What is Virial theorem: Definition and 52 Discussions

In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. Mathematically, the theorem states










{\displaystyle \left\langle T\right\rangle =-{\frac {1}{2}}\,\sum _{k=1}^{N}{\bigl \langle }\mathbf {F} _{k}\cdot \mathbf {r} _{k}{\bigr \rangle }}
for the total kinetic energy ⟨T⟩ of N particles, where Fk represents the force on the kth particle, which is located at position rk, and angle brackets represent the average over time of the enclosed quantity. The word virial for the right-hand side of the equation derives from vis, the Latin word for "force" or "energy", and was given its technical definition by Rudolf Clausius in 1870.The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in statistical mechanics; this average total kinetic energy is related to the temperature of the system by the equipartition theorem. However, the virial theorem does not depend on the notion of temperature and holds even for systems that are not in thermal equilibrium. The virial theorem has been generalized in various ways, most notably to a tensor form.
If the force between any two particles of the system results from a potential energy V(r) = αrn that is proportional to some power n of the interparticle distance r, the virial theorem takes the simple form







{\displaystyle 2\langle T\rangle =n\langle V_{\text{TOT}}\rangle .}
Thus, twice the average total kinetic energy ⟨T⟩ equals n times the average total potential energy ⟨VTOT⟩. Whereas V(r) represents the potential energy between two particles, VTOT represents the total potential energy of the system, i.e., the sum of the potential energy V(r) over all pairs of particles in the system. A common example of such a system is a star held together by its own gravity, where n equals −1.
Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step.

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  1. G

    A Virial theorem and translational invariance

    According to the virial theorem, $$\left\langle T\right\rangle =-{\frac {1}{2}}\,\sum _{k=1}^{N}{\bigl \langle }\mathbf {F} _{k}\cdot \mathbf {r} _{k}{\bigr \rangle }$$ where ##N## is the number of particles in the system and ##T## is the total kinetic energy. It is often claimed that this...
  2. G

    Use the Virial theorem to show the following...

    The expression ##\langle \cal H \rangle_k## is the expected value of the canonical ensemble. The Hamiltonian is defined as follows, with the scaling ##\lambda## ##\lambda \cal H ## : ##\lambda H(x_1, ...,x_N)=H(\lambda^{a_1}x_1,....,\lambda^{a_N}x_N)## As a hint, I should differentiate the...
  3. Barbequeman

    Virial Theorem for an expanding globular cluster

    I attached a file which shows my attempt to resolve this problem with the possible two pair interaction which gives us the kinetic energy of the cluster in an expanding system, at least I think so. But to be honest I´m more or less completely stuck with this question and I would be glad if...
  4. C

    Decrease of Solar radius per year using Virial Theorem

    Hello, I am trying to solve this question: Assume that the Sun's energy production doesn't happen by fusion processes, but is caused by a slow compression and that the radiated energy can be described by the Virial Theorem: $$L_G = - \frac{1}{2} \frac{GM^2}{R^2} \frac{dR}{dt} $$ How much must...
  5. A

    I Help with the Virial Theorem please

    Anyone able to help if I provide you with more details?
  6. B

    I Basic Virial Theorem Help

    Hello, This term in university I'm taking a second year intro to astrophysics course and my professor talks a lot about different situations and then solves a problem using the virial theorem. The reason I'm confused is because the range of topics that he applies this theorem to vary in many...
  7. A

    A Helium atom, variation method and virial theorem

    I need to calculate the energy of the ground state of a helium athom with the variational method using the wave function: $$\psi_{Z_e}(r_1,r_2)=u_{1s,Z_e}(r1)u_{1s, Z_e}(r2)=\frac{1}{\pi}\biggr(\frac{Z_e}{a_0}\biggr)^3e^{-\frac{Z_e(r_1+r_2)}{a_0}}$$ with ##Z_e## the effective charge considered...
  8. N

    Virial Theorem and Simple Harmonic Oscillator

    Homework Statement Show that the virial theorem holds for all harmonic-oscillator states. The identity given in problem 5-10 is helpful. Homework Equations Identity given: ∫ξ2H2n(ξ)e-ξ2dξ = 2nn!(n+1/2)√pi P.S the ξ in the exponent should be raised to the 2nd power. So it should look like ξ2...
  9. JMatt7

    I Virial Theorem and the Ergodic hypothesis

    Hi! I'm currently reading "The Virial Theorem in Astrophysics" by G.W. Collins (the book is available as a free web edition at http://ads.harvard.edu/books/1978vtsa.book/) in which the author claims the importance of the ergodic hypothesis when applying the virial theorem to astrophysical...
  10. W

    A Is this an application of the Virial theom?

