quantum computer, a qubit serves as the fundamental unit of information, and it may represent any integer between 0 and 1, including zero.

Question 2: Define Stefan’s Law of Radiation

Answer: The total radiant heat power output from a surface is proportional to its fourth absolute temperature power, according to Stefan law. Boltzmann’s rule only applies to black bodies, which are fictional surfaces that absorb heat radiation from all activities.

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Planck’s Law – Planck’s Hypothesis

Blackbody radiation.

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1.2: Quantum Hypothesis Used for Blackbody Radiation Law

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Learning Objectives

• To understand how energy is quantized in blackbody 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.

One 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 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 (Figure 1.2.1 dashed line). There was no agreement between theory and experiment in the ultraviolet region of the blackbody spectrum.

Quantizing Electrons in the Radiator

In 1900, 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 1.2.1 , 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.

Quantization

Although quantization may seem to be an unfamiliar concept, we encounter it frequently in quantum mechanics (hence the name). 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. It is also similar to going up and down a hill using discrete stair steps rather than being able to move up and down a continuous slope. Your potential energy takes on discrete values as you move from step to step. Even electrical charge is quantized: an ion may have a charge of −1 or −2, but not −1.33 electron charges.

Planck's quantization of energy is described by the his famous 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

\[h=6.626070040(81) \times 10^{−34}\, J\cdot s \nonumber \]

However, for our purposes, its value to four significant figures 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 radiant energy increases. By assuming that energy can be emitted by an object only in integral multiples of \(hν\), Planck devised an equation that fit the experimental data shown in Figure 1.2.1 . 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 1.2.1 , 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.

Max Planck explain the spectral distribution of blackbody 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:

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

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

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

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

With a wavelength of maximum energy density at:

\[ \lambda_{max}=\frac{hc}{4.965kT} \nonumber \]

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

Max Planck (1858–1947)

Planck made many substantial contributions to theoretical physics, but his fame as a physicist rests primarily on his role as the originator of quantum theory. 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.

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 1.2.1

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 near perfect agreement of this formula with precise experiments (e.g., Figure 1.2.3 ), and the consequent necessity of energy quantization, was the most important advance in physics in the century. His blackbody 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 did not sink in. Planck was not 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

\[E = nh \nu \nonumber \]

where \(n\) is an integer, 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.

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 and Attributions

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

David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski (" Quantum States of Atoms and Molecules ")

• Electromagnetism
• Planck Equation

Planck's Equation

What is Planck’s Equation?

Max Planck discovered a theory that energy is transferred in the form of chunks called quanta, assigned as h. The variable h holds the constant value of 6.63 x 10 -34 J.s based on the International System of Units, and the variable describes the frequency in s-1. Planck’s law helps us calculate the energy of photons when their frequency is known.

If the wavelength is known, you can calculate the energy using the wave equation to calculate the frequency and then apply Planck’s equation to find the energy.

What is Planck’s Constant?

Put differently, Plank’s constant describes the relevancy between the energy per quantum (photon) of electromagnetic radiation and its frequency.

Solved Example:

Green light has a wavelength of 525 nm. Determine the energy for the green light in joules.

To find the Frequency;

As we know that, \(\begin{array}{l}c=\lambda \times v\end{array} \)

\(\begin{array}{l}v=\frac{3 \times 10^{8}}{525}\end{array} \)

Hence, \(\begin{array}{l}v = 5.71 \times 10^{14}/s\end{array} \)

To find the Energy;

As we know that \(\begin{array}{l}E = h \times \nu\end{array} \)

\(\begin{array}{l}(6.626 \times 10^{-34 })\times (5.71 \times 10^{14})\end{array} \)

\(\begin{array}{l}3.78 \times 10^{-19} J/photon\end{array} \)

Planck’s Law:

It states that electromagnetic radiation from heated bodies is not emitted as a continuous flow but is made up of discrete units or quanta of energy, the size of which involves a fundamental physical constant (Planck’s constant).

Mathematically,

\(\begin{array}{l}B_{\lambda }(T)=\frac{2hc^{2}}{\lambda ^{5}}\frac{1}{e^{\frac{hc}{kT\lambda }}-1}\end{array} \)

h = Planck’s Constant \(\begin{array}{l}=6.62\times 10^{-34}Js\end{array} \)

k = Boltzmann’s Constant = 1.381 × 10 -23 J/K

Frequently Asked Questions – FAQs

What is the significance of planck’s equation, what is the unit of planck’s constant, what is the planck constant, what is the length of planck’s constant, which of the following scientific quantities has the same dimensional formula as planck’s constant.

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Planck’s Constant Definition and Value

Planck’s constant is one of the fundamental constants in physics that sets the scale for quantum effects. It is the proportionality constant that relates the energy of a photon to the frequency of its corresponding electromagnetic wave. The symbol for Planck’s constant is h . It is also known as the Planck constant.

