How does Planck derive the law of blackbody radiation?丨Paper volume

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German physicist Planck derives Planck's law by interpreting the black body radiation curve. Since then, physics has entered the quantum age, so Planck has also been recognized as the "father of quantum theory", although he himself was stillA champion of the classics. As one of the greatest physicists of the 20th century, Planck has outstanding academic achievements, not only laying the foundation of quantum theory, but also a master in the field of thermodynamics, understanding of concepts such as "entropy"Transcend the age. But as a patriot, his life is complicated. When the Nazis were in power, he firmly supported Einstein and funded persecuted Jewish scientists; but he also signed Germany during World War IThe "Civilization Declaration" defended the occupation of Belgium. In a long life, Planck has encountered many misfortunes. For an important figure in the history of human civilization, Planck's research and himself need to learn and interpret."Planck's Biography: Identity Crisis and Moral Dilemma" is just such a book. This article is only an excerpt from a short node in Planck's career, which is just a well-known milestone in the development of modern physics. For more content, please refer to the original book..

This article is an authorized excerpt from Chapter 10 of "Planck's Biography: Identity Crisis and Moral Dilemma" Rising Star Press. The content has been deleted, and the title and subscript in the article are added by the editor. Go to "Return to Park",Click "Read the original text" at the end of the article to purchase the book. Click "Looking at" and post your thoughts to the message area. As of September 19, 2021 at 12:00 noon, we will select 3 messages, and each person will give 1 book.

Written by Brandon R. Brown Brandon R. Brown, Professor of Physics, University of San Francisco

Translation丨Yin Xiaodong and Zhang Shuo

July 1944-celebration

When Planck recovered his health after the operation, the Allied forces captured Rome on June 5 and landed on the beaches of Normandy on the 6th. Although the Nazis were still propagating, most Germans could see that the war was moving forward.Step by step towards the end.

In early July, Max Planck once again put on the tuxedo he used to wear, and went to Berlin to participate in a science celebration. The Prussian Academy of Sciences decided to hold an anniversary celebration on a night that temporarily escaped the darkness of the surroundings. The top physicist of the EmpireHeisenberg, the head of the atomic bomb research project, organized the event. Heisenberg went to the hotel where Max Planck was staying and planned to take Planck and others to the banquet hall, but the dilapidated scene before him was no longer.Neither of them could make out the streets of Berlin. After asking for directions several times, the group found the right place, but they thought they were lost again. Heisenberg recalled: "We ended up parked in a bunch of cars.In front of the rubble, the concrete block was inserted with twists and turns of steel bars." After further investigation, they saw a zigzag path in the dust and broken bricks and tiles, and they climbed up the ruins of the empire.Then finally came to an open door. As we walked through the door, the surviving banquet hall was full of vitality, as intact as memory. When Planck stepped into the banquet hall, everyone was quiet.Heisenberg said: "Everyone pays tribute to Planck with respect, and you can clearly see that there is so much respect and love coming to this person." "Moreover, you can also feel that he himself is very happy to see it again.To these familiar faces. A string quartet sounded in the banquet hall, and in those one or two hours it seemed to have returned to the once academically prosperous and culturally long Berlin. There, Planck was a natural leader." A large group of friends andThe scientists who stayed welcomed the distinguished guest and toasted together for the 50th anniversary of the inauguration of the Privy Council and Professor Dr. Planck at the Prussian Academy of Sciences. At the main table, Max was very happy to find that his seat was in Egypt.Next to Irwin.

Why did Planck become such a well-known name in the scientific world after fifty years? Although he was almost almighty when he was a young student, he did not have any outstanding talents. How did he gain such an influence today?What about?

Supporter of "Atom Theory"

Although Planck has an extraordinary mind, it is hard to call him a genius after all. Later, he commented on himself: "Unfortunately, I was not given the ability to respond quickly to intellectual stimulation." In his student days, PuPlanck has never been among the best in the class-on the contrary, his teacher noticed his ability in interpersonal communication. As a scientist, Planck has extraordinary concentration and unparalleled rigor of thinking. He is in faith.His concentration is constantly forged in the furnace. He believes from the bottom of his bones that nature operates through a certain logical system, and he defies difficulties and recklessly dedicated himself to revealing this system.

