Science and Math Articles

Here is the collection of expert STEM articles that span all the sciences and mathematics. Some examples include physics, mathematics, biology, chemistry, cosmology, and earth sciences. For math, we cover algebra to advanced calculus.  These articles are all peer-reviewed by the community.

Star Equations

Equations of State for Photon Gas and Relativistic Electron Gas

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0 Comments
This Insight develops equations of state that are useful in calculations about cosmology and about the insides of stars. The first calculation is for a…
rules of quantum mechanics

The 7 Basic Rules of Quantum Mechanics

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45 Comments
For reference purposes and to help focus discussions on Physics Forums in interpretation questions on the real issues, there is a need for fixing the common…
Quantum Particles

Clarifying Common Issues with Indistinguishable Particles

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15 Comments
Commonly there is a lot of imprecision in talking about ''indistinguishable'' (or ''identical'') particles, even in serious work. This Insight article…
Renewable Energy Power Grid

When Renewable Energy Meets Power Grid Operations

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55 Comments
The electric power industry faces much turmoil in the coming decades.  The business model of the electric utility company (public or private) may not…
Maupertuis Principle

A Pure Hamiltonian Proof of the Maupertuis Principle

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Here is another version of proof of Maupertuis's principle. This version is pure Hamiltonian and independent of the Lagrangian approach.The proof…
create a universe story

How to Create a Universe: Instructions for an Apprentice God

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11 Comments
(-: ##~~~## A fantasy to be read at leisure time ##~~~## :-) Written October 7, 1999 to relax from the strains of working on a new interpretation of quantum…
quantum qubit

A Classical View of the Qubit

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2 Comments
This Insight article is part of my paper Foundations of quantum physics III. Measurement, featuring the thermal interpretation of quantum physics. See…
The Sum of Geometric Series from Probability Theory

The Sum of Geometric Series from Probability Theory

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12 Comments
Here I present a simple (but to the best of my knowledge, new) derivation of the formula for the sum of the infinite geometric series. The derivation is…
quantum mechanics is incomplete

9 Reasons Quantum Mechanics is Incomplete

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45 Comments
I argue that all interpretations of quantum mechanics (QM) are incomplete, each for its own reason. I also point out that for some interpretations (those…
against quantum interpretations

Against “interpretation”

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197 Comments
I am against "interpretations" of Quantum Mechanics (QM) in a sense in which John Bell [1] was against measurement in QM and Travis Norsen [2] is against…
lie algebra representations

Lie Algebras: A Walkthrough The Representations

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1 Comment
  Part III: Representations  10. Sums and Products. Frobenius began in ##1896## to generalize Weber's group characters and soon investigated…
abacus introduction

An Introduction to the Traditional Abacus of China and Japan

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7 Comments
Introduction Abacuses are commonly seen in ancient civilizations around the world as a convenient tool to do simple calculations. In this article, we…
Hooke's Law Experiement

An Example of An Accurate Hooke’s Law Laboratory

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16 Comments
Learning ObjectivesGain confidence and experimental care in making accurate measurements. Understand the relationship between force and spring…
Lie Algebra Structure

Learn Lie Algebras: A Walkthrough The Structures

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  Part II: Structures5. Decompositions.Lie algebra theory is to a large extend the classification of the semisimple Lie algebras…
lie algebra basics

Learn Lie Algebras: A Walkthrough The Basics

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  Part I: Basics 1. Introduction. This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems,…
Bernoulli's equation

Demystifying the Often Misunderstood Bernoulli’s Equation

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19 Comments
Introduction Bernoulli's equation is one of the most useful equations in the field of fluid mechanics, owing to its simplicity and broad applicability.…
Magnetic Field Lines

Are Magnetic Field Lines Real?