    The problem concerns a liquid column (assume infinitely long) with radius R connected to an electric potential V, the liquid thus has certain surface charge density. A small perturbation may change the surface profile of the cylinder (small compared to R) and thus change its surface charge...
  11. eme

    How do long-range interactions between atoms affect the virial theorem?

    Homework Statement Suppose that there are long-range interactions between atoms in a gas in the form of central forces derivable from a potential. $$V(r) = \frac k r^m $$ where r is the distance between any pair of atoms and m is a positive integer. Assume further that relative to any given...
  12. Dadface

    I Virial theorem as applied to hydrogen atom

    In the virial theorem the numerical value of the average potential energy within a system is exactly twice that of the average kinetic energy. I know the theorem is proved mathematically but to me it seems a coincidence that one value is exactly twice the other value. I find that interesting. I...
  13. D

    How to find the average potential energy given V(x,y) and E?

    Homework Statement A classical particle with total energy E moves under the influence of a potential V(x,y) = 3x3+2x2y+2xy2+y3. What is the average potential energy, calculated over a long time? Homework EquationsThe Attempt at a Solution I think that this can be solved using Virial Theorem...
  14. R

    I Deriving the Virial Theorem

    Hello everyone! I am reviewing the derivation of the Virial Theorem from an introductory Astrophysics book (Carroll and Ostlie's) and found a step I couldn't follow. I've attached a photo of the step. Can anyone explain how Newton's Third Law brings about eqn 2.41? I don't see how that first...
  15. J

    Virial Theorem Question

    Homework Statement Using paper, pencil and the Virial theorem, calculate the position uncertainty (an estimate of the vibration amplitude) of the H atom in its ground state C-H stretching mode. In more precise language, calculate the bond length uncertainty in a C-H bond due to the C-H...
  16. throneoo

    I "Moment of Inertia" in Virial Theorem

    Moment of inertia is supposed to be defined with respect to a rotational axis such that for a system of point masses, I=∑miri2 where ri 's are the perpendicular distances of the particles from the axis. However, in some derivations of the virial theorem (like the one on wiki), the so-called...
  17. A

    I Why does temperature always rise in stars?

    Hello everyone, I know that pre-main sequence stars do heat up because of gravitational contraction, and the increase in internal energy (and so in temperature) comes from this shrinking and is governed by the virial theorem (...
  18. S

    Virialized dark matter halos

    Dear all, I am new in the field of galaxy formation, so I am sorry if my questions are a bit simple. -what does virialized halo mean? does it mean they obey the virial theorem concerning their kinetic and potential energy? \begin{equation} 2<T>=n<V> \end{equation} -Why should the halos be like...
  19. Peeter

    Conditions for the X dot P expectation to be constant?

    Homework Statement Under what conditions is \left\langle{{\mathbf{x} \cdot \mathbf{p}}}\right\rangle a constant. A proof of the quantum virial theorem starts with the computation of the commutator of \left[{\mathbf{x} \cdot \mathbf{p}},{H}\right] . Using that one can show for Heisenberg...
  20. B

    Integration problem deriving virial theorem

    Hello there, I'm reading a section of my textbook that is deriving the virial theorm from the hydrostatic equilibrium equation. In part of the derivation it states that $$-\int_0^M\frac{Gm(r)}{r}dm(r)=E_{GR}=-\frac{GmM}{r}$$ When I perform this integral I get the wrong answer. Here's my...
  21. F

    Using Virial Theorem for V(x)=A*|x| Potential

    Hey guys, I was wondering if i can use the virial theorem for a potential of the form V(x)=A*|x| Got some trouble at the point x=0. Best regards
  22. K

    Exploring the Virial Theorem: Understanding the Derivative of G Over a Period T

    Homework Statement In the Virial theorem The scalar virial G is defined by the equation: $$G=\vec{p}\cdot \vec{r}$$ Where ##\vec{p}## is the momentum vector and ##\vec{r}## the location vector. When i take the mean of the derivative ##\bar{\dot{G}}## over a whole period T it equals 0. why...
  23. T

    Application of virial theorem

    How do you use the virial theorem to derive the van der Waals equation of state?
  24. T

    Virial Theorem and Energy of a Two-Body System

    Hi all, I want to make sure I am understanding this correctly. Say we have two identical particles orbiting (in circles) about their center of mass. We know that the average potential energy is -2 times the average kinetic energy. So the average total energy is negative the average...
  25. S