Planck’s drastic change occurred in 1894, the year he went from a firm but not outstanding physicist to a solemnly recorded Max Planck in history. His previous professor and friend HayMholtz nominated Planck to become a full member of the Berlin Academy of Sciences. Helmholtz said that the 36-year-old Max “has made a significant contribution to physics.” Helmholtz wrote: “The most noticeable thing is that,He has made achievements in the field of physical chemistry. He did not rely on the concept of atoms as sadly as chemists.” Planck uses pure thermodynamics, assuming that there is only continuous matter no particles or atoms, and uses pen and paper calculations to explore pureThermodynamics-everything from phase transitions to the conduction behavior of fluids. Until 1894, his entire career has been in the field of thermodynamics, especially the second law of thermodynamics and entropy.

In fact, the work and thinking in chemistry changed Max Planck's thinking. Unlike most physicists, he quietly and methodically accepted "atomism". In his many important academic thoughts changesSupporting the existence of the atom is the first important change. Planck’s flexibility is widely praised and has become one of his most famous characteristics. Too many scholars are deeply rooted in personal favorite fields and beliefs, and indulge in them., But Planck is able to chase the sun and the most fertile soil, constantly transplanting himself to a more sensible position.

In 1882, the young physicist still held a common view. "Atomic theory," he said, "will have to be abandoned eventually, and the matter continuity hypothesis will be supported." In the same year, he completed it.The paper "Vaporization, Melting and Sublimation" Vaporization, Melting and Sublimation pointed out that "thermodynamic theory is incompatible with the finite atom hypothesis." This statement is very popular with mainstream people. Although in the 21st century, people will not talk about "continuous matter".The concept of ”is believed to be true, but in the previous hundreds of years, it has passed the “sampling test”: Most scientists believe that as long as a knife is sharp enough and small enough, a tiny object can be cut into smaller pieces infinitely.Fragments. One of the advantages of this hypothesis lies in mathematics, because Newton’s calculus has the supreme status, especially when used to describe continuous matter. But in 1884, the Swedish chemist Arrhenius proposed a special method to describeA breakthrough claim of electrical properties. He believes that atoms in water can be ionized and charged as a result. In the same year, Jacobs Vanter Hoff published "Studies in Chemical Dynamics" Studies in Chemical Dynamics also based on atomic assumptions.Therefore, by 1887, Planck, who was struggling with such subjects, changed his views. In 1890, he wrote to a colleague: "Physicists have no choice: if they want to test certainThey must accept atoms and molecules as prerequisites.”

In 1894, Max was already a full professor. When he was promoted to the Berlin Academy of Sciences, he must feel like climbing to the top of a mountain every summer when hiking. He was finally recognized by the system. At this time, even if his reputation is not great, But it is also reliable enough. For five years, he and his family have stabilized in Berlin. He and Mary ushered in the first birthday of their last child, Erwin. The city of Berlin is gradually being lit by electric lights.With telephone ringing. The audience watched in amazement as Ottomar Anschütz's silent movie showed images of athletes and soldiers on horseback. The wealthy class in Berlin enjoyed new technology and looked forward to moreNew stuff. The science lecture hall is full of students like never before.

At the same time, German physics experienced a sudden leadership vacancy. By the end of 1894, the famous Helmholtz and the rising genius Hertz had both passed away.

As mentioned earlier, Max Planck is ready to meet new challenges. He is now fully studying the black body radiation problem that hides the true meaning of the universe. Looking at Berlin, Planck has seen the world’s top expertsCollecting blackbody measurement data, only waiting for a theory to explain their results. The most important thing is that he discovered a problem that he considers to be the most important in physics: how to reconcile thermodynamics and mechanics. The former is his profession.A time arrow can be defined. According to the second law of thermodynamics, in the process of gradual increase in entropy, a moment irreversibly follows the previous moment. However, the new "gas theory"-a microscopic dynamics, does notIn line with the second law. The statistical method of Maxwell and Boltzmann is a bet with reality-placing individual particles in different configurations of beating, colliding, and rotating gas molecules. Here, physicists canDiscuss the average value of hundreds of millions of tiny particles, or the possible distribution of particles. Under the gas theory, it is not so much to take a snapshot of the system, and thus accurately record its irreversible entry into the next inevitableThe situation of the process is rather the probability of this chaotic system from state A to state B, but at the same time it is also possible to reach state C or state D. Sometimes, from state A to state C or state D will break the secondThe law. In 1895, Boltzmann wrote: “The probability of this happening is not zero mathematically, but the probability is extremely small.” On the other hand, Max is very loyal to Clausius’ views.At that time, I thought that the principle of entropy increase was valid without exception, just like the principle of conservation of energy," he wrote in his later years, "For Boltzmann, this principle is just a law of probability.Exceptional circumstances."

