Physics, Math and Science Articles

Independent Research: Thinking Outside The Box Versus Knowing What’s In The Box

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In Memory of Dr. Thomas J. LeCompte (1964-2025), Detector Designer and Champion of Education and Science. Prologue Someone who shows interest in science is initially a welcome development. So are fresh ideas from unexpected quarters. In contrast, there is a scientific community that is meticulously organized down to the last detail, allowing little to no…

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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…
manifold2

A Journey to The Manifold SU(2): Representations

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3 Comments
Part 1 Image source: [23]  Some useful bases of ##\mathfrak{su}(2,\mathbb{C})## Notations can differ from author to author: 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…
10mathtips

10 Math Tips to Save Time and Avoid Mistakes

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25 Comments
Exam situations are always situations of stress. It comes with our endeavor to be as good as possible and our fear of failure. Some students handle these…
solarimaging

Solar System Imaging Guide: Planets, Moon & Sun Tips

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4 Comments
Some of the most visible features in our sky are solar system objects: the Sun, the Moon, and the planets. These objects are generally much brighter than…
fourierseries2

Learn Further Sums Found Through Fourier Series

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6 Comments
In an earlier insight, I looked at the Fourier series for some simple polynomials and what we could deduce from those series. There is a lot more to be…
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”…
Parsevalstheorem

Learn an Integral Result from Parseval’s Theorem

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5 Comments
Introduction: In this Insight article,  Parseval's theorem will be applied to a sinusoidal signal that lasts a finite period of time.  It will be shown…
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…
whatisatensor

What Is a Tensor? The mathematical point of view.

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59 Comments
Introduction Let me start with a counter-question. What is a number? Before you laugh, there is more to this question than one might think. A number can…
jimbiology

From Biology to Sysadmin: Career Insights & Advice

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17 Comments
Interview: From Biology to Programming, Sysadmin, and Life in New Mexico What got you interested in biology? I was an amateur naturalist as a kid. I…
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…
name_the_scientist

Name the Scientist Quiz and Trivia

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27 Comments
Time to test your scientist photo recognition skills! As always please post your score in the comment thread at the end.Ready for your next quiz?…
pamelagay

Pamela Gay Interview — Astronomy, Citizen Science & Outreach

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2 Comments
Please give us a bit of background into your educational and professional life. I'm a public school kid through and through. After going to the local…
SpectralParadox

Exploring the Spectral Paradox in Physics

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16 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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70 Comments
We frequently get questions such as these:"What is energy?""What are Cooper Pairs?""What is conservation of momentum?"etc...etc.And…
processing

The Joy of Processing

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11 Comments
In the early days of the personal computer revolution, computers were small, simple, and easy to operate. It was always great fun to write BASIC games…
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…
BondiKcalculus

Learn Relativity Using the Bondi K-calculus

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20 Comments
Although Special Relativity was formulated by Einstein (1905), and given a spacetime interpretation by Minkowski (1908) [which helped make special relativity…
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.…
frasercain2

Q&A With Universe Today’s Astronomer Fraser Cain

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4 Comments
Video Q&A: Fraser Cain (Universe Today) Special thanks to Universe Today founder Fraser Cain for this fascinating video Q&A. All questions were…
lensfocus

Lenses and Pinholes: What Does “In Focus” Mean?

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11 Comments
In introductory physics, the optics unit often teaches about virtual and real images, focal lengths, indexes of refraction, etc. Some questions that are…
sabine1

Sabine Hossenfelder: Dark Matter, Gravity & LIGO Interview

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18 Comments
We are pleased to introduce Sabine Hossenfelder. Sabine is a theoretical physicist at the Frankfurt Institute for Advanced Studies. She also authors the…
deriitive5

The Pantheon of Derivatives – Important Theorems (V)

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Implicit Function Theorem [1] Jacobi Matrix (Chain Rule). Let ## (x_0,y_0 ) ## be a point in$$U_1 \times U_2 = \{x \in \mathbb{R}^k\,\vert \,||x-x_0||<…
RelativityVariables

Relativity Variables: Velocity, Doppler-Bondi k, and Rapidity

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8 Comments
Traditional presentations of special relativity place emphasis on "velocity", which of course has an important physical interpretation... carried over…
deriitive4

The Pantheon of Derivatives – Lie Derivatives And Others (IV)

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6 Comments
  Lie Derivatives A Lie derivative is in general the differentiation of a tensor field along a vector field. This allows several applications…
deriitive3

The Pantheon of Derivatives – Sections, Pullbacks And Pushforwards (III)

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3 Comments
  Some Topology Whereas the terminology of vector fields, trajectories, and flows almost by itself suggests its origins and physical relevance,…
cameralense

Ultra-Wide Lenses: Field of View, Design & Tips for Photographers

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7 Comments
Note, unless otherwise specified, all focal lengths are in terms of the 35mm image format. Overview: why ultra-wide-angle lenses matter Recently there…
deriitive2

The Pantheon of Derivatives – Manifolds And Vector Fields (II)

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9 Comments
  Generalizations Beyond ##\mathbb{R}## and ##\mathbb{C}## As mentioned in the section on complex functions (The Pantheon of Derivatives - Part…
deriitive