    Virial theorem, hamiltonian

    Homework Statement A quantum particle, i.e. a particle obeying Schrodinger equation and moving in one dimension experiences a potential ˆV (x). In a stationary state of this system show that ⟨x∂/∂x(ˆV(x)⟩ = ⟨ˆp2/2m⟩ Hint: Consider the time dependence of ⟨ˆxˆp⟩. Homework Equations...
  26. L

    Using the Virial Theorem to Calculate Expectation Values in Quantum Mechanics

    Hi everybody Homework Statement I showed the following: \frac{1}{\hbar i} [H,xp]=x \frac {dV}{dx} - \frac {p^2}{m} Now I want to use this to show that: \langle \frac {p^2}{2m} \rangle = \frac{1}{2} \langle x \frac {dV}{dx}\rangle whereas |E> is an eigenstate to H with the...
  27. shounakbhatta

    Explaining the Virial Theorem in Astrophysics

    Hello, I am reading Virial theorem from Wikipedia. It states in astrophysics, virial relations are: 3/5 GM/R=3/2... I want to know how this 3/5 and 3/2 value comes from? If somebody can explain the step by step deduction of the Virial theorem in Astrophysics... Thanks, --...
  28. J

    Help Deriving Hydrostatic Equilibrium and Virial Theorem

    Hey all, Long story short, for my Modern Physics course, we have to do a research paper on a physics topic we didn't cover in class. Since I've always been interested in astronomy and the cosmos, I figured I'd do star formation / life cycle of stars. The paper has to have mathematical and...
  29. D

    Virial Theorem and Hartree Fock

    Hi, short question: Why is the virial threorem valid in the Hartree Fock approximation? Some authors just mention this fact incidentally, but the don't explain it.
  30. F

    Puzzling problem (using virial theorem)

    Consider a star of Radius R and mass M, with a pressure gradient given by \frac{dP}{dr} = \frac{4\pi}{3}G\rho2r exp(-\frac{rr}{\lambda\lambda}) where \rho is the central density. calculate the gravitational energy, using the Virial theorem. Show that in the limit \lambda « R this energy is...
  31. 1

    Virial theorem for an inverse-square law force

    Calculate the time-average of i) potential and ii) kinetic energy of a particle orbiting on ellipse in an inverse-square-law force field f =(k/r^2) (K<0) Express your answers in terms of a ( semi-major axis of the ellipse) and k (constant in the force given) have no idea how to do this...
  32. P

    Exploring the Virial Theorem for Atomic Physics

    So my Atomic Physics professor today was talking about some called the "Virial theorem" in relation to Bohr Model of the H-atom, which in my 5 years of college physics I had never heard of >_<;; . It turns out that I had seen equations that use concept of the Potential "==" Kinetic Energy, I...
  33. D

    Virial Theorem and Equipartition Theorem

    Hi folks, in this book (correct page should open, if not: p.199): http://books.google.com/books?id=12DKsFtFTgYC&lpg=PP1&dq=greiner%20thermodynamics&pg=PA199#v=onepage&q&f=false it says (formula (7.168)): <\sum_i \vec{r}_i \vec{F}_i > = -p \oint{\vec{r} d\vec{S}} It is explained, why...
  34. P

    Virial Theorem derivation

    Currently working through a derivation of the Virial Theorem relating average internal pressure to gravitational potential energy. So I've got to -3\int^V_0 Pdv=-\int^M_0 \frac{Gm}{r}dm which is meant to give 3 \langle P \rangle V=-E_{grav} But if I'm right in saying that E_{grav} is...
  35. H

    Validity of virial theorem in QM

    In the standard derivation of the virial theorem, one assumes to be working in the energy basis. One then gets <T> = n/(n+2) <H>. This relation doesn't hold for the continuous spectrum of Coulomb potential where <T> > 0, <H> > 0, n/(n+2) = -1. So, where in the derivation did we use the fact we...
  36. D

    The Virial Theorem problem

    Homework Statement A particle is moving along the x-axis in the potential: \[V\left( x \right)=k{{x}^{n}},\] where k is a constant, and n is a positive even integer. \left| \psi \right\rangle is described as a normed eigenfunction for the Hamiltonoperator with eigenvalue E. Show through the...
  37. S

    Understanding Virial Theorem: Comparing Equations

    In my astrophysics book, it says it's \bar{P}=-\frac{1}{3}\frac{E_{gr}}{V} This Wikipedia article has a different equation. http://en.wikipedia.org/wiki/Virial_theorem Can someone explain the difference?
  38. T

    Derivation of velocity dispersion from virial theorem?