In terms of professionalism, Max’s relationship with the emotional Boltzmann was barely maintained. In the early 1890s, even though he turned to support atomism, he still believed that many of Boltzmann’s work "hasExcessive reputation.” In essence, he thinks the gas theory looks very impressive, but up to then it has been nothing more than a tumult and commotion. The sensitive and fickle Boltzmann felt that he was offended., And to Max’s surprise, he publicly questioned Planck in a subsequent meeting.

Aside from some of the above-mentioned awkward interactions, Planck is also attracted by this question now. How can he reconcile his favorite second law of thermodynamics with increasingly successful and more reasonable statistical methods?On the one hand, even though few people continue to delve into this predicament, Max is still content with this relative obscurity. "This is a weird joke of fate," he later wrote, "My colleagues lacked interest in this subject...…It turns out that this has become my gospel…Since the importance of the concept of entropy has not been fully understood, no one will notice the method I have adopted. In this case, I can go leisurely, earnestly and thoroughly.Perform calculations without worrying about anything that will interrupt or exceed me.”

Blackbody radiation curve-the perfect battlefield for thermodynamics and new mechanics

Planck turned all his attention to the mysterious and universal curve of black body radiation, which he regarded as the perfect battlefield between thermodynamics and new mechanics. Because the brightness and color of the light emitted by the object depends on the temperature., So thermodynamics will be the key. He also needs some new methods of Boltzmann and Maxwell. The molecules in the black body involve some statistical facts. Their energy and vibration are different, but they produce precise emission in some way.Light spectrum. For example, if a football game is going on in a crowded stadium, if we want to fully understand the shouting in the stadium, then physicists can start by building a model of the fan seat. He gives a fanThe statistical distribution of, whether the fans are standing or sitting, screaming or applauding, and different volume levels, so this physicist can begin to estimate the sound we hear from a certain distance. At this levelAbove, the thermal radiation spectrum seems to be mocking Planck. Why does this spectrum the shouting sound have nothing to do with the material of the black body the exact stadium, location and the size of the crowd? Also, why the spectrum of light emitted by these objectsLines happen to have only one shape, but no other shapes? The fascination with this universality pushed him to the study of thermal radiation.

When Planck began to study this problem, he needed to face the light seriously for the first time. If there were two things that Planck thought he could count on in 1894, they were the second law of thermodynamics and it had just been provenThe theory of light propagation. The highly respected Hertz the two positions that Planck has received, the first being Hertz has published his evidence on electromagnetic waves in 1888, and Planck has also begun to follow his research work."Dear colleague, sorry for taking your time, I only have a short question," Planck wrote to Hertz in the summer of 1890, "I am glad to read your article "Electrodynamics of Stationary Objects" Electrodynamics.for stationary bodies.” Then, Planck asked Hertz whether he had written a negative sign wrong, and whether there was a typographical error in Formula 8. "I sincerely hope that I won’t take you too much time." Planck saidThis rigorous and demanding way of reading is "notorious", which makes him an excellent editor of the "Annual Book of Physics".

He used his usual methodical method to deal with the problem of blackbody radiation. At first, like his elderly teacher Kirchhoff, he constructed a hollowed-out object-a cavity. In 1860, Kirchhoff publishedHe wrote the original first article on thermal radiation detection. He has an incredible ability to absorb a large amount of information and transform it into a thorough and unified summary, and uses this ability in the emitted light of the objectIn terms of radiation spectrum. In his own experiments, he noticed that the ratio of light absorbed and emitted by an object has nothing to do with the material itself;Perfect "black body", and also declared that the universal function is suitable for thermal radiation. Kirchhoff put forward the universal characteristics of thermal radiation very early, although there is no sufficient and complete evidence to support this view, butThis is still surprising today.