The Pantheon of Derivatives – The Direction (I)

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3 Comments
  Differentiation in a Nutshell I want to gather the various concepts in one place, to reveal the similarities between them, as they are often…
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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9 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…
integraltrick

Trick to Solving Integrals Involving Tangent and Secant

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6 Comments
This little trick is used for some integration problems involving trigonometric functions is probably well-known, but I only learned it yesterday. So…
linearacceleration

Frames of Reference: Linear Acceleration View

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2 Comments
My previous Insight, Frames of Reference: A Skateboarder's View, explored mechanical energy conservation as seen from an inertial frame moving relative…
scavengerhunt2

PF Insight Scavenger Hunt 2

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4 Comments
We have an amazing count of over 320 Insights published at Physics Forums. Now we test to see how much you remember from some of them. If you need a hint…
computerlanguages

Types of Programming Languages: Uses, Examples, Overview

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67 Comments
In this Insights Guide, the focus is entirely on types of languages, how they relate to what computers do, and how they are used by people. It is a general…
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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27 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…
nasa

NASA Quiz and Trivia: Test Your NASA Knowledge

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27 Comments
Since the mid 20th century NASA has been a force in aerospace research and programs. Let's find out just how much you know about it's history, programs…
jimhardy

Jim Hardy: Nuclear Engineer — Energy Insights & Hobbies

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9 Comments
Give us a little background on Jim Hardy Born and raised in Miami, Florida, to parents with roots in Springfield, Missouri. So at heart I'm a small‑town…
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,…
steven

Interview with Physics Mentor: Stevendaryl

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5 Comments
Give us a brief history on stevendarylI was born and raised in the American South, mostly living in Georgia. It's a Southern tradition, for some reason,…
inferometer

Fabry-Perot and Michelson Interferometry: A Fundamental Approach

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14 Comments
Fabry-Perot Effect: The Fabry-Perot effect is usually treated in most optics textbooks as the interference that results from multiple reflections of the…
FourierSeries

Using the Fourier Series To Find Some Interesting Sums

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4 Comments
Preliminaries If f(x) is periodic with period 2p and f’(x) exists and is finite for -π<x<π, then f can be written as a Fourier series: [itex]f(x)=\sum_{n=-\infty}^{\infty}a_{n}e^{inx}…
SchwarzschildGeometry4

The Schwarzschild Geometry: Physically Reasonable?

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33 Comments
 In the last article, we looked at various counterintuitive features of the Schwarzschild spacetime geometry, as illustrated in the Kruskal-Szekeres…
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…
SchwarzschildGeometry2

The Schwarzschild Geometry: Coordinates

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2 Comments
 At the end of part 1, we looked at the form the metric of the Schwarzschild geometry takes in Gullstrand-Painleve coordinates:$$ ds^2…
fakenews

Spotting Fake Science News: Wendelstein 7‑X & Media

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107 Comments
Fake News, Data, and the Need to Check Sources During and even after the recent US election campaign, numerous reports of fake or misleading news were…
SchwarzschildGeometry1

The Schwarzschild Geometry: Key Properties

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22 Comments
 Not long after Einstein published his Field Equation, the first exact solution was found by Karl Schwarzschild. This solution is one of the…
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.…
stephenhawkings

Stephen Hawking Quiz: Think You Know Stephen Hawking?

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23 Comments
Stephen Hawking has been a giant in the science world for decades. He has been a accessible figure with his willingness to reach out to the public in various…
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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107 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…
anyon

Learn The Main Conceptual Ideas of Anyon Particles

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55 Comments
Every quantum physicist knows that all particles are either bosons or fermions. And the standard textbook argues that this so does not depend on the number…
tripods

Camera Tripod Guide: Choose Stable Tripods for Photos

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1 Comment
Why use a tripod? A tripod holds a camera still while the shutter is open, preventing motion blur. It must maintain stability on uneven ground, in wind,…
dslr

DSLR Camera Guide: Lenses, Aberrations & Tips—Enthusiasts

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15 Comments
I'm ready for a DSLR! These cameras were originally designed for professionals. Users of DSLRs are expected to already understand basic digital photography…
partial-differentiation

Learn Partial Differentiation Without Tears

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6 Comments
Differentiation is usually taught quite well. Perhaps that's because it is the first introduction to calculus, which is considered a big step in a student's…
frameofreference2

Frames of Reference: A Skateboarder’s View

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22 Comments
My essay Explaining Rolling Motion raised some commentary about frames of reference and their equivalence when solving physics problems. I wish to pursue…
relativitygauge

Exploring General Relativity as a Gauge Theory

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24 Comments
The fundamental interactions of the Standard Model are described by Yang-Mills theory. This is a gauge theory, which means the following: (1) Choose a…
astro_advanced

Astrophotography Mounts, Tracking & Post-Processing Tips

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4 Comments
"I’ve tried astrophotography and want to know how to improve." Choosing a mount Here’s where I assume you are contemplating the purchase of a…
astro_inter

