What is Phi: Definition and 130 Discussions

Phi (; uppercase Φ, lowercase φ or ϕ; Ancient Greek: ϕεῖ pheî [pʰé͜e]; Modern Greek: φι fi [fi]) is the 21st letter of the Greek alphabet.
In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voiceless bilabial plosive ([pʰ]), which was the origin of its usual romanization as ⟨ph⟩. During the later part of Classical Antiquity, in Koine Greek (c. 4th century BC to 4th century AD), its pronunciation shifted to that of a voiceless bilabial fricative ([ɸ]), and by the Byzantine Greek period (c. 4th century AD to 15th century AD) it developed its modern pronunciation as a voiceless labiodental fricative ([f]).
The romanization of the Modern Greek phoneme is therefore usually ⟨f⟩.
It may be that phi originated as the letter qoppa, and initially represented the sound /kʷʰ/ before shifting to Classical Greek [pʰ]. In traditional Greek numerals, phi has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter Ef (Ф, ф) descends from phi.
As with other Greek letters, lowercase phi (encoded as the Unicode character U+03C6 φ GREEK SMALL LETTER PHI) is used as a mathematical or scientific symbol. Some uses, such as the golden ratio, require the old-fashioned 'closed' glyph, which is separately encoded as the Unicode character U+03D5 ϕ GREEK PHI SYMBOL.

View More On Wikipedia.org
  1. J

    Rotate Points on Sphere by Theta and Phi

    Dear Forum, My goal is to rotate several points on a sphere by a theta and phi. For example, I have a sphere where the elevation is theta (90 to -90) and the azimuthal is phi (-180 to 180). I have the following points on the sphere: theta = [45 45 45 45] phi = [-180 90 90 180] This generate...
  2. I

    Deriving the Solution to $$\frac{\partial}{\partial \phi^{*}}$$

    The solution can be viewed here on page 41 https://usermanual.wiki/Document/Steven20H20Simon2020The20Oxford20Solid20State20Basics2C20Solution20ManualOxford20University20Press202015.1463186034/view What I have is $$\frac{\partial}{\partial \phi^{*}} (\frac{\sum_{n,m} \phi_{n}^{*}...
  3. Diracobama2181

    A Perturbative Renormalization in Phi 4 Theory

    For example, after the Lagrangian is renormalized at 1-loop order, it is of the form $$\mathcal{L}=\frac{1}{2}\partial^{\mu}\Phi\partial_{\mu}\Phi-\frac{1}{2}m^2\Phi^2-\frac{\lambda\Phi^4}{4!}-\frac{1}{2}\delta_m^2\Phi^2-\frac{\delta_{\lambda}\Phi^4}{4!}$$. So if I were to attempt to find the...
  4. Istiak

    Understanding Direction of Unit Vectors r roof & phi roof

    The unit vector r roof points in the direction of increasing r with phi fixed; phi roof points in the direction of increasing phi with r fixed. Unlike x roof, the vectors r roof and phi roof change as the position vector r moves. What I was thinking of the image is Although, I was thinking why...
  5. MichaelTam

    Help Needed: Solving y = A sin({2*10^6 ({\pi / 3 -\pi t }) + \phi })

    I know so ## y = A sin({2*10^6 ({\pi / 3 -\pi t }) + \phi }) ## There still some unknown I cannot find, can anyone give me some hint please?
  6. R

    Show that a projectile lands at a distance ##R = \frac{2v_0^2 sin \theta cos(\theta + \phi)}{g cos^2 \phi}##

    ##V_x = V_0 cos \theta ## ##x = V_0 cos \theta t## ##V_y = V_0 cos \theta ## ##y = V_0 cos \theta t## ##F_x = m\ddot{x}## ##-mgsin \phi = m\ddot{x}## ##\dot{x} = -gtsin\phi + V_x## ##x = -\frac{1}{2} gt^2 sin \phi + V_x t## ##x = -\frac{1}{2} gt^2 sin \phi + v_0 cos\theta t## ##F_y =...
  7. ole cram

    Physics equations with the mathematical constant Phi?

    I know Phi appears often when modelling exponential growth and, probably because of that, also in Biology/Ecology. But does it appear spontaneously in the mathematical description of some fundamental physics phenomenon at all? (As does Pi, the ubiquitous irrational number) Hope I'm posting on...
  8. Diracobama2181

    Phi 4 Theory Propagator Question

    I know in the Heisenburg picture, $$\Phi(\vec{x},t)=U^{\dagger}(t,t_0)\Phi_{0}(\vec{x},t)U(t,t_0)$$ where $$\Phi_{0}$$ is the free field solution, and $$U(t,t_0)=T(e^{i\int d^4x \mathcal{L_{int}}})$$. Is there a way I could solve this using contractions or Feynman diagrams? Because otherwise, it...
  9. Diracobama2181

