Discontinuous energy in Blackbody Radiation and Specific Heat

2022-06-16 0 By

Read and create/Shenzhen Business Daily reporter Chen Shu At 12 o ‘clock on February 4th, “Zhang Chaoyang’s physics class” the 25th period as scheduled broadcast.Sohu founder, chairman of the Board of Directors and CEO Zhang Chaoyang sit in sohu video broadcast room, began to talk about harmonic oscillator problem.Has experienced a few lessons before detailed solution of the schrodinger equation, this class first reviewed the first two body schrodinger equation of hydrogen atom in the center of mass sports and the decomposition of the relative motion, angular and radial decomposition, finally obtaining complete operator set of eigen function together, the wave function of hydrogen atom and the corresponding level, and expand to other multi-electron atoms from hydrogen atoms.Then Zhang began to discuss the harmonic oscillator model, which is as important as the hydrogen atom problem, introduced its importance and general bias, and predicted the harmonic oscillator energy levels to be derived in the next class and their applications.On the fourth day of the first lunar month, Zhang chaoyang sent New Year’s greetings to net friends in the live broadcast room, “We spent a lot of lessons on quantum mechanics and hydrogen atoms.””Today,” he said, “we’re going to talk about harmonic oscillators, which are as important as the hydrogen atom.Let’s recognize its importance and universality.”Looking back on the hydrogen atom problem Jump out of the details and look at the big picture Zhang Chaoyang took the netizens to review the key content that had been said before.”Hydrogen is a two-body system composed of electrons and protons attracted by the Coulomb potential.””The Schrodinger equation for such a system is not easy to solve directly,” he said.As the course has shown, the solution is not complicated.Since the Coulomb potential is only related to the relative distance between the electron and the proton, the stationary Schrodinger equation of the two-body system can be decomposed into the moving part of the center of mass and the relative moving part.The center of mass part is easy to solve directly and has nothing to do with the internal structure of the hydrogen atom.Focus only on the stationary schrodinger equation of relative motion.In order to solve the problem more conveniently, Zhang introduced the concept of angular momentum operator. He said, “The Hamiltonian operator of relative motion can be expressed as the combination of radial kinetic energy and the square of angular momentum operator.”Through the derivation of complicated formulas, he and his friends came to the conclusion that the Hamiltonian operator, the square of the angular momentum operator, and the z component of the angular momentum operator are reciprocal and form a complete set of operators.”In other words, the solution of the steady-state Schrodinger equation can be obtained only by finding the common eigenfunctions of the set of operators.”He said.He went on to explain that since the Coulomb potential between an electron and a proton has nothing to do with the orientation between them but only with the relative distance, the stationary Schrodinger equation for the relative moving part can be divided into the angular part and the radial part.By solving the common eigenfunctions of the square of the angular momentum operator and the z component of the angular momentum operator, the angular wave function and the corresponding discrete quantum number can be obtained.As for the radial part, the radial Schrodinger equation can be deformed and simplified by analyzing the asymptotic behavior and boundary conditions when R approaches 0 and r approaches infinity. Finally, the radial wave function and corresponding quantum number can be obtained by series expansion, and the energy level of hydrogen atom can be obtained at last.”According to the separation of variables method, we can multiply the angular wave function and the radial wave function to obtain the complete hydrogen atom wave function, thus completing the solution of the Hydrogen atom Schrodinger equation,” he told netizens.After completing the solution of the schrodinger equation for hydrogen atom and obtaining the wave function and energy level of hydrogen atom, we can know how the electrons outside the nucleus move around the nucleus.He explained that the hydrogen wave function was described by three quantum numbers: the principal quantum number N, the angular quantum number L and the magnetic quantum number M.The energy level is only related to N. At the same energy, the angular quantum number can take n values from 0 to n-1. The orbits of electrons moving around the nucleus are named as NS, NP, Nd, NF….;At a certain angular momentum L, the magnetic quantum number m takes 2L +1 values, from -L to + L.Zhang extended these electron orbitals of hydrogen atom to the multi-electron system.”Every atom other than hydrogen has multiple electrons with repulsion coulomb interactions between them. It is very difficult to solve these schrodinger equations directly and quantitatively,” he pointed out the problem and the solution. “Here we make some approximations and assumptions to discuss qualitatively.””Let’s start with the simplest atom after hydrogen, helium.”He took the net friend step by step derivation.Helium has two electrons in the 1s shell, assuming they’re spherical around the nucleus, and because of the Coulomb repulsion between the electrons, the number of nuclear charges felt by one of the electrons will be shielded, and by variational calculation, the number of nuclear charges felt by the helium electrons will not be 2, but 27/16.The electrons can then be considered equivalent to being independent of each other, except that the number of nuclear charges they feel is reduced by shielding.If you look at lithium, it has three electrons, and according to the Pauli exclusion principle and the range of quantum numbers discussed above, the 1s orbital can only be filled with two electrons with different spin orientations, and the third electron can only be squeezed into the 2s orbital.But because the 2 s orbital outside the 1 s, the inner two electrons will feel strong shielding 2 s electron number of nuclear charge, through