Fibonacci Scarf Finally Done.

Wave Function

A wave function in quantum mechanics describes the quantum state of an isolated system of one or more particles. There is one wave function containing all the information about the entire system, not a separate wave function for each particle in the system. Its interpretation is that of a probability amplitude. Quantities associated with measurements, such as the average momentum of a particle, can be derived from the wave function. It is a central entity in quantum mechanics and is important in all modern theories, like quantum field theory incorporating quantum mechanics, while its interpretation may differ. The most common symbols for a wave function are the Greek letters ψ or Ψ

The image above shows the comparison of classical and quantum harmonic oscillator conceptions for a single spinless particle. The two processes differ greatly. The classical process (A–B) is represented as the motion of a particle along a trajectory. The quantum process (C–H) has no such trajectory. Rather, it is represented as a wave. Panels (C–F) show four different standing wave solutions of the Schrödinger equation. Panels (G–H) further show two different wave functions that are solutions of the Schrödinger equation but not standing waves.

A classic formula for pi has been discovered hidden in hydrogen atoms

This is truly amazing.

For the first time, scientists have discovered a classic formula for pi in the world of quantum physics. Pi is the ratio between a circle’s circumference and its diameter, and is incredibly important in pure mathematics, but now scientists have also found it “lurking” in the world of physics, when using quantum mechanics to compare the energy levels of a hydrogen atom.

Why is that exciting? Well, it reveals an incredibly special and previously unknown connection between quantum physics and maths.

“I find it fascinating that a purely mathematical formula from the 17th century characterises a physical system that was discovered 300 years later,” said one of the lead researchers, Tamar Friedmann, a mathematician at the University of Rochester in the US. Seriously, wow.

The discovery was made when Carl Hagen, a particle physicist at the University of Rochester, was teaching a class on quantum mechanics and explaining to his students how to use a quantum mechanical technique known as the ‘variation principle’ to approximate the energy states of a hydrogen atom.

While comparing these values to conventional calculations, he noticed an unusual trend in the ratios. He asked Friedmann to help him work out this trend, and they quickly realised that it was actually a manifestation of the Wallis formula for pi – the first time it had even been derived from physics.

“We weren’t looking for the Wallis formula for pi. It just fell into our laps,” said Hagen. “It was a complete surprise,” added Friedmann. “I jumped up and down when we got the Wallis formula out of equations for the hydrogen atom.”

Since 1655 there have been plenty of proofs of Wallis’s formula, but all have come from the world of mathematics, and the new results have people freaking out. The results have been published in the Journal of Mathematical Physics.

Continue Reading.

Visualization showing the flow of first 10,000 digits of Pi, revealing how the number Pi looks like. Circle is divided in 10 segments, from 0 to 9. Then path is traced, going from the third segment to the first segment. From 1, the path jumps to 4, then back to 1, then 5 and so on. After a while, Pi appears in front of your eyes. The final illustration is beautiful.

the last few days has seen me getting into very basic electronics, so here is my small robot son that yells if you get close to them

I really just want someone who’s in the same shoes as me to tell me it’s okay. I mean. All my friends are smart and yeah, they say stuff like “you’re not dumb!!” But im just thinking “well im not smart either”

my first post..

and I should be writing an essay. seems about right

When weather.com has advice for chem sophomores

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The Gaussian Integral is a beautiful integral for which the area between the e^(-x^2) and the x-axis from negative infinity to positive infinity perfectly equals the square root of pi. Image sources: 1, 2.

I had some really nice asks about math encouragement, and so I wanted to share some things from my responses:

Since I was never a math person before my mid-twenties, I had a TON of catching up to do to get my mathematics degree. This meant I couldn’t afford to compare myself to my classmates, because I was so hilariously behind that I had to accept (and even embrace) being dead last in the ranking.  I had to acknowledge that as a necessary pre-requisite to attempting a mathematics degree, and I truly believe that attitude is one of the main reasons I was able to complete that degree. Because of this attitude, I was kind to myself. I applauded myself for just being able to be in each room, enrolled in each class, because it was so improbable that a former math illiterate could have gotten there in the first place. I still endeavor to celebrate every victory from ‘Yay! You showed up with your shoes tied and everything’ to ‘Look at you go! You took that really hard test and got a WHOLE THIRD of the points! That so much more than none of the points!’ That mindset kept me going, kept me from being competitive with my colleagues, and kept me feeling fulfilled and proud of my work. I never was a modern-day Gauss, but that’s fine with me. I don’t do math to glorify myself. I do math because it is important, because I love it more than I have ever loved anything. I study mathematics because the only thing I need in this life is to know more about mathematics today than I did yesterday.