Planck assumed that the cavity is a perfect sphere in the calculation, the cavity is completely vacuum, there are no impurities, or even a gas molecule, and then the overall study of the electromagnetic wave light inside the cavity, as well as the electromagnetic wave light and the cavityThe interaction of the body wall. Since the actual material of the cavity wall is no longer important—that is, as mentioned above, whether it is mud, chocolate, or metal, the heat radiation is exactly the same—Planck usesHe assumed that there are a series of "harmonic oscillators" on its inner surface. Although from the perspective of modern people, we may unconsciously interpret the specific physical meaning of Planck's harmonic oscillators, but Planck himself is veryHe may just be looking for the simplest possible constituent elements for his blackbody theory. He quoted a paper by Hertz six years ago and borrowed the concept of electronic harmonic oscillators from Hertz’s work. Therefore, although Hertz harmonic oscillators have electricityHowever, Planck may not care about its true nature. Interestingly, the idea that molecular vibration emits radiation has been mentioned in the early years Maxwell proposed in English, but Planck may not haveSee. In the end, Planck and others speculated that harmonic oscillators might be electrons, but at the beginning of the work, he did not write out their specific physical meaning.

Planck's harmonic oscillators each have a natural frequency, which means that each harmonic oscillator can and can only interact with a certain corresponding color light. If we return to the aforementioned football field analogy, Planck willIt is assumed that each fan can only hear and emit one tone. Like many initial models in physics, this hypothesis is realistic for the cavity wall and for football fans. Planck’s goal is toThe abstract electronic resonator first absorbs light and then emits the light into the cavity. He gave his first lecture on this subject at the Prussian Academy of Sciences, entitled Absorption and Emission.He hopes that the final radiation spectrum he calculated can match the observed experimental measurements, and it can also prove that the spectrum maximizes the entropy of the cavity, thus confirming and supporting the second law of thermodynamics.

Inspiration from Boltzmann

In the first few years, he tried to use classic and continuous technical means to overcome this problem. His first few bold papers established a framework with little complexity, but subsequent papers graduallyIncreased the complexity of the model. He aims to show that no matter what kind of light fills the cavity first, the activity of the harmonic oscillator will eventually follow the discovered black body radiation law. He wants to expose a single-pointed physics, fromStarting from an arbitrary starting point points to a uniform ending, so that he can explain how arbitrary objects produce the same experimental results. However, since the end of 1897, some doubts began to linger in his mind.

"Any unidirectionality discovered by Professor Planck," Boltzmann wrote in a devastating criticism, "must...because he chose the unidirectional initial conditions." BoltzMann’s point of view is that Planck’s attempt to reveal a series of unified and inevitable events is not unified at all. Boltzmann correctly discovered that Planck’s theory has other theoretical support hidden in it.The theory itself has not yet reached a balance and consensus. Nature can obviously start anywhere and reach the same end. However, Planck's theory requires a carefully chosen starting point, otherwise it will overturn the whole situation. A sound theory should notSensitive to the so-called initial conditions, and each starting point A, B, or C should point to point Z, which is the blackbody spectrum obtained in the laboratory. Initially, Planck was humorous and hurtful to BoltzmannThe remarks were rebutted, and after a lot of fierce talk, Planck finally compromised at the end of that year. He had to try something new.

Planck understands and respects Boltzmann’s work, although Planck struggled to evaluate the utility of the latter. When Boltzmann accepted probability calculus as his favorite toolMost physicists in the 1890s must have been shocked, and they thought it was esoteric, extreme, and even confusing. But by 1898, Planck’s attitude towards Boltzmann’s views gradually eased.

