Heisenberg has made great achievements in theoretical physics all his life, and his greatest contribution is undoubtedly the establishment of matrix mechanics. Prior to this, he did a lot of research work in fluid mechanics, anomalous Zeeman effect, molecular model and dispersion theory. These works prepared him for establishing matrix mechanics. Especially the cooperation with Born made him feel the urgency of establishing a new quantum theory.
Born used to study lattice dynamics. 192 1 became a professor of theoretical physics at the university of g? ttingen and began to study atomic theory. He asked students to learn some quantum physics and wanted to compete with sommerfeld. Hilbert, a famous mathematician in G? ttingen, advocated that mathematicians and physicists should unite to study physics. He and Born jointly organized a seminar on "Material Structure". In addition, there are various discussion classes around Born, such as "beginner's class", "advanced class" and "atomic mechanics I writing group", which have a very strong academic atmosphere. In order to prosper science, Born often invites famous scholars from all over the world to visit and give lectures. This greatly broadens students' horizons. Bonn is friendly and informal to his students. After class, he often walks, picnics and plays music with students. Born encouraged questions and criticisms in the discussion class he presided over. Therefore, Born is often surrounded by a large number of talented students, and Heisenberg is one of them.
If Heisenberg received strict training on Bohr's theory in Sommerfeld, he learned more about Bohr's theory in Born. When the Born School doubted the correctness of Bohr's theory, the Sommerfeld School also believed that Newtonian mechanics could solve the problems in the atomic field as long as the quantum conditions proposed by Planck, Bohr and Sommerfeld were attached. Heisenberg believes that Born is more convinced than Bohr that there should be a complete set of mathematically unified quantum theory, rather than wandering and reconciling Newtonian mechanics, quantum conditions and optical quantum assumptions.
1922, Born and his assistant Pauli discussed deeply the application of perturbation theory in atomic theory, and developed the general method of energy representation of perturbation theory. 1923, born and Heisenberg cooperated to apply perturbation theory to helium atom. Although the theoretical results are qualitatively consistent with the experimental results, there is still a big gap in quantitative aspects. This convinced them that the foundation of physics must be fundamentally changed.
1924, in gottingen's discussion class, born emphasized that it is incorrect to attribute the difficulties of quantum theory only to the interaction between radiation and mechanical systems. He believes that mechanics must be reformed and replaced by some kind of quantum mechanics, which provides a basis for understanding atomic phenomena. Born even called the expected new theory "quantum mechanics" for the first time in a paper in 1924. At this time, Born had some vague understanding of the new theory he expected. Heisenberg found a mathematical method to describe this theory.
Heisenberg is actually a teaching assistant of Bos, although he is only a scholar in G? ttingen. He cooperated closely with Born and tried to establish a new mechanics from the symbolic mechanics model. A year later, he obtained university teaching qualification and became an unpaid lecturer with a paper entitled "The Change of the Formal Law of Quantum Theory on the Abnormal Zeeman Effect". In September of the same year, Heisenberg came to Copenhagen, Denmark as a Rockefeller scholar, and his creative work such as Matrix Mechanics actually took root in Copenhagen.
Heisenberg worked mainly with Dutch physicist Cramer in Copenhagen. Cramer has been Bohr's assistant since 19 16, helping Bohr to do a lot of work in developing quantum theory. He is versatile. He can not only speak five foreign languages, but also play the cello. He often plays Heisenberg's piano after work. And they are extremely strict with students in their studies. At the beginning of 1924, Bohr, Krams and Slater published a theory which had a great influence on the future, also called BKS theory, which was mainly based on the idea put forward by Slater, an American physicist who came to work in Copenhagen at that time. The central idea of this theory is that a virtual radiation field can be generated by introducing a group of virtual oscillators to each atom, and each virtual oscillator has a transition frequency (that is, the transition frequency of atoms). This links the discontinuous atomic process with the continuous radiation field, so that we can use the correspondence principle and adopt a method similar to the classical theory to deal with the dispersion problem of quantum theory. It is with this idea that Krams deduced his dispersion formula.