    A Scattering Amplitudes for Phi 4 Theory

    I know $$ i\mathcal{M}(\vec {k_1}\vec{k_2}\rightarrow \vec{p_1}\vec{p_2})(2\pi)^4\delta^{(4)}(p_1 +p_2-k_1-k_2) $$ =sum of all (all connected and amputated Feynman diagrams), but what is meant by 1 loop order? In other words, when I take the scattering matix element...
  10. H

    Integration ## f(\theta, \phi) = \frac{sin \theta}{4\pi}##

    Hi, I have this formula ## f(\theta, \phi) = \frac{sin \theta}{4\pi}## I have this statement that say if I integrate this formula above on a sphere then p = 1. what does integrate on a sphere means? I know ##\int_0^{2\pi} ## is used for the circle.
  11. Diracobama2181

    How Does Time-Reversal Invariance Affect Phi 4 Theory Renormalization?

    Let ##\phi_{+}=\frac{d^3k}{2\omega_k (2\pi)^3}\int(\hat{a}^{\dagger}(\overrightarrow{k})e^{ikx})## and ##\phi_{-}=\frac{d^3k}{2\omega_k (2\pi)^3}\int(\hat{a}(\overrightarrow{k})e^{-ikx})##. Then ##\phi^4=\phi_{1}\phi_{2}\phi_{3}\phi_{4}=(\phi_{1+}+\phi_{1-})(\phi_{2+}+\phi_{2-}...
  12. Peeter

    Calculating the decay rate for h -> phi phi process

    Homework Statement Given a coupling h \; \partial_\mu \phi^a \partial^\mu \phi^a , meant to model the first order interaction of the Higgs field h to boson fields \phi^a , compute the width \Gamma(h \rightarrow \phi^3 \phi^3) of the Higgs particle to decay to two longitudinal (say)...
  13. nomadreid

    I Physically relevant: fractals, phi?

    There are two subjects which pop up a lot as having physical examples (or, more precisely, where their approximations have), but many (not all) of them seem rather indirect or forced. For example: [1] phi (the Golden ratio) or 1/phi: (a) trivia: sunflowers and pineapples giving the first few...
  14. T

    MHB Five, Phi and Pi in one integral = −5ϕπ

    $$\int_{0}^{\pi\over 2}{\ln(\sin^2 x)\over \sin(2x)}\cdot \sqrt[5]{\tan(x)}\mathrm dx=-5\phi \pi$$ $\phi$ is the golden ratio Any help, please. Thank you!
  15. hilbert2

    A Constant Solutions of Real Scalar Field

    Suppose I have a self interacting real scalar field ##\phi## with equation of motion ##\partial^i \partial_i \phi + m^2 \phi = -A \phi^2 - B\phi^3##, and I attempt to find constant solutions ##\phi (x,t) = C## for it. The trivial solution is the zero solution ##\phi (x,t) = 0##, but there can...
  16. C

    I Solve Mystery of Phi-Based Equation: Help Colin

    If you know phi it is about 1.618...=2cos36. The equations when x=phi which is equal to 0 is x^2-x-1=0. I took the first derivative squared and the second derivative cubed. The equation with x=phi is: [2x-1]^2+2^3=13 Check for yourself, if you fill in phi you get 13. Anyway, I do not know what...
  17. H

    B Phi Ratio in Physics | G Factor Calculation

    Hello it's me again, I looked into connections between Pure Mathematics and Physics again and came across this interesting Web Page that proposes time in Quantum Physics may be related to Phi << Dubious Link Deleted by Mentors >> it's not a lot to read and I was wandering what you people think...
  18. 10Exahertz

    Orbital Potential Energy to find r and phi in terms of t.

    Homework Statement A particle in central force field has the orbit r=cφ^2, c is a constant. Find the potential energy, Find r and phi in terms of t. I get how to find the potential energy and found it to be U=-l^2/mu (2c/r^3+l/2r^2) l is angular momentum and mu is the reduced mass But how do I...
  19. R

    PHI 600 Scanning Auger Multiprobe Problem

    Hi, Does anyone have any experience with the Perkin Elmer PHI 600 Scanning Auger Multiprobe? I ran into a problem with my system. Please respond so we can discuss this in detail. Thanks.
  20. thegirl

    I Why does cos(phi1- phi 2 +kX)=cos(kx)?

    In the case of two fields interfering with each other when calculating the total electric field, cos (phi1-phi2 + kx) = cos( kx) where kx is the path difference between the two fields. How does cos (phi1-phi2 +kx)=cos(kx) Isit just algebra?
  21. B

    Angular Momentum - rotating rod makes an angle phi with axis

    Homework Statement A thin rod of length l and mass M rotates about a vertical axis through its center with angular velocity ω. The rod makes an angle φ with the rotation axis. Determine the magnitude and direction of L (angular momentum). So we're given: mass - M, length - l, angular velocity...
  22. Elvis 123456789

    How to go from a*cosx + b*sinx TO A*cos(wt - phi)?