Astrophotography Level 2: Camera Setup & Techniques

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2 Comments
Level 2: I know how to use a camera — what do I need to know for astrophotography? Image processing, optics, and sensor basics Image-processing methods…
astro_intro

Astrophotography Guide for Beginners — Essential Tips

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2 Comments
For this Insight series (three total), I assume you either have a camera (with an attached lens) or already know how to attach your camera to your…
davidhestenes

David Hestenes on Geometric Algebra: Applications & Learning

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14 Comments
For those who don't know the great David Hestenes, he is the inventor of the geometric algebra formalism of physics. Here we go! 1) What is the best application…

Interview with Science Advisor DrChinese

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6 Comments
We are pleased to share with you an interview with PF Science Advisor and one of our founding members: DrChineseWho is DrChinese and share some early…
govandteachers

Blaming Government for Teacher and Scientist Failures in Integrity

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46 Comments
The article, "Governmental policy is wrecking science," makes some interesting points but is fundamentally in error, because government policy is only…
cubicfunction

Learn How to Solve the Cubic Equation for Dummies

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5 Comments
Everybody learns the "quadratic formula" for solving equations of the form [itex]A x^2 + B x + C = 0[/itex], even though you don't really need such a formula,…
digitalcamera3

Digital Camera Buyer’s Guide: Bridge Cameras

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5 Comments
This article is geared towards:I've never owned a camera, but I want to get a 'real' camera (that doesn't cost too much). I've owned an entry-level…
rollingmotion

Learn The Basics of Rolling Motion

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50 Comments
Although rolling wheels are everywhere, when most people are asked "what is the axis of rotation of a wheel that rolls without slipping?", they will answer…
davidgriffths

David J. Griffiths Interview: Teaching Quantum Mechanics

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35 Comments
We are pleased to introduce David J. Griffiths, one of the most successful physics textbook authors. Odds are if you studied physics in college, you've…
quantumparticle

Spectral Standard Model and String Compactifications

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29 Comments
 The Connes-Lott-Chamseddine-Barrett model is the observation that the standard model of particle physics -- as a classical action functional,…
digitalcamera2

Ultimate Digital Camera Buyer’s Guide: Compacts, DSLR and Tripods

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2 Comments
Introduction First, congratulations! Camera technology has advanced to the point where a complete novice, using an entry-level camera right out of the…
cmbprimer4

A Poor Man’s CMB Primer: Cosmic Acoustics

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1 Comment
 Before decoupling, photons and charged particles were in good thermal contact: though the primordial plasma might have varied in temperature…
digitalcamera1

Digital Camera Buyer’s Guide — Choose the Best Camera

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75 Comments
Introduction First, congratulations! Camera technology has advanced so much that a complete novice using an entry-level camera straight out of the box…
spacetimecon

Struggles with the Continuum: Spacetime Conclusion

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5 Comments
 We've been looking at how the continuum nature of spacetime poses problems for our favorite theories of physics --- problems with infinities.…
11dgravity

11d Gravity From Just the Torsion Constraint

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17 Comments
 It is familiar that Einstein gravity may be formulated in terms of a vielbein field together with a "spin connection", subject to the constraint…
spacetime7

Struggles with the Continuum: General Relativity

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2 Comments
 Combining electromagnetism with relativity and quantum mechanics led to QED. Last time we saw the immense struggles with the continuum this…
namethephoto

Name the Science Photo Quiz

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16 Comments
Summer is almost over, but the fun doesn't have to stop! Take a look at the photos below and see if you can identify them.  Post your score in the comment…
ohmslaw

Learn Why Ohm’s Law Is Not a Law

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41 Comments
At first, I wanted this title to say “Ohm’s law is not a Law.”  But someone else used that phrase in a recent PF thread, and a storm of protest…
carcapacity

Why Road Capacity Is Almost Independent of the Speed Limit

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30 Comments
Let us start with a familiar situation. Take a familiar part of a road, for example, the one from your home to work. How long will it take you to drive…
quantumhard

Why Quantum Mechanics Feels Difficult — Mathematical Bridge

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253 Comments
QM's formalism Strangely enough, quantum mechanics (QM)'s formalism isn't any more difficult than other areas of physics. The mathematics of the "standard"…
introphysics

So I Am Your Intro Physics Instructor

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14 Comments
I posted this elsewhere (on my blog), and someone mentioned that maybe it might also be useful here on PF. So I'm reproducing the entire entry here in…
entropy

How to Determine the Change in Entropy

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9 Comments
How do you determine the change in entropy for a closed system that is subjected to an irreversible process?Here are some typical questions we get…
Supersymmetry

Learn About Supersymmetry and Deligne’s Theorem

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55 Comments
 In 2002, Pierre Deligne proved a remarkable theorem on what mathematically is called Tannakian reconstruction of tensor categories. Here…
Schwarzschild

Learn Orbital Precession in the Schwarzschild and Kerr Metrics

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7 Comments
The Schwarzschild Metric A Lagrangian that can be used to describe geodesics is [itex]F = g_{\mu\nu}v^\mu v^\mu[/itex], where [itex]v^\mu = dx^\mu/ds[/itex]…