If Krams's dispersion theory actually destroyed the basis of the concept of electron orbit, it can be said that Heisenberg prefers to abandon the electron orbit model and express Bohr's correspondence principle with the correct mathematical formula. Together with Krams, he studied dispersion with Born's method, and co-wrote a paper "On the Scattering of Radiation by Atoms".
Heisenberg returned to G? ttingen in April 1925. On the basis of the above work, he wants to further solve the problem of spectral line intensity of hydrogen atoms, but he has encountered great difficulties in mathematics. So, he turned to solve the problem fundamentally, that is, to find the equation of motion of the electrons in the hydrogen atom corresponding to the classical equation of motion. However, according to classical mechanics, this equation should describe the trajectory of electrons in atoms, but the atoms are too small, and the electron orbits can neither be seen nor touched, that is to say, they cannot be observed. So, how to test the correctness of the equation from the experiment?
Heisenberg was confused when he got hay fever. This is an allergy caused by some toxic pollen, which needs to be treated at the seaside. While recuperating in Helan Island in the North Sea, he was suddenly inspired by Einstein's theory of relativity. Einstein thought that the concepts of absolute velocity of objects and absolute simultaneity of events in two different places were meaningless, because these concepts were actually unobservable. So Heisenberg thinks that since the electron orbit that determines the radius and rotation period in Bohr's principle is unobservable, it is meaningless. What people can observe in the experiment is only the frequency and intensity of the spectral line.
Therefore, Heisenberg started from Bohr's correspondence principle, "trying to establish a theoretical quantum mechanics, which is similar to classical mechanics, and in this quantum mechanics, only the relationship between observable measurements appears." He saw signs of doing this in Bohr's frequency condition and Krams's dispersion theory. According to Bohr's frequency condition, the interaction between electrons in atoms can be expressed by the characteristic amplitude of electrons. Using Krams's quantum dispersion theory, starting from the classical equation of motion, we can get a quantum mechanical equation of motion based only on observable measurement. Theoretically, the solution of this equation should be able to give the fully determined frequency and energy values of the atomic system and the fully determined transition probability of quantum theory.
After several days of intense calculation, he used the obtained equations to deal with a relatively simple anharmonic quantum mechanical system and the movement of electrons around the core, and he succeeded twice.
When he finally finished the calculation, it was already past three in the morning. At this time, he was very excited and sleepy. He ran out of the room and climbed a rock tower near Shanghai until sunrise. He later recalled the mood at that time and said, "At first, I was deeply surprised. I feel that I am looking at a wonderful inner world through the surface of the original phenomenon. I am dazzled by the rich mathematical structure that is so generous to nature. "
Heisenberg lived on Helan Island for more than a week, and finally wrote the article "Reinterpretation of Quantum Theory on Kinematics and Dynamics". He found that the difference between quantum mechanical quantities and classical mechanical quantities is that quantum mechanics does not obey the exchange law of general multiplication, and they are not reciprocal, that is, AB≠BA. Starting from his equations, we can naturally get a solution that satisfies the quantum conditions, without having to attach a few assumptions like Bohr. He knew that "this very obvious but complicated physical problem can only be solved through more thorough mathematical research." However, his theory is still in the primary stage in mathematical processing and can only be applied to some simple examples. Therefore, he is not fully sure about his paper and hesitates whether to send it for publication immediately.
After much reflection, Heisenberg sent the completed paper to Pauli, his most strict critic, on July 9, and said, "I took the liberty to send my manuscript directly to you, because I believe it contains some real physics, at least in terms of criticism or negation. At the same time, I'm sorry, because I have to ask you to return the manuscript to me in two or three days. I must finish it or burn it in the last few days of my stay here. "
Pauli enthusiastically supported Heisenberg's theory and said, "I salute Heisenberg's brave hypothesis." It was because of Pauli's encouragement and support that Heisenberg made up his mind to send the paper to the teacher he was born for.
