Archive of posts in category “physics”
The duality between two plane trajectories related by a conformal map
The conformal map transforms the trajectory with energy in potential into the trajectory with energy in potential . I will prove this beautiful result and show some implications of it.
- Categories: physics
- Tags: complex, classical mechanics, canonical transformation, kepler problem, mathematical physics, vector analysis, long paper
You can replace with in the Schrödinger equation?
When someone asks you why it is here instead of or the other way around, you can say that this is just a convention. My professor of quantum mechanics once asked the class similar a question, and I replied with this letter.
- Categories: physics
- Tags: quantum mechanics, letter, complex
A measure-theoretic formulation of statistical ensembles (part 2)
For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms.
- Categories: physics
- Tags: mathematical physics, statistical mechanics, functional analysis, measure theory, probability, long paper
A measure-theoretic formulation of statistical ensembles (part 1)
For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms.
- Categories: physics
- Tags: mathematical physics, statistical mechanics, functional analysis, measure theory, probability, long paper
Even solutions to bound states in an odd number of potential wells
We try solving the even function solutions to the time-independent Schrödinger equation for the potential such that (bound states).
- Categories: physics
- Tags: from zhihu, quantum mechanics
Defend our earth against aliens’ bullets!
The aliens intiated their attack to the earth! They shoot bullets with mass and speed from a far-awar planet. To defend, humans built a field that can repel the bullets. What regions are safe? The answer turns out to be the interior of a circular paraboloid.
- Categories: physics
- Tags: calculus, classical mechanics, kepler problem, from zhihu
How to construct mechanics in higher dimensions?
We can derive the equation of motion for mechanical systems in a Galileo universe with time dimensions and space dimensions by generalizing the principle of relativity and Hamilton’s principle.
- Categories: physics
- Tags: from zhihu, classical mechanics, imagination
When a supersonic airplane flies over your head
Suppose a supersonic airplane has Mach number . It flies horizontally. At some time, it flies past over your head at height . Then, the distance between you and it when you have just heard it is .
- Categories: physics
- Tags: from zhihu
Finding curvature radius physically
Sometimes the curvature radius of a curve can be found by using physical methods although it seems that you must use calculus to find it. In this article, the curvature radius of the curve at the point where the curvature radius is smallest is found by using physical methods without using calculus (with only high school knowledge). The answer is that the smallest curvature is exactly , and the point with smallest curvature is the vertex.
- Categories: physics
- Tags: from zhihu
A whirling point charge
In the vacuum, inside a fixed ring of radius with fixed charge uniformly distributed, there is a point charge with charge and mass moving in the plane of the ring due tue the electrostatic force. It moves in the small region around the center of the ring, and the motion is periodic along a closed curve. The area of the region enclosed by the curve is . Denote the distance from the center to the point charge as , and . Find the magnetic induction at the center of the ring.
- Categories: physics
- Tags: from zhihu