    Derivation of velocity dispersion from virial theorem?? Hey, Im probably being a bit dim here but could anyone help me derive the velocity dispersion from the virial theorem. I've got 2K+U=0, K/m~sigma^2 and U/m~GM/R. From rearranging I get a negative velocity^2? Or maybe its the...
  39. V

    Proving 2<T> = <x\frac{dV}{dX}> and Virial Theorem

    Homework Statement I must prove that: \frac{d}{dt}<xp> = 2<T> - <x\frac{dV}{dX}> And use the virial theorem to prove that <T> = <V> Homework Equations 2<T> = <x\frac{dV}{dX}> \frac{d}{dt}<Q> = \frac{i}{h(bar)}<[H, Q]> + <\frac{\partial Q}{\partial t}> Where Q on the...
  40. P

    Origin to the 'virial theorem' by Clausius

    Hello, I wonder if there is a common origin to the 'virial theorem' by Clausius and the 'virial expansion' or equation of state used to express the compressibility of a gas. I am just wondering about this because the two concepts do not seem to have a common mathematical origin (that I can...
  41. maverick280857

    Exploring the Physical Significance of the Virial Theorem

    Hi everyone I have a question regarding a step in the proof of the Virial Theorem. Specifically suppose |E\rangle is a stationary state with energy E, i.e. \hat{H}|E\rangle = E|E\rangle Now, [\hat{r}\bullet\hat{p},\hat{H}] = i\hbar\left(\frac{p^2}{m} - \vec{r}\bullet\nabla...
  42. cepheid

    Virial Theorem in Stellar Dynamics

    After reading in Longair's Galaxy Formation through the derivation of the virial theorem in the context of a dynamical system in equilibrium consisting of "point masses" interacting only through gravity, I proceeded to try to understand his comments on how the theorem can be applied in order to...
  43. O

    Virial Theorem and Frictional Forces

    In the virial theorem, why do velocity-dependent frictional forces disappear? I've seen this stated a number of times, but never any explanation. (And, obviously, haven't been able to figure it out myself!)
  44. N

    Dark Matter Halo + Virial theorem Q.

    A year ago we had a HW problem about galactic rotation curves: If the dark matter density is \rho= \frac{\rho_0}{1+\left(\frac{r}{r_0}\right)^2}, Then how does velocity depend radius at large r ( r>>r_o)? You want to use the virial theorem here, so you calculate M(r) and then finally...
  45. C

    Alternate approach to proving the virial theorem?

    Hi guys, In quantum mechanics, the virial theorem for a system in its ground state is proved by a very nice scaling technique (Nielsen and Martin, PRB, 1985). I was trying to do something similar in classical mechanics and arrived at the virial theorem but i am not sure about why it should...
  46. G

    Universal quality of virial theorem

    "Distances among galaxies in clusters on average are no greater than their diameter". Luminosity, color, and other qualities are used to obtain distances - and these values are connected to masses that are observed by motions. If a system is in equilibrium then the theorem is: 2T+U=0 Mv^2/2...
  47. H

    How does the Virial Theorem derivation for <K>=-<delta(U)> work?

    Can somebody explain me how does the derivation for <K>=-<delta(U)> goes under the viral theorem?
  48. U

    Calculating Average Values and Proving Inequality for Particle Potential - N^nX

    [SOLVED] Virial Theorem Homework Statement A particle has a potential \lambda X^n and Hamiltonian H = \frac{P^2}{2m} + V(x) Knowing that the commutator of H and XP is i\hbar(n\lambda X^n - \frac{P^2}{m}), find the average values <T> and <V> and verify that they satisfy: 2<T>=n<V>...
  49. J

    Applying the Virial Theorem to Stellar winds

    Hello, In the process of revising for an exam I have, I am having difficulty with this question. "Consider an isotropic stellar wind of mass density rho, pressure p, temperature T and velocity v that has reached to a distance r=R_w from the centre of a star. The star has mass M* and...
  50. N

    Three-Dimensional Virial Theorem (Quantum Mechanics)

    Homework Statement (a) Prove the three-dimensional virial theorem: 2<t> = <r\cdot \nabla V> (for stationary states) Homework Equations Eq. 3.71 (not sure if this applies to 3 dimensions, but I think so) \frac{d}{dt}<Q> = \frac{i}{\hbar}<[\hat H, \hat Q]> + \left<\frac{\partial...