How to solve the Schrödinger equation of infinite deep potential wells? At 12 o'clock on January 9, the nineteenth issue of "Zhang Chaoyang's Physics Class" began broadcasting. Zhang Chaoyang, founder, chairman of the board and CEO of Sohu, sat in the live video broadcast room of Sohu, carefully analyzed the difference between phase speed and group speed, and clarified that group speed is the speed of transmitting energy and information. And the separation of variables method, to obtain the fixed Schrödinger equation, and then take the infinite depth well problem as an example, specifically solve the discrete energy level and energy eigenstatic wave function. Finally, the physical significance of the Fourier transform is introduced and given, and the uncertainty principle is once again explored, and the world of quantum mechanics is glimpsed.
Different from the gradual derivation formula of previous live broadcasts, Zhang Chaoyang went straight to the theme at the beginning and wrote down Schrödinger's equation on the "live whiteboard". "It's equivalent to Newton's laws in the microscopic world." Zhang Chaoyang said, "Mathematicians have to exhaust all possibilities, but people who are engaged in physics will guess several special situations, so today we will guess its solution." He also reminded netizens, "To learn quantum mechanics, you need to master the basics of calculus, linear algebra, Fourier transforms and so on." This knowledge is also used in lectures.
Solve the stationary Schrödinger equation using the separation variable method
Zhang Chaoyang first took netizens to review the concept of wave velocity mentioned in the previous lesson, and discussed phase velocity and group velocity. Subsequently, he began to solve Schrödinger's equations by separating variables. The wave function in the equation contains both a partial derivative of time t and a partial derivative of position x, and takes the method of separating variables to separate time t from position x, that is, it is assumed that the wave function has the following form:
Substituted into Schrödinger's equation, the term containing the position x and the term containing the time t are separated to the two sides of the equal sign, i.e.:
Zhang Chaoyang introduced, since this equation is true for any position x and time t, so the left (or right) of the equal sign is a constant that has nothing to do with both position x and time t, and if we assume that this constant is E, we can decompose the original complex binary differential equation into two unary differential equations:
"This makes it much less difficult to understand the equations. If you simplify it again, you can get the schrödinger equation. Zhang Chaoyang said as he wrote.
(Zhang Chaoyang derives the schrödinger equation on the "live whiteboard")
Explore infinitely deep potential traps to obtain discrete energy levels
In order to concretely show the complete process of solving Schrödinger's equation, Zhang Chaoyang gave the example of an infinitely deep potential well. He introduced that when the position x is between 0 and a, the potential energy U(x) is zero, and the potential energy is infinite elsewhere, so the wave function is zero outside the area of position 0 to a, and because the wave function has a continuous nature, the wave function is also zero at x=0 and x=a. Thus, when x is between 0 and a, its fixed Schrödinger equation can be written:
The corresponding boundary conditions are:
This differential equation is very classical and easy to solve, Zhang Chaoyang solved this unary differential equation, and got the expression of energy level E:
And the corresponding energy eigenstatic wave function:
where n takes the positive integer.
(Zhang Chaoyang solves the energy level of infinite deep potential well on the "live whiteboard")
"Very different from the continuity of classical theory, it can be seen that energy E is discrete." Zhang Chaoyang concluded.
Giving the Fourier Transform Physical Significance Explores the uncertainty principle
In the subsequent live broadcast, Zhang Chaoyang also introduced the relationship between the Fourier transform of mathematics and the representation of quantum mechanics. "Momentum is actually an operator in quantum mechanics." He deduced step by step on the whiteboard:
He introduced that if a quantum state satisfies the following formula:
This indicates that this quantum state is the eigenstat of the operator, and the value p is the eigenvalue corresponding to this eigenstat. This is the wave function that describes particles with momentum of p.
The traditional mathematical Fourier transform is:
"This is consistent with the uncertainty principle of quantum mechanics." Zhang Chaoyang said.
"We present today the application of Schrödinger's equations in a simple case. In the future, we will continue to explain the Schrödinger equation to solve the hydrogen atom problem. At the end of the live broadcast, Zhang Chaoyang told netizens, "The next class will further explore quantum mechanics." ”
Sohu video power knowledge live broadcast spread knowledge to play science
Since November 5, "Zhang Chaoyang's Physics Class" has been broadcast live for 18 issues. In the first and second lessons, Zhang Chaoyang popularized "force" and "speed" and calculated the number of daily flight circles around the ground of spacecraft and space stations; the third and fourth lessons were related to "vibration" and the basic knowledge of popular science visible light. Lessons five and six sparked a big discussion about the speed of sound and temperature. Lessons 7, 8, and 9 revisit the "two dark clouds" of classical mechanics; lessons 10 and 11 focus on the blackbody radiation curve and its applications; lessons 12, 13, and 14 attempt to enter Einstein's world of thought, derive the famous formula "E=mc", and demonstrate the clock slow scale-down effect. Lesson 15 explains the structure of atoms and the decay of atomic nuclei. Lesson 16 begins with quantum mechanics, discussing wave-particle duality of light, Compton scattering, Heisenberg uncertainty, and Schrödinger's equations.
From these more than ten physics lessons, it can be seen that the live broadcast style of "Zhang Chaoyang's Physics Class" is unique - by observing the phenomena of daily life, using the topics that netizens are most familiar with to enhance interest, and then explaining the physical principles behind it in the way of formula derivation, "seeing the essence through the phenomenon", and then solving similar problems in life in turn.
Zhang Chaoyang believes that studying the natural world is a particularly interesting thing, and he hopes that the audience of physics classes can maintain curiosity, "driven by curiosity, understand the mysteries of nature and understand the truth of our survival in this world." In subsequent courses, he will continue to explain common physical phenomena in life in this live broadcast style, so that ordinary people can understand and professionals can recognize, stimulating the upsurge of scientific learning. The course is broadcast live on Sohu video every Friday and Sunday at 12:00.
In addition to "Zhang Chaoyang's Physics Class", Sohu Video also invited a number of general education broadcasters to popularize science knowledge and convey value. Dr. Chen Zheng, a teacher at the School of Science of Beijing Jiaotong University, played a "strange scientific experiment" and walked into the "wave-particle duality of light"; Bao Kun, a doctor of physical chemistry at Cornell University, incarnated as "Bao Daren play science" to teach ordinary people to understand the 2021 Nobel Prize; and Liu Boyang, a doctor of astrophysics, popularized "How a total solar eclipse is produced". In the future, more knowledge broadcasters will settle in and play science together.