    Moved from non-homework forum section, so homework template is not present. Express Cos(t - pi/8) + Sin(t - pi/8) in the form A*Cos(wt - phi). I got sqrt(2)*Cos(t-3pi/4). Not sure if that's right though
  23. H

    Must the 1st derivative of phi be undetermined at V=infinity

    I think it is not true that a discontinuous ##\nabla^2\psi## implies a discontinuous ##\nabla\psi##, because a continuous function can have a discontinuous derivative, eg. ##y=|x|##. Is it true that ##\nabla\psi## must always be undetermined at the boundary where ##V=\infty##? Attached below...
  24. Saurabh Sharma

    Why phi component is not taken into account for vector R ?See attachment.

    Why phi component is not taken into account while calculation electric field intensity due to line charge?See attachment for details.https://physicsforums-bernhardtmediall.netdna-ssl.com/data/attachments/74/74211-22d26b27211fd4b186955b890458804e.jpg...
  25. S

    Feynman diagrams for phi phi -> phi phi

    Homework Statement Compute the matrix element for the scattering process \phi \phi \to \phi \phi Homework Equations The Lagrangian is given by L = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi + \frac{\alpha}{2} \phi \partial_{\mu} \phi \partial^{\mu} \phi + \frac{\beta}{2} \phi^2...
  26. Calpalned

    Spherical coordinates - phi vs theta

    Homework Statement My textbook states that when ##\theta = c ## where c is the constant angle with respect to the x-axis, the graph is a "half-plane". However, when ##\phi = c ## it is a half-cone. The only difference I see is that ##\phi## is the angle with respect to the z-axis, rather than...
  27. slatts

    Is Phi Squared Inflation Compatible with Latest Planck Data?

    Looking at blogs about the 2-5-15 Planck data release, I noticed a couple of them claiming that it rules out some of the simplest models of inflation, including one called "phi squared inflation". I can't read the specialized characters in physics notations, but I'm figuring this is the same as...
  28. A

    Time derivatives of sin and cos phi

    Homework Statement By using chain rule of differentiation, show that: $$ \frac{\mathrm{d} sin\phi }{\mathrm{d} t} = \dot{\phi} cos\phi , \frac{\mathrm{d} cos\phi }{\mathrm{d} t} = -\dot{\phi} sin\phi , $$ Homework EquationsThe Attempt at a Solution I got this right for a homework problem...
  29. Elroy

    Qubit Math: Partitioning Phi into BetaA and BetaB

    Hi All, I'm gradually teaching myself quantum computing, being an excellent programmer and fair at linear algebra and geometry. To learn it, I'm writing a quantum computer emulator (complete with graphical representations). I'll share it when I get to some level of completion. The following...
  30. H

    Yaw Pitch & Roll to spherical Theta & Phi

    hi guys, can someone please tell me how to find Theta and Phi from Yaw Pitch and Roll? I use my smartphone orientation sensor on a project and i need to calculate the smartphone's normal vector projection.
  31. N

    Name of something that is forced to look a certain way

    I was looking up information about phi and how it seems apparent everywhere in nature, debatable. I stumbled upon this photo which is quite funny: This got me thinking about how people try to fit and force data to match whichever hypothesis they have. This is more jokingly, I think. However...
  32. F

    How to find phi in simple harmonic motion

    Given a position vs time graph of simple harmonic motion of an object and using the equation x(t) = xo = A sin (ωt +Φ), how am I supposed to find Φ? I can easily find A and the period (T). I also know T = 2 pi / ω, so I can find ω. But I don't know x(t) and t. I tried finding random points in...
  33. J

    How Do You Calculate the Phase Constant φ from a Wave Displacement Graph?

    1. The figure shows the displacement y versus time t of the point on a string at x = 0, as a wave passes through that point. The scale of the x axis is set by ys = 18.0 mm.The wave has form y(x, t) = ym sin (kx - ωt+φ). What is φ...
  34. Whovian

    Apparent indiscrepancy in the average value of phi function

    So, while solving a problem a friend came up with involving the Totient function, I ended up doing a bit of research into the average asymptotics of the function. On page 268 of Introduction to the Theory of Numbers, it's mentioned that "The average of order of ##\phi\left(n\right)## is...
  35. D

    Simple faraday's law problem (phi = what?)

