In mechanics, the virial theorem relates the average over time of the total kinetic energy, , of a stable system consisting of several particles bound by potential forces, with that of the total potential energy, .

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Why is the Virial Theorem not a Special Case of the Ergodic Theorem? What is their Relationship?

The virial theorem involves the time-averages of the potential and kinetic energies if the motion of the system is bounded to a finite region of space. An ergodic theorem relates the time and space ...
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Why is Kinetic Energy = (-) Total Energy and Potential Energy = 2 $\times$ Total Energy?

I came across this relation while reading on the Bohr atomic model. Are there any other forces for which these relations hold good?
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How does the total change in energy being negative indicate the direction of the Force?

To be more precise consider the following situation: Suppose a satellite is orbiting at a distance of $r_1$ from the centre of the earth. After some time, rockets are fired such that the new distance ...
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What happens when we use the Virial Theorem iteratively?

Say I want to model the formation of structure in the Universe as a series of events whereby already virialised systems are brought together to create a larger virialised system. I will take the ...
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Interpretation of Virial theorem in the case n=-2

For potentials $V$ that fulfil $V(\lambda \vec{r}_1, \lambda \vec{r}_2, \ldots, \lambda \vec{r}_N) = \lambda^n V(\vec{r}_1,\vec{r}_2,\ldots, \vec{r}_N)$ one can rewrite the virial theorem as ...
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Proving the Virial theorem

Consider the expectation in the canonical ensemble defined by $$\left\langle x_i\frac{\partial \mathcal{H}}{\partial x_j} \right\rangle=\frac{1}{Z}\int d\Gamma x_i\frac{\partial ...
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Why is the total energy of an orbiting system negative?

Assume it's an circular orbit. Object A orbits around object B. Take object B as frame of reference. .$E=KE_a + GPE$ .$E=\frac 12m_av_a^2 +(-\frac {GM_bm_A}r)$ .$E=\frac 12m_a(GM_br)+(-\frac ...
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How do I find the average kinetic energy and average potential energy of a hydrogen electron in the ground state?

How do I find the average kinetic energy and average potential energy of a hydrogen electron in the ground state? In my modern physics class, we are wrapping up the 3D Schrödinger equation, and I am ...
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For $N$ particles acting under gravity, how long until they settle into a virial equilibrium?

As the title says, if I have a system of particles interacting only due to gravity, over what timescale do we expect them to fall into a virial equilibrium? By virial equilibrium I mean a system that ...
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Energy in harmonic oscillator [closed]

The expectation value of the potential energy is exactly half the total according to Griffiths. Is that case always true for quantum harmonic oscillator? Is that the case also for classical harmonic ...
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How does dark matter collapse?: Entropy considerations

Inspired by this question. I believe that the usual explanation that preserves the second law of thermodynamics as an astrophysical gas cloud collapses under gravity is that the gas must heat and ...
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How to show that the gravitational potential energy of a two particle system is -2 times the total kinetic energy, without using the virial theorem?

Consider a two-particle system with identical masses, orbiting in circles about their center of mass. I'm supposed to prove that: $$U_p = -2U_k$$ With $U_p$ potential energy of the system, and $U_k$ ...
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Classical vs. quantum energy of the hydrogen atom

If I have an electron and a proton and calculate the classical energy which I get by bringing the electron from infinity to the distance of a Bohr radius to the proton, I get 27.2 eV, but the electron ...
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Origins of many-particle interactions

The internal potential energy of an $N$ particle system is a general function of the coordinates of the particles: $U(r_1,...,r_N)$. In some approximations and expansions - e.g. virial expansion - it ...
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Virial theorem and variational method: a question

I have an hydrogenic atom, knowing that its ground-state wavefunction has the standard form $$ \psi = A e^{-\beta r} $$ with $A = \frac{\beta^3}{\pi}$, I have to find the best value for $\beta$ ...
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Virial of a system

I had obtained $$\overline{E_{kin}} = -\frac{1}{2}\overline{\sum_j\mathbf{r}_j\cdot\mathbf{F}_j}$$ and was asked to show that if the forces are conservative then $$\overline{E_{kin}} = ...
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The virial theorem and a delta function potential

So the virial theorem tells us that: $2\langle T\rangle = \langle \textbf{r}\cdot\nabla V\rangle$. Now I was wondering what would happen if V has te form: $V(\textbf{r}-\textbf{r}') = ...
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Gravitational Potential Energy (GPE) to Kinetic Energy (KE) transfer in Satellite Orbits

I am stumped by a mundane A Level Physics question (teacher of physics here obviously a bit short of practice!). My colleagues and I are stumped and were wondering if any one could help us. It ...
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Average Energy of the Quantum Harmonic Oscillator

In Griffiths, the average potential energy for the quantum harmonic oscillator is given as $$\langle V\rangle~=~\frac{1}{2}\hbar \omega(n+\frac{1}{2}).$$ Is the potential energy of the quantum ...
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Virial theorem and the energy in a gas

I clearly am interpreting the Virial Theorem incorrectly, but I don't know how. In dipole gases, the molecules can exhibit five kinetic modes, while they can only experience 2 potential modes. Doesn't ...
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Intuition behind classical virial theorem

I am continuing to brush up my statistical physics. I just want to gain a better understanding. I have gone through the derivation of the classical virial theorem once more. I have thought about it, ...
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Relating generalized momentum, generalized velocity, and kinetic energy: $2T~=~\sum_i p_{i}\dot{q}^{i}$

According to equation (6) on the first page of some lecture notes online, the above equation is used to prove the virial theorem. For rectangular coordinates, the relation $$ 2T~=~\sum_i ...
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Is the gravitational potential of a planet in orbit always equal to minus the squared velocity?

Say a planet (mass $m$) is orbiting a star (mass $M$) in a perfect circle, so it is in circular motion. $F=ma$ and the gravitational force between two masses $F=\frac{GMm}{r^2}$ so ...
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Is there some connection between the Virial theorem and a least action principle?

Both involve some 'averaging' over energies (kinetic and potential) and make some prediction about their mean values. As far as the least action principles, one could think of them as saying that the ...
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Large-Scale-Structure (LSS) and the Fingers-of-God

In the Large-Scale-Structure (LSS) artifacts named fingers-of-god are apparent in the redshift space and justified by "The large velocities that lead to this effect are associated with the ...
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Stellar Viscosity in Galaxies

Is there such as thing as the viscosity of stars in a galaxy, along the lines of gravitational attraction between stars changing the dynamics. If so, how is that put in terms of the Virial Theorem?