A song rose in Shen Qi's heart, and his brush moved with divine skill:
Using oxy as the coordinate, the air is divided into many thin layers parallel to the ground.
The refractive index of the thin layer containing point P is n1.
The angle between the ray and the ground normal is θ1.
The following layers are, in order: n2, θ2, n3, θ3...
According to the law of refraction and geometric relations, we have:
(Dy/dx)^2=1/sin^2θ-1
After taking the square root, dy/dx takes a negative value.
Therefore: dx=-2n1/kαnp*sinθ1*dφ/(φ^2-1)^1/2
Here, a point needs to be integrated, and the final integral constant C is expressed as:
C=2n1sinθ1/kαnp*arcosh(ke^α/2*H)
This is the equation for the trajectory of light that produces a mirage.
Through explanations in physics and a little mathematical processing, we can clearly see that the characteristic of a mirage is that "it changes as I change, and eventually disappears."
If one day you happen to be at the beach or in the desert and are lucky enough to witness a mirage, don't move. Just stay where you are and quietly admire it. At this romantic moment, confess your love to the girl next to you, and she will surely accept your affection.
According to the equation of light propagation trajectory C=2n1sinθ1/kαnp*arcosh(ke^α/2*H), if you lose your composure during such a romantic moment of courtship and move away, the dreamlike mirage will appear and disappear intermittently until it vanishes completely. Then the girl will run away, and you'll have nothing left; your dream will be shattered. What a pity!
The wonder of physics lies in its ability to explain complex and fantastical phenomena using simple and easy-to-understand equations.
If physics and mathematics work together, this powerful, technical method for attracting women can turn you into a heartthrob.
Learn more knowledge; it will come in handy someday.
When you reach the pinnacle of human knowledge, hehe... Shen Qi laughed, and wrote the answer on the exam paper: C=2n1sinθ1/kαnp*arcosh(ke^α/2*H)
After finishing the first fill-in-the-blank question, Shen Qi felt some of the passion he had during last year's math competition, and he also gained a deeper understanding of the beauty of both mathematics and physics, as well as their close connection.
Physics attempts to bypass complex mathematical calculations, relying on qualitative descriptions and rough quantitative methods to establish its dominance, but this is impossible. The more advanced the field of physics, the more refined the mathematical processing required. Physics is like a devastatingly powerful cannonball; the launch medium determines its range, accuracy, and destructive power.
Mathematics mostly exists on paper and in the minds of mathematicians; it's like a powerful cannon, only without a shell inside. Once loaded, bang! bang bang! Anything is possible. The most suitable shell for the mathematical cannon is, of course, physics.
Shen Qi understood this principle well: modern mathematical artillerymen must be proficient in the characteristics of various shells in order to fire excellent shots in various ways.
The first question in the CPhO semi-finals was a fill-in-the-blank question, and the next seven questions were all calculation questions.
The semi-finals were much more difficult than the preliminary rounds. There were no multiple-choice questions, which greatly reduced the chances of guessing.
Question 2, Calculation Problem, 16 points.
This is a Quantum Physics problem, a basic Quantum Physics.
In the CPhO competition, high school contestants only need to know some basic concepts of Quantum Physics and be able to apply them simply; they don't need to delve into the principles, nor do they have the ability to do so. If they were to delve into that stuff, they would either win an award, go insane, or win an award while in a state of madness.
The problem statement provides a bunch of data and constants, including Electron Charge, Electron Mass, Bohr Radius, Rydberg Energy, Proton Rest Energy, etc.
In summary, this collection of emoticon-like data describes a physical phenomenon: in a sufficiently hot gas discharge, various ions are present, one type of which is an atom with a nuclear charge of Z stripped down to just one electron.
Without a diagram, Shen Qi needed to find some useful clues from the large amount of data in the problem statement, ultimately calculate the value of Z, and write down what element the ion was.
In physics problems involving mechanics, acoustics, and electromagnetism in physics competitions, the diagrams often contain a lot of usable information, and reading the diagrams is an important step in understanding the question. Quantum Physics is different from these; whether or not a diagram is provided makes little difference, and most of the work depends on the test taker's imagination.
There are many mind-bending branches of physics, and Quantum Physics is definitely among the top 5.
There is no connection between macroscopic optical refraction and microscopic ion trapping, or between mirages and the electronic ground state.
Physics encompasses so much that Shen Qi switched to quantum mode and began solving the 16-point calculation problem.
The semi-finals started with two major obstacles, and the second question was no easy feat either.
First, Shen Qi needs to find inspiration from Heisenberg.
Heisenberg is not a place name; he was a distinguished German physicist whose contributions to Quantum Theory were second only to Einstein.
Heisenberg was a talented individual, even a physics genius; he won the Nobel Prize in Physics at the age of 31. Einstein, on the other hand, was already in his forties when he received the Nobel Prize in Physics.
Historically, Heisenberg's reputation is controversial. During World War II, he conducted scientific research for the German government, working on the atomic bomb. Of course, it was the Americans who first developed the atomic bomb and used it in combat.
Regardless of Heisenberg's political leanings, his "Heisenberg Uncertainty Principle" holds a very high position in academia.
Shen Qi first used the "Heisenberg Uncertainty Principle" to launch a surprise attack, assuming that the only electron in A^(Z-1)+ is in the ground state.
With slight manipulation of this state, the average value of the square of the distance from the electron to the center of the atomic nucleus, r0^2, can be obtained.
This is a simple mathematical operation.
High school students participating in the Physics Competition only need to know that r0^2 is defined as the sum of the squares of the uncertainties in position coordinates: (△x)^2, (△y)^2, and (△z)^2.
The requirement for an excellent high school physics competition participant is the ability to apply the Heisenberg Uncertainty Principle in a basic way, without needing to understand it in depth. A deep understanding is for university students; we can discuss that later.
Following the same method, Shen Qi obtained the average value of the square of the electron momentum in this state, p0^2.
When an A^(Z-1)+ ion captures an electron, it emits a photon. This process necessarily obeys the laws of conservation of energy and momentum.
Both conservation relations involve the angular frequency ω0 of the emitted photon, and they form a system of equations that include ω0.
By Heisenberg Uncertainty Principle:
(△x)(△px)≥1/2
(△y)(△py)≥1/2
(△z)(△pz)≥1/2
The energy conservation equation can be specifically expressed as:
1/2meve^2+1/2(M+me)v^2+Eli=1/2(M+2me)μ^2+E'li+ω0
The next step requires a series of slightly complex mathematical operations, which are not difficult for Shen Qi:
O(∩_∩)Omeowo(╥╥)o...
(The above formula will appear as gibberish in Word; use your imagination, the author couldn't help it.)
To outsiders, mathematical and physical studies, when taken to a certain level, may not seem much different from metaphysics.
Mathematicians and physicists don't need to express their thoughts in any written language; they simply throw out a bunch of symbols at the drop of a hat, and you can look at them for yourself. Once you understand them, then we can talk.
After a series of deductions and calculations, Shen Qi finally obtained the value of Z.
Z=4
"This... Z equals 4." Shen Qi thought for a moment, silently counting in his mind: hydrogen, helium, lithium, beryllium, boron, carbon, nitrogen, oxygen, fluorine, neon...