That year, Planck adopted a more radical statistical method to solve the problem of thermal radiation, calculating the average value of the entire spectrum of possible resonance frequencies and corresponding color lights. He introduced "natural radiation" The concept of “natural radiation” introduced the “molecular disorder” concept mentioned by Boltzmann in his first gas theory paper into electromagnetics. Planck’s 1899 paper announced a completeIn this concluding article, he frankly admitted that Boltzmann and gas theory provide the best path to understanding the divine entropy and the second law of thermodynamics.

At the same time, some mistakes occurred in the physics laboratory in Berlin. For several years, the black body radiation has been fairly calm in terms of empirical evidence.I worked together as an editor in the "Annual Book of Physics" and constructed a formula that fits all the experimental data of the black body. Most physicists accepted Wien’s formula, even if it was not a gospel truth, but also regarded it as a kind ofThe experience is similar to the facts. However, in 1899, a new method of measuring at longer wavelengths such as infrared light emerged, and the measurement results deviated from Wien’s prediction. The intensity of infrared light emitted from the hot cavity exceeded expectations., So the Wien formula is obviously not the one adopted by nature. The new experiment will deepen the research into the infrared field. In 1899 and mid-1990, every few months, Planck’s desk in Berlin will be updatedAs a result. Planck began to think about a new empirical fitting to replace his friend’s formula. For many years, he wanted to build a new theory from scratch, but he suffered many setbacks. So he decided to go back to the previous ones.Work. If he can intuitively figure out the correct natural heat radiation formula, it will provide him with an accurate compass for exploring the direction of basic physics.

The turmoil that lasted for several weeks finally brought him to the front of a breakthrough. The autumn of 1900 witnessed the death of two German "Williams". In August, after the death of the worker leader William Liebknecht, more than 100,000Workers held a solemn silent parade in the streets of Berlin, and the working class has since appeared. Even when the emperor condemned Liebknecht as a public enemy of Germany, this time it was called "iron silence".The parade still cooled the city and caused the huge class divisions to emerge at the time. And in September, another "William" who was close to Max also passed away. He was Max’s father and died in Munich.

After preparing the funeral in Munich, Planck returned to Berlin to continue his well-arranged lectures. In the weeks after his father passed away, an epiphany flashed in Planck’s mind. This sudden inspiration occurred inOn his way back to his office in Berlin? During a lecture? Even on his train from Munich back to Berlin? In either case, Planck clearly saw the form of mathematics that nature has always longed to reveal.This elegant form impressed him. He wrote a new formula that seemed to avoid the problems that Wien had previously conjectured. On October 19, Max briefly introduced his results at the regular meeting of the German Physical Society.His speech was not included in the meeting schedule, but in the final "discussion and comment" section. His oral comment only took up a little more than two pages. He admitted: "I finally started to construct an expression of the entropy of an arbitrary harmonic oscillator.The formula." He then wrote down his results to the people present. He said: "As far as I know, this formula satisfactorily conforms to the published observational data and is the best available.Spectral formula." The announcement is shown in the figure below:

This formula accurately gives the radiant energy E measured at the wavelength λ of the black body at the temperature T therefore also gives the radiant intensity. The uppercase C and lowercase c are only added to make the formula conform to the measured dataThe constants on the above. This type of constant is a measurement made by physics when using mathematical tools to fit data. Planck knew that one day, he must actually assign values ​​to C and c.

How did his audience react to this? It is speculated that only a small group of people who are really interested in it such as the experimenters Prinsheim, Kurbaum, and Rubens wrote in their notebooksThis new formula was laid down. Early the next morning, Rubens appeared happily at the door of Planck’s house, holding a few of the results obtained from his laboratory in his hand. All his measurements, including the latest infrared measurementsAll values ​​conform to Planck’s new curve, like beads strung on a thin thread. Rubens is convinced that Planck has discovered the law of heat in nature.

Other colleagues are not so convinced. Wien heard of the speech from a distance and wrote that he did not want to offend anyone, but the new formula is undoubtedly a serious mistake. Planck replied that he was convinced of the two of themDisagreements can be resolved, and Wien’s achievements in this field will gradually appear and improve in the days to come, rather than disappear. He proposed that the two meet during the Christmas holidays.

Quantum or classical?