    Homework Statement A toroid with N windings and radius R has cross sectional radius x (x<<R). The current running through the wires is given by ##I = I_0\sin (\omega t)##. There is a magnetic field in the center of the toroid. A loop of wire of radius three times that of the...
  36. Math Amateur

    MHB Tensor Products - D&F page 369 Example 3 - The map phi

    I am reading Dummit and Foote, Section 10.4: Tensor Products of Modules. I am currently studying Example 3 on page 369 (see attachment). Example 3 on page 369 reads as follows: (see attachment) ------------------------------------------------------------------------------- In general...
  37. E

    Derivation of Phi-Hat wrt Phi in Spherical Unit Vectors

    Homework Statement I just want to know how to get from this: ∂ø^/∂ø = -x^cosø - y^sinø to this: = -(r^sinθ+θ^cosθ) Homework Equations All the equations found here in the Spherical Coordinates section: http://en.wikipedia.org/wiki/Unit_vector The Attempt at a Solution I've...
  38. U

    Abstract algebra, show that phi is a homomorphism

    Homework Statement The Attempt at a Solution I'm very new to this kind of maths, so don't quite know how to get started. If I understood the question at all we have g_i \mapsto \phi_i and so I have a homomorphism if I can show that \pi(g \cdot g_i) = \pi(g) \circ \pi(g_i) I'm thinking...
  39. T

    MHB Eulers phi function, orders, gcd

    Let a, k , l , m e Z>1 and let a^k=1 (mod m) and a^l= 1 (mod m). Let d=gcd(k,l) Prove that a^d=1 (mod m). I get already confused at the start: Is it true that k|phi(m) (Lagrange) but k can also be a multiple of the order of a (mod m) and then it can be the other way round. Can anybody clarify...
  40. matqkks

    How Is the Euler Totient Function Applied in Real Life?

    What is most motivating way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but is this function used elsewhere.
  41. matqkks

    MHB Exploring the Practical Applications of the Euler Totient Function

    What is most motivating and tangible way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but is this function used elsewhere.
  42. D

    Finding angles of Phi. Very challenging.

    1) Given: two ropes of negligible weight suspend an object weighing 50N from the ceiling There are two angles of PHI that form from as a measure from the vertical. PHI1 + PHI2= 60°. The tension of the left rope is given to be 80N and the tension of the left rope is given to be 70N. The entire...
  43. T

    Vector Calculus Proof: Curl V = 0 -> V = grad phi

    If ∇ x v = 0 in all of three dimensional space, show that there exists a scalar function ##\phi (x,y,z)## such that v = ∇##\phi##. (from Walter Strauss' Partial Differential Equations, 2nd edition; problem 11; pg 20.) I'm not really sure where to begin with this problem. I asked a few of my...
  44. jim mcnamara

    Is Phi the Secret to Perceiving Beauty in Human Faces?

    This whole post is just for fun -- here is an article about applying phi (Golden ratio) to human faces and esthetics of human beauty. http://www.goldennumber.net/face/ If you don't know much about Phi try:: http://en.wikipedia.org/wiki/Golden_ratio The authors constructed 1.6... x 1...
  45. S

    Find the potential fucntion, phi

    Homework Statement Consider the vector field given by F(x, y, z) = yz \hat{i} + xz \hat{j} + (xy + 3z^{2})\hat{k} a. Calculate ∇xF and show that F is a conservative field. Done, result = <0,0,0> which implies the vector field is conservative. b. The way we were taught this is to set...
  46. M

    What happens when a neutral pion decays?

    The neutral phi meson decays to make two photons after around 8 x10-17 seconds Is this just straight annihilation of the quark anti quark pair? If it is this does it take this amount of time due the quark colour, as in do you need a red up and a red anti up for annihilation? Or does this...
  47. P

    What is the Proof for the Summation of Integers using Euler's Phi Function?

    Homework Statement Prove: \sum_{k<n} k= \frac{n \phi (n)}{2} gcd(k,n)=1 \phi is Euler's phi function or Euler's totient function The Attempt at a Solution So the sum should be 1+2+3+...+(n-2)+(n-1) which will equal \frac{n^2-n}{2} using the formula for the sum of integers by...
  48. I

    Finding phi constant based on initial position and initial velocity

    x(t)=Acos(ωt+ϕ)\\ v(t)=-Aωsin(ωt+ϕ)\\ \\ Let\quad n=integer,\quad A=0.5,\quad ω=2.\\ \\ Given\quad initial\quad position\quad (0,\quad 0.25)\quad and\quad initial\quad velocity\quad (0,\quad 1),\quad find\quad ϕ.\\ \\ x(t)=0.5cos(2t+ϕ)\\ 0.25=0.5cos(ϕ)\\ ϕ=±1.0471975512+2πn\\ \\...
  49. E

    Understanding the 'ERR: Nonreal Ans' Error with Psi and Phi

    I just had a question involving both psi and phi. I know that: Ψ= (1-√5)/2 = -0.618033989... Φ= (1+√5)/2 = 1.618033989... And out of boredom, I decided to put into my calculator: (Φ^Ψ) = 0.7427429446... But my question rose from there: What happens if you do (Ψ^Φ)? I plugged it in and...