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31 Comments
We recently had a question in the relativity forums that mentioned the behavior of magnetic field lines and reminded me of my confusion at school about…
vehicle power

When Vehicle Power Dictates Acceleration

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33 Comments
One interesting problem when dealing with a vehicle of a certain mass is to determine what is required to get the maximum acceleration while going from…
Quantum Eraser

The Quantum Mystery of Wigner’s Friend

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30 Comments
In this Insight I will introduce the quantum mystery called ``Wigner's friend'' using Healey's version [1] of Frauchiger and Renner's (FR's) version [2]…
Calculus Therom

A Numerical Insight for the Fundamental Theorem of Calculus

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9 Comments
The purpose of this article is not to provide some rigorous statement, neither a rigorous clever proof of the fundamental theorem of calculus. It is rather…
Super Quantum

The Unreasonable Effectiveness of the Popescu-Rohrlich Correlations

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3 Comments
In this Insight, I will show how the Popescu-Rohrlich (superquantum) correlations provide an unreasonable advantage in a particular "quantum guessing game''…
killing Fields and Black Holes

Exploring the Math in Killing Fields and Black Holes

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Killing Vector Field The Killing vector field is a vector field on a differentiable manifold that preserves the metric over spacetime (from this I assume,…
LHC Energies

Exploring the Energies in the Large Hadron Collider (LHC)

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The Large Hadron Collider has produced collisions at 7 TeV. For collisions at 7 TeV, protons need to be ‘ramped’ to 3.5 TeV, the proton has a mass…
Falling in Black Holes

Black Holes and the Properties of In-Falling Radial Plungers

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From 'Exploring Black Holes' by John Wheeler and Edwin Taylor; can apply to any object falling radially towards a static spherical mass (where the mass…
stock options math

Learn a Simplified Synthesis of Financial Options Pricing

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19 Comments
Financial options (the right to purchase ("call") or sell ("put") stock (or other assets)) at a fixed price at a future date have been around for a long…
why quantum

Why the Quantum | A Response to Wheeler’s 1986 Paper

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397 Comments
Wheeler's opening statement in his 1986 paper, ``How Come the Quantum?'' holds as true today as it did then [1] The necessity of the quantum in the construction…
tensor calculus

The 10 Commandments of Index Expressions and Tensor Calculus

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16 Comments
Having more than 10 years of experience in teaching vector and tensor calculus and special and general relativity, I have noted that many people have the…
speed of sound

Direct Echo-Based Measurement of the Speed of Sound

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42 Comments
Figure 1: Distance vs. time a firecracker report echoed off of a building at different distances on both a cold and a warm day.The speed of sound…
information definition

How to Better Define Information in Physics

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11 Comments
When I ask questions about the conservation of information I frequently get the reply, “It depends on what you mean by information.”   So, I researched…
bizarro univerise

Lessons From the Bizzaro Universe

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23 Comments
The terms Bizarro and Bizarro World originated in Superman comics, where strangely imperfect versions of Superman, other action characters, and even Earth…
sun_getting_smaller

Either the Sun Is Getting Smaller or Gravity Is Getting Weaker

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36 Comments
Paper discussion: Solar system expansion and strong equivalence principle as seen by the NASA MESSENGER mission. Antonio Genova, Erwan Mazarico, Sander…
statmech2

Learn Statistical Mechanics: The Ideal Gas

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11 Comments
Read Part 1: Equilibrium Systems The Ideal Gas: Boltzmann's Approach (The Microcanonical Ensemble) Consider a monatomic gas of ##N## non-interacting…
classicalphysics

Addressing the “Classical Physics Is Wrong” Fallacy

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188 Comments
One of the common questions or comments we get on PF is the claim that classical physics or classical mechanics (i.e. Newton's laws, etc.) is wrong because…
hilbertspaces2

Learn the Basics of Hilbert Spaces and Their Relatives: Operators

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4 Comments
  Operators. The Maze Of Definitions. We will use the conventions of part I (Basics), which are ##\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}##,…
hilbertspaces

Learn the Basics of Hilbert Spaces and Their Relatives

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2 Comments
  Basics Language first: There is no such thing as the Hilbert space.Hilbert spaces can look rather different, and which one is used in…
qft_renormalization

Learn Renormalization in Mathematical Quantum Field Theory

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9 Comments
This is one chapter in a series on Mathematical Quantum Field TheoryThe previous chapter is: 15. Interacting quantum fields. 16. Renormalization In…

Learn Interacting Quantum Fields in Mathematical Quantum Field Theory

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This is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 14. Free quantum fields.The next chapter is 16. Renormalization. 15.…
qft_freequantum

Learn Free Quantum Fields in Mathematical Quantum Field Theory

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This is one chapter in a series on Mathematical Quantum Field TheoryThe previous chapter is 13. Quantization.The next chapter is 15. Interacting…
bb_infinity2

Intro to the Big Bang and Infinity Concepts

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34 Comments
In the cosmology section of the Physics Forums, I encountered many Big Bang and related infinity problems. Hence, in this article, I want to explain the…
journeycosmos3