At the same time, Planck is now facing a question worth millions of Deutsche Marks: Where did this delightful formula come from? He later wrote that his new formula is "just a little fortunate intuition."The law of eternity brought", "From the day I clarified this law, I began to devote myself to giving it real physical meaning." He set out to find a way to prove this new by tracing the roots of physics.Formula. He must turn his conjecture into a real theoretical point of view.

"After the most stressful weeks of work in my life, the darkness gradually receded, and some unforeseen sceneries began to appear." Planck realized that if he wanted to continue on the right path, You need to rely on Boltzmann’s statistical methods. Planck learned from a relatively early paper by Boltzmann—On the Relation between the Second Law of Probability in 1877.In Law of Thermodynamics and the Theory of Probability, the key mathematics part was introduced. In "The Connection between the Second Law of Thermodynamics and Probability Theory", Boltzmann used combinatorial mathematics to enumerate the possible structures of gas molecular groupsThe method of combinatorial mathematics is to study all possible ways of permutation and combination of a large number of substances gas molecules, socks, volleyball players and playing cards, etc.. Combinatorics is a branch of discrete mathematics and has a great relationship with Planck’s storyThey are all studying single, separated objects, such as molecules, marbles, etc. This kind of mathematics has nothing to do with the continuous light waves emitted from the thermal cavity, really? Max is desperate, thinking that only combinatorial mathematics can lead to his new formulas.

In December 1900, he presented his own new evidence and new theory. From a historical point of view, physicists believed that this marked the birth of quantum theory. According to reports, his audience did not take it seriously, and it was difficult for themFollow Planck’s ideas and walk through his brand-new forest of mathematics. In order to obtain this "October equation", Max divides the energy of the harmonic oscillator into equal small shares: "In any case, we take this view-And this is the most important point in the whole calculation-think energy E is composed of some completely certain, finite and equal parts." Once he treats the energy of the harmonic oscillator as a set of socks, cards orTo deal with objects separated by molecules like this, you can use Boltzmann's method in 1877 to march towards the new black body formula on a broad road.

However, the new "critical step" is radical, thorough, and earth-shaking. This is the first time someone wants to treat the divine energy in such a gritty way. Just like some more advanced physicistsProposing that matter atoms and electricity charges are both "lumpy", Planck also advocated the existence of similar energy granularity granularity. In 1900, he did not call it a new type of physics-he evenThe word “quantum” was not used. He later said that this historic step was “purely a formal hypothesis. I actually didn’t think too much about it.Produce a positive result".

Planck believes that he has successfully preserved the classical physics and solved the mystery of black body radiation. It is certain that he is very excited, he can see this success, but there is no sign that he smelled the change he ignitedFlame. Within thirty years, quantum physics will completely rewrite our understanding of light and matter. Most of this new knowledge comes from the writings of Einstein, Niels Bohr, Heisenberg and others. However, in thatFor a period of time, Planck mostly played the role of a worried patriarch, asking everyone to slow down and be cautious.

Now, we know that Planck’s derivation itself is flawed, not because he assumed discrete energies, but because of his exposition of the existence of discrete energies. There is no path from classical physics to precision.The black body spectrum. Younger physicists, such as Einstein and Paul Ellenfest, were the first to doubt and criticize Planck’s black body mathematics, and when Planck finally returned to this question, He changed a new derivation method. He published related papers between 1911 and 1913, calling this method the "second theory." Here, he still hopes to prevent the new quantum view from infiltrating the beautiful and beautifulIn continuous light waves.

In retrospect, 1900 will be the pinnacle of Planck and New Physics. Later, Erwin recalled their father and son walking in the garden in late autumn. He said that his father quietly announced to his 7-year-old son.This made one of his scientific breakthroughs. Max said that his recent work will be the greatest discovery in physics. 27 In Erwin’s narrative, Planck is no longer the familiar Planck personality image, and it is also missing.Some humility: Maybe it’s that Erwin’s distant childhood memories have been blurred and distorted over time. But it’s more likely that Max only allowed such flashy words to flow into his son’s ears. Even so, there is no self in the existing letters.Bragging words. In his work, there is only one thing that can make him so confident and excited. In the process of deriving the new formula, Planck explained two basic constants: k and h. The former is called BoltzmannThe latter was named Planck’s constant almost immediately.

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