simple calculation and analysis, although a 2 s electron of nuclear charge number is not simply reduce 2, but the shielding effect of by far than helium atom is at the same layer of the electronic shielding effect is much larger.By explaining helium and lithium atoms, Zhang told netizens that after the shielding of electrons in the same shell and the shielding of electrons in different shells, we can know qualitatively how the electrons in other multi-electron atoms are arranged, thus obtaining a preliminary explanation of some chemical problems.Vibration, Phonon and Specific Heat From harmonic oscillator model “In addition to hydrogen atom, harmonic oscillator is also a very important research object in quantum mechanics, and its Schrodinger equation has an analytical solution just like hydrogen atom.Zhang Chaoyang continued to explain that in the harmonic oscillator model, the restoring force when the object vibrates is proportional to the distance from the equilibrium position of the object, and the force direction always points to the equilibrium position.For a system with a stable equilibrium point, the potential near the minimum point can often be approximated as a harmonic oscillator potential, which is very common in physics.For example, in classical mechanics, the vibration of air in sound waves, the vibration of piano strings, small Angle pendulum;For example, black-body radiation in quantum mechanics, vibration of diatomic molecules, phonons in superconductors, and so on.Zhang chaoyang also predicted that in the next class we would solve the schrodinger equation of harmonic oscillator and get the corresponding energy level.He explained that n is an integer greater than or equal to zero, which means that the energy levels of the harmonic oscillator are discrete and equally spaced, which is very different from classical mechanics, but consistent with Planck’s assumption when deriking the black-body radiation, and hence the correct black-body radiation formula.It should also be noted that the minimum energy of the harmonic oscillator is not zero, which can explain the specific heat of diatomic molecules.Diatomic molecule has three centroid translational dof and two rotational degrees of freedom, a degree of freedom vibration, in addition to the kinetic energy of the degrees of freedom, a degree of freedom vibration between diatomic molecules corresponding potential energy can be stored energy, also contribute to a degree of freedom, so a total of seven degrees of freedom, can according to the equipartition theorem, the total internal energy should be 7/2 kt,But experiments measuring its specific heat capacity show that its internal energy is 5/2kT.This is precisely because the minimum value of vibration energy is not 0, and is larger than the minimum excitation energy of other degrees of freedom. At room temperature, the energy allocated to each degree of freedom is only enough to excite the energy of other degrees of freedom. The degree of freedom related to vibration is frozen, so the effective degree of freedom is changed from 5 to 7.He said, “In short, solving the Schrodinger equation of harmonic oscillator, not only can we get familiar with the solution of the Schrodinger equation again, but also the result has many physical applications, so we will study it in depth in the next class.”Sohu video sends force knowledge to broadcast knowledge to play science since November 5th, “Zhang Chaoyang’s Physics Lesson” has been broadcast more than 20 times.In the first and second lessons, Zhang popularized “force” and “speed” and calculated the number of times that Musk’s spacecraft and China’s space station flew around the earth every day.Lesson three and four are related to vibration and teach basic knowledge of visible light.Lessons five and six led to a big discussion about the speed of sound and temperature.Revisit two dark clouds of classical physics in lessons 7, 8, and 9.Lessons 10 and 11 focus on the blackbody radiation curve and its application.Lesson 12, 13, 14 Try to enter Einstein’s world of thought, deduce the famous formula “E= MC ²”, and demonstrate the effect of slow clock shrinkage;Lesson 15 covers the structure of atoms and the decay of nuclei;Lesson 16 begins with quantum mechanics, discussing the wave-particle duality of light, Compton scattering, Heisenberg uncertainty principle, and schrodinger equation.Can be seen from more than 20 issue of the physics class, physics class of zhang live style, by observing the phenomenon of everyday life is, the net friend the most familiar topics to improve interest, explain it in the form of formula derivation behind it, the physical principle of “see the essence through the phenomenon”, which, in turn, to solve the similar problems in life.Zhang believes that studying the natural world is particularly interesting, and he hopes that physics students will have curiosity to “understand the secrets of nature and the reason why we live in this world.”In the following courses, he will continue to explain common physical phenomena in daily life in such a live broadcast style, which can be understood by ordinary people and recognized by professionals, and stimulate the enthusiasm of scientific learning.The course is broadcast live on Sohu every Friday and Sunday at 12 PM.At the same time, users can search for “Zhang Chaoyang” in sohu video “following stream” to watch the full replay of the previous video.In addition to “Zhang Chaoyang’s physics class”, Sohu video also invited the head of various professional fields to enter, live popular science knowledge, transfer value.Dr. Chen Zheng, a teacher of the School of Science, Beijing Jiaotong University, played with “curious scientific experiments” and walked into “the wave-particle duality of light”.Dr. Bao Kun of Physics and chemistry of Cornell University, personified as “Bao Adults play science”, teaches ordinary people to understand the 2021 Nobel Prize;Liu Boyang, a doctor of astrophysics, popularized the science of “How is a total solar eclipse produced”, and Zhou Siyi, a postdoctoral fellow in theoretical physics who is popular recently, also opened a live class of “String Theory World” on Sohu video.In the future, there will be more knowledge broadcasters to play science together.Read: Yu Fanghua