Value of Planck’s Constant in SI Units

In SI units, the value of Planck’s constant is defined:

h = 6.62607015×10 −34 m²·kg/s = 6.62607015×10 −34 J·Hz -1 = 6.62607015×10 −34 J·s

Value of Planck’s Constant in eV

In terms of electron volts (eV), the value is approximately:

h = 4.135667696×10 −15 eV·s

Significance and Importance

Planck’s constant is pivotal in the realm of quantum mechanics, the branch of physics dealing with the behavior of particles at the atomic and subatomic levels. Without Planck’s constant, quantum theory would be mathematically incoherent. It sets the scale for a multitude of phenomena, from the behavior of electrons in atoms to the properties of the early universe.

Relating Photon Energy and Wave Frequency

Planck’s constant h relates the energy E of a photon to the frequency of its corresponding electromagnetic wave f :

By relating frequency and wavelength λ, the equation becomes:

E = h ⋅ c / λ

The Dirac Constant or Reduced Planck Constant

The Dirac constant or reduced Planck constant ℏ (h-bar) is h /2 π . Dividing Planck’s constant by 2π makes it easier working in radians rather than hertz. This constant is especially useful when dealing with angular momentum in quantum systems. The value of ℏ in SI units is approximately 1.0545718×10 −34 m²·kg/s. It plays a crucial role in the Schrödinger equation, which governs how quantum systems evolve over time.

The constant was first postulated by Max Planck in 1900. He introduced it to explain the ultraviolet catastrophe, a divergence in the predictions of classical physics when describing the electromagnetic spectrum of radiation in a black body. With the introduction of h , Planck provided a groundbreaking solution that laid the groundwork for quantum theory.

Max Planck received the Nobel Prize in Physics in 1918 for his discovery of the energy quanta, which essentially laid the foundation for quantum theory. His introduction of the Planck constant revolutionized our understanding of atomic and subatomic processes. The Nobel Prize recognized the immense significance of his work, which marked a watershed moment in the history of physics and set the stage for the development of quantum mechanics. Planck’s work deeply influenced subsequent generations of physicists and led to groundbreaking theories and applications, ranging from quantum mechanics to quantum field theory and beyond.

Relation to the Photoelectric Effect

Albert Einstein used the concept of Planck’s constant to explain the photoelectric effect in 1905. He showed that light could be thought of as a stream of photons, each with energy E = h ⋅ f . This explanation won Einstein the Nobel Prize in Physics in 1921 and provided early experimental evidence in favor of quantum theory.

Atomic Structure

The Bohr model of the hydrogen atom was one of the first applications of Planck’s constant in atomic physics. The quantization of angular momentum in the model is directly related to Planck’s constant, and this quantization explains phenomena like atomic spectra.

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle , formulated by Werner Heisenberg in 1927, states that the position x and the momentum p of a particle cannot both be known exactly at the same time. The principle is mathematically represented as:

Δ x Δ p ≥ ℏ​/2

Here, Δ x and Δ p are the uncertainties in position and momentum, respectively, and ℏ is the reduced Planck constant.

Fixed Definition

In 2019, the International Committee for Weights and Measures redefined the kilogram in terms of Planck’s constant, thereby “fixing” its value. This redefinition is significant because it provides a stable and universal basis for mass, which was previously based on a physical artifact. This makes all of the SI base units defined.

Determining Planck’s Constant Before 2019

Before 2019, Planck’s constant was determined through experiments like the Kibble balance and Josephson voltage standards, along with comparisons to the mass of the International Prototype of the Kilogram. A 2011 experiment at the Large Hadron Collider also determined the value of the Planck constant experimentally.

Additional Facts

• Planck’s constant also appears in the expression for the energy levels of a quantum harmonic oscillator.
• It is used to calculate the Planck length, time, and mass, which are the scales below which the classical notions of space, time, and mass cease to exist.
• Planck units, derived using Planck’s constant along with other fundamental constants, provide a natural unit system particularly useful for cosmology and high-energy physics.
• Barrow, John D. (2002). The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe . Pantheon Books. ISBN 978-0-375-42221-8.
• Einstein, Albert (2003). “Physics and Reality”. Daedalus . 132 (4): 24. doi: 10.1162/001152603771338742
• International Bureau of Weights and Measures (2019). Le Système international d’unités [ The International System of Units ] (in French and English) (9th ed.). ISBN 978-92-822-2272-0.
• Kragh, Helge (1999). Quantum Generations: A History of Physics in the Twentieth Century . Princeton University Press. ISBN 978-0-691-09552-3.
• Planck, Max (1901). “Ueber das Gesetz der Energieverteilung im Normalspectrum”. Ann. Phys . 309 (3): 553–63. doi: 10.1002/andp.19013090310

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