The Virial Theorem: Difference between revisions

	Quantum Mechanics A
	Schrödinger Equation; The most fundamental equation of quantum mechanics; given a Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , it describes how a state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|\Psi\rangle} evolves in time.
	Physical Basis of Quantum Mechanics
	Basic Concepts and Theory of Motion; UV Catastrophe (Black-Body Radiation); Photoelectric Effect; Stability of Matter; Double Slit Experiment; Stern-Gerlach Experiment; The Principle of Complementarity; The Correspondence Principle; The Philosophy of Quantum Theory
	Schrödinger Equation
	Brief Derivation of Schrödinger Equation; Relation Between the Wave Function and Probability Density; Stationary States; Heisenberg Uncertainty Principle; Some Consequences of the Uncertainty Principle
	Operators, Eigenfunctions, and Symmetry
	Linear Vector Spaces and Operators; Commutation Relations and Simultaneous Eigenvalues; The Schrödinger Equation in Dirac Notation; Transformations of Operators and Symmetry; Time Evolution of Expectation Values and Ehrenfest's Theorem
	Motion in One Dimension
	One-Dimensional Bound States; Oscillation Theorem; The Dirac Delta Function Potential; Scattering States, Transmission and Reflection; Motion in a Periodic Potential; Summary of One-Dimensional Systems
	Discrete Eigenvalues and Bound States
	Harmonic Oscillator Spectrum and Eigenstates; Analytical Method for Solving the Simple Harmonic Oscillator; Coherent States; Charged Particles in an Electromagnetic Field; WKB Approximation
	Time Evolution and the Pictures of Quantum Mechanics
	The Heisenberg Picture: Equations of Motion for Operators; The Interaction Picture; The Virial Theorem
	Angular Momentum
	Commutation Relations; Angular Momentum as a Generator of Rotations in 3D; Spherical Coordinates; Eigenvalue Quantization; Orbital Angular Momentum Eigenfunctions
	Central Forces
	General Formalism; Free Particle in Spherical Coordinates; Spherical Well; Isotropic Harmonic Oscillator; Hydrogen Atom; WKB in Spherical Coordinates
	The Path Integral Formulation of Quantum Mechanics
	Feynman Path Integrals; The Free-Particle Propagator; Propagator for the Harmonic Oscillator
	Continuous Eigenvalues and Collision Theory
	Differential Cross Section and the Green's Function Formulation of Scattering; Central Potential Scattering and Phase Shifts; Coulomb Potential Scattering

Latest revision as of 10:43, 16 August 2013

We will now derive the quantum mechanical virial theorem. For a Hamiltonian of the form,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}=\frac{\hat{\mathbf{p}}^2}{2m}+\hat{V}(\hat{\mathbf{r}})=\hat{K}+\hat{V}(\hat{\mathbf{r}}),}

this theorem gives the expectation value of the kinetic energy in a stationary state in terms of the potential energy. To derive this relation, we consider the expectation value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{r}}\cdot\hat{\mathbf{p}}.} The time derivative of this expectation value is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{d}{dt}\langle\hat{\mathbf{r}}\cdot\hat{\mathbf{p}}\rangle&=-\frac{i}{\hbar}<[\hat{\mathbf{r}}\cdot\hat{\mathbf{p}},\hat{H}]>=-\frac{i}{2m\hbar}\langle[\hat{\mathbf{r}}\cdot\hat{\mathbf{p}},\hat{\mathbf{p}}^2]\rangle-\frac{i}{\hbar}\langle[\hat{\mathbf{r}}\cdot\hat{\mathbf{p}},\hat{V}(\hat{\mathbf{r}})]\rangle \\ &=\frac{\langle\hat{\mathbf{p}}^2\rangle}{m}-\langle\hat{\mathbf{r}}\cdot\nabla\hat{V}(\hat{\mathbf{r}})\rangle=2\langle\hat{K}\rangle-\langle\hat{\mathbf{r}}\cdot\nabla\hat{V}(\hat{\mathbf{r}})\rangle. \end{align} }

For a stationary state, the expectation value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{r}}\cdot\hat{\mathbf{p}}} is constant in time. This gives us the relation,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\langle\hat{K}\rangle=\langle\hat{\mathbf{r}}\cdot\nabla\hat{V}(\hat{\mathbf{r}})\rangle.}

This is the virial theorem.

As an example of its application, let us consider the isotropic three-dimensional harmonic oscillator,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{V}(\hat{\mathbf{r}})=\tfrac{1}{2}m\omega^2\hat{r}^2.}

The right-hand side of the virial theorem is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{r}}\cdot\nabla\hat{V}(\hat{\mathbf{r}})=m\omega^2\hat{r}^2=2\hat{V}(\hat{\mathbf{r}}).}

Therefore,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\hat{K}\rangle=\langle\hat{V}\rangle.}

@@ Line 1: / Line 1: @@
-Consider
+{{Quantum Mechanics A}}
+We will now derive the quantum mechanical virial theorem.  For a Hamiltonian of the form,
+<math>\hat{H}=\frac{\hat{\mathbf{p}}^2}{2m}+\hat{V}(\hat{\mathbf{r}})=\hat{K}+\hat{V}(\hat{\mathbf{r}}),</math>
+this theorem gives the expectation value of the kinetic energy in a stationary state in terms of the potential energy.  To derive this relation, we consider the expectation value of <math>\hat{\mathbf{r}}\cdot\hat{\mathbf{p}}.</math>  The time derivative of this expectation value is
 <math>
 \begin{align}
+\frac{d}{dt}\langle\hat{\mathbf{r}}\cdot\hat{\mathbf{p}}\rangle