Learn About the FLRW Metric and The Friedmann Equation

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1 Comment
Previous Chapter: A Journey Into the Cosmos – The Friedmann Equation                Chapter 2- FLRW Metric and The Friedmann Equation In…
qm_interpretations

The Fundamental Difference in Interpretations of Quantum Mechanics

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194 Comments
A topic that continually comes up in discussions of quantum mechanics is the existence of many different interpretations. Not only are there different…
qtf_quantization

Learn Quantization in Mathematical Quantum Field Theory

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15 Comments
The following is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 12. Gauge fixing.The next chapter is 14.…
expandinguniverse

Explore Coordinate Dependent Statements in an Expanding Universe

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In this Insight, we will discuss the general Robertson-Walker (RW) universe, in which a set of co-moving observers with proper time ##t## observe the universe…
gps3

The Schwarzschild Metric: A Newtonian Comparison

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A Short Proof of Birkoff's Theorem derived the Schwarzschild metric in units of ##G = c = 1##:\begin{equation} ds^2 = -\left(1 - \frac{2M}{r}\right)dt^2…
gps2

The Schwarzschild Metric: The Photon Sphere

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 A Short Proof of Birkoff's Theorem derived the Schwarzschild metric in units of ##G = c = 1##:\begin{equation} ds^2 = -\left(1 - \frac{2M}{r}\right)dt^2…
qft_gauge

Learn Gauge Fixing in Mathematical Quantum Field Theory

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The following is one chapter in a series of Mathematical Quantum Field Theory.The previous chapter is 11. Reduced phase space.The next chapter…
gps1

The Schwarzschild Metric: GPS Satellites

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 A Global Positioning System (GPS) device gives your precise location by receiving light pulses from satellites with synchronized clocks then…
cosmosintro

Learn About the Friedmann Equation and the Cosmos

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6 Comments
This is an introduction to cosmology for someone who has some knowledge of calculus and basic physics. In this tutorial, we will take a journey into the…
qtf_phased_space

Learn Reduced Phase Space in Mathematical Quantum Field Theory

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4 Comments
The following is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 10. Gauge symmetries.The next chapter is…
workethic

Why Work Ethic Is a Key to Success for Science Majors

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25 Comments
After earning a 4.0 GPA in his first semester as a Physics major at an esteemed state university, our son (and homeschool grad) attributes his…
dice

Learn About Intransitive Dice with a Twist

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6 Comments
Intransitive dice are sets of dice that don't follow the usual rules for "is better/larger than". If A<B and B<C, then A<C. If Bob runs faster…
qtf_guagesym

Learn Gauge Symmetries in Mathematical Quantum Field Theory

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18 Comments
This is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 9. Propagators.The next chapter is 11. Reduced phase…
qft_prapagators

Learn Propagators in Mathematical Quantum Field Theory

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8 Comments
This is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 8. Phase space.The next chapter is 10. Gauge symmetries. 9.…
Maxwell equations

What is the Homopolar Generator: An Analytical Example

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8 Comments
Introduction It is surprising that the homopolar generator, invented in one of Faraday's ingenious experiments in 1831, still seems to create confusion…
qtf_phasespace

Learn Phase Space in Mathematical Quantum Field Theory

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7 Comments
The following is one chapter of a series on Mathematical Quantum Field Theory.The previous chapter is 7. Observables.The next chapter is 9. Propagators. 8.…
qtf_observables

Learn Observables in Mathematical Quantum Field Theory

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8 Comments
The following is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 6. Symmetries.The next chapter is 8. Phase…
dna_stretch

Learn the Amazing New Research on Untying Knots in Polymers

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4 Comments
Polymer Key PointsThe published paper studied how knots in polymers respond to an external flow field and how the knot affects the response of the…
qtf_symmerities

Learn Symmetries in Mathematical Quantum Field Theory

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4 Comments
The following is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 5. Lagrangians.The next chapter is 7. Observables.  6.…
qft_lang

Learn Lagrangians in Mathematical Quantum Field Theory

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This is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 4. Field Variations.The next chapter is 6. Symmetries.  5.…
qtf_field_variations2

Learn Field Variations in Mathematical Quantum Field Theory

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6 Comments
This is chapter 4 of a series on Mathematical Quantum Field Theory.The previous chapter is 3. Fields.The next chapter is 5. Lagrangians.  4.…
dna_twist

How to Use the Spaghetti-Twist Method to Align DNA

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A new paper from a collaboration of Canadian physicists has demonstrated how simply twirling a wire through a solution of DNA molecules can align the molecules…
qtf_fields

Learn the Fields of Mathematical Quantum Field Theory

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82 Comments
The following is one chapter in a series on Mathematical Quantum Field TheoryThe previous chapter is 2. Spacetime.The next chapter is 4. Field…
qft_spacetime

Learn Spacetime in Mathematical Quantum Field Theory

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9 Comments
The following is one chapter in a series on Mathematical Quantum Field Theory.The previous chapter is 1. Geometry.The next chapter is 3. Fields  2.…
qtf_geometry

Learn the Geometry of Mathematical Quantum Field Theory

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33 Comments
This is the first chapter in a series on Mathematical Quantum Field Theory.The next chapter is 2. Spacetime. 1. Geometry The geometry of physics…
quantumfieldtheory

A First Idea of Quantum Field Theory – 20 Part Series

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8 Comments
These notes mean to give an expository but rigorous introduction to the basic concepts of relativistic perturbative quantum field theories, specifically…
circuitanalysis

The What and Why of Circuit Analysis Assumptions

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15 Comments
Probably everyone reading this article already has education on elementary circuit analysis (CA) and many of us are much more advanced. Typically, the…

A Journey to The Manifold SU(2): Differentiation, Spheres, and Fiber Bundles

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3 Comments
Part 2  Differentiation, Spheres, and Fiber Bundles Image source: [24]The special unitary groups play a significant role in the standard…
perturbation

Introduction to Perturbative Quantum Field Theory

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110 Comments
This is the beginning of a series that gives an introduction to perturbative quantum field theory (pQFT) on Lorentzian spacetime backgrounds in its…
quantumpeace

How I Stopped Worrying and Learned to Love Orthodox Quantum Mechanics

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289 Comments
Many people here know that I am a "Bohmian", i.e. an adherent of a very non-orthodox interpretation of quantum mechanics (QM). Indeed, in the past, I have…

15 of the Best PF 2017 Eclipse Photos

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69 Comments
The 2017 August Eclipse really was something special. Fortunately Physics Forums has great photographers and snapped some remarkable shots. Here are a…
inflation5

A Poor Man’s CMB Primer: Quantum Seeds

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11 Comments
 The CMB establishes a record of ancient acoustic oscillations in the baryon-photon plasma. We've been studying how these primordial sound…
mathoperators

How to Tell Operations, Operators, Functionals, and Representations Apart

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24 Comments
 All these concepts belong to the toolbox of physicists. I read them quite often on our forum and their usage is sometimes a bit confusing.…
epsresults

Find Out the Results Shown at 2017 CERN EPS Conference

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7 Comments
The EPS conference was a week ago, and many new results and future plans were shown. A good general overview has been collected by Paris Sphicas in his…
m-theory

Super p-Brane Theory Emerging from Super Homotopy Theory

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6 Comments
  A notorious open problem: What is the non-perturbative theory formerly known as Strings?We still have no fundamental formulation of “M-theory”…
EdmondHamilton

Assessing the Physics in Edmond Hamilton’s “The Second Satellite”

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2 Comments
This article won't contain much about physics that typical readers of this site don't already know. However,  deconstructing improper physics in literature…
entanglement_entropy2

Lesson on Entanglement Entropy in Quantum Field Theory

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3 Comments
Part 1: Quantum Mechanics[This article expects the reader to have some experience with quantum field theory and path integrals][I should also…
dampedforces

Learn Damped Motion in Classical and Quantum Mechanics

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21 Comments
This Insights article is about dissipative forces (friction, air resistance, viscosity...).Damping by friction forces is one of the concepts that is…
SpectralParadox

Exploring the Spectral Paradox in Physics

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17 Comments
In terms of wavelength, peak solar radiation occurs at about 500 nm.  Interestingly, this is well within the range of human vision.  When solar radiation…
undergradresearch

Finding Niches for Publishable Undergraduate Research

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6 Comments
Undergraduate interest in research is a good thing; it's even better if they aspire to publish their work for review and consideration from a broader audience.…
coldquestioning

How to Ask Questions Effectively and Avoid Cold Questions

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71 Comments
We frequently get questions such as these:"What is energy?""What are Cooper Pairs?""What is conservation of momentum?"etc...etc.And…
entanglement_entropy

Learn Entanglement Entropy in Quantum Mechanics

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22 Comments
Quantum Mechanics(QM) is one of the greatest intellectual achievements in human history. Not only because it describes the world at the microscopic level…
physicslab

Revamping Undergraduate Introductory Physics Laboratory

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19 Comments
This is a compilation of my blog entry on this singular "project" to change the way we conduct the introductory physics laboratory in most US universities.…
cameralense

Learn About New High Performance Ultra-wide Angle Camera Lenses

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7 Comments
Note, unless otherwise specified, all focal lengths are in terms of the 35mm image format.Recently, there has been a flurry of new high-performance…
diffractiontheory

The Diffraction Limited Spot Size with Perfect Focusing

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12 Comments
Purpose The purpose of this Insights article is to give the reader a brief introduction to the principles behind diffraction-limited focusing. The reader…
learned

What Is The Most Important Thing That You Learned?

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10 Comments
I mentioned in an earlier Insight what I think are the most important things you can learn by being on Physics Forums. This time, I tackled the question…
cornuspiral

Learn Frenet Equations in 2-D Which Result in the Cornu Spiral

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3 Comments
This Insights article is intended as an introduction to the Frenet-Serret equations by showing an interesting application that results from a case in two-dimensional…
experimenthistory

Learn About History’s First Science Experiments

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11 Comments
In this Insight, I'm going to talk about two of the first science experiments in recorded history. One was allegedly performed by the Prophet Elijah in…
boatramps

Real Research in Unexpected Places; Boat Ramps

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4 Comments
Read part 1 -Backyards, Barns, BayousWhen I was offered a job at the Air Force Academy, it seemed like an opportunity to exercise both my commitment…
geometrysimple

When Simple Geometry Unveils Deep Math

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28 Comments
Introduction It is a remarkable fact that consideration of very elementary concepts in geometry often leads quickly into deep and unexpected mathematical…
quantumflux

Learn About Vacuum Fluctuations in Experimental Practice

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41 Comments
This Insight Article is a sequel of the Insight Articles ”The Physics of Virtual Particles”, “Misconceptions about Virtual Particles“ , and ''The…
magnet

Permanent Magnets Explained by Magnetic Surface Currents

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38 Comments
Introduction: The purpose of this Insight is to explain permanent magnets in a way that is in agreement with advanced textbooks on the subject, and that…
highercategorytheory

Why Higher Category Theory in Physics?

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30 Comments
 This here is my personal story. For an alternative introduction see my talk: Higher Structures in Mathematics and Physics.Initially,…
SchwarzschildGeometry3

The Schwarzschild Geometry: Spacetime Diagrams

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12 Comments
 When we left off in part 2, we were looking at the metric for the Schwarzschild geometry in Kruskal-Szekeres coordinates:$$ ds^2 = \frac{32…
fakenews

The Problem of Fake News and Science Reporting

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107 Comments
During and even after the recent US election campaign, numerous reports of fake news, made-up news, and other kinds of news were deemed either outright…
quantumapplitudes

Learn About Quantum Amplitudes, Probabilities and EPR

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83 Comments
This is a little note about quantum amplitudes. Even though quantum probabilities seem very mysterious, with weird interference effects and seemingly nonlocal…
fishresearch

Real Research in Unexpected Places: Backyards, Barns, Bayous

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2 Comments
21st-century science labors under the myth that good experimental science requires expensive lab facilities in a university or institutional setting.…
representations

Linear Representations and Why Precision is Important in Math

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2 Comments
First of all: What is a representation? It is the description of a mathematical object like a Lie group or a Lie algebra by its actions on another space…
angledimensions

Can Angles be Assigned a Dimension?

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106 Comments
1. Some Background on Dimensional Analysis ... if you are not already familiar with it. 1.1 Dimensions Dimensional Analysis is a way of analyzing…
whystudyphysics

Why Study Physics? A Bit Goes a Long Way!

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6 Comments
If you don’t plan to become a scientist or an engineer, is there a good reason to study physics?  I say, “Yes,” because knowing just a little bit…
vaccummyth

Explore The Vacuum Fluctuation Myth in Quantum Theory

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230 Comments
This Insight Article is a sequel of the Insight Articles ”The Physics of Virtual Particles” and “Misconceptions about Virtual Particles“ which…