After reading Heisenberg's paper, Born soon realized the great significance of his student work. At this time, because Heisenberg went to Copenhagen again, on the one hand, he recommended Heisenberg's landmark papers to the Journal of Physics for publication, and on the other hand, he cooperated with student Iordan, trying to further develop Heisenberg's thoughts into the theory of quantum mechanical systems in mathematics.
After a week of hard thinking, it suddenly occurred to Born that if Bohr wrote each steady-state energy level horizontally and vertically, he would get a matrix. Where diagonal positions correspond to states and off-diagonal positions correspond to transitions. Then, the considerable measurement of Heisenberg can be represented by these arrays, and these arrays are just matrices! The operation method of this matrix is the same as Heisenberg's algorithm. It's really "I can't find a place to get it, I can get it without effort". Mathematicians have already prepared mathematical tools for physics, only to see which physicist can get there first. It is no accident that Born, who has worked in the capital of mathematics for a long time and is deeply interested in mathematics, will reap the fruits of victory.
Born was so excited about this discovery that he immediately threw himself into the tense calculation with Iordan. It took only a few days to write a paper on quantum mechanics. In this paper, they expounded the rules of matrix operation, applied the correspondence principle, started from the classical Hamiltonian canonical equation, applied the matrix form to Heisenberg theory, and obtained the matrix equation equivalent to Heisenberg quantum condition. According to this equation, the law of conservation of energy and Bohr frequency can be further derived and successfully applied to the quantum mechanical system of harmonic oscillator and anharmonic oscillator.
In February of the following year, they cooperated with Heisenberg and published the famous article "About Quantum Mechanics Ⅱ" in the name of three people, which extended the quantum mechanics developed according to Heisenberg's way to systems with any number of degrees of freedom, completed the perturbation theory of non-simple systems and a large class of simple systems, and derived the conservation laws of momentum and angular momentum, selection rules and strength formulas. Finally, the theory is applied to the statistical problem of eigenvibration of blackbody cavity.
In this paper, Heisenberg's original thought has been greatly developed in matrix form, and finally a complete system of quantum mechanics in matrix form has been formed. It is a new mechanical system based on particle images of microscopic objects. Because it uses matrix mathematics, it is also called matrix mechanics.
Soon, Pauli first applied this new mechanics to the spectrum of hydrogen atoms and calculated the steady-state energy value of hydrogen atoms. The results are completely consistent with Bohr's conclusion, thus confirming the correctness of the new theory. Then, physicists used quantum mechanics to deal with many puzzling atomic problems in the past, and they all succeeded. As a result, Gottingen's victory soon spread in physics. Einstein said humorously, "Heisenberg laid a lot of eggs." Cambridge, Berlin and Copenhagen all invited Heisenberg to talk about his new quantum mechanics.
In his later years, Heisenberg continued to explore on the road of quantum mechanics and achieved fruitful results. His "uncertainty relation" became one of the important principles of quantum mechanics, and 1932 won the Nobel Prize in physics. Heisenberg is recognized as one of the founders of quantum mechanics because of his great contribution.
Matrix mechanics is regarded as a successful attempt to replace qualitative correspondence principle with quantitative relation. In the process of establishing this theory, Heisenberg relied on an important methodological principle, that is, the observability principle. This principle requires theoretically abandoning those actually unobservable quantities and directly adopting observable quantities.
Heisenberg was lucky enough to learn from some first-class physicists, such as Sommerfeld, Born and Bohr. He later recalled that he learned physics from Sommerfeld, mathematics from Born and philosophy from Bohr. But he never followed blindly. He dares to doubt and criticize. He often asks the teacher sharp questions and has a profound discussion with him. His famous saying is: "Science is rooted in discussion." He dares to innovate when solving new physical problems. It is from this spirit of scientific exploration that he founded matrix mechanics and made great contributions to science. He once said: "At every new stage of understanding, we should always follow Columbus's example. He dared to leave his familiar world and discovered a new continent on the other side of the ocean with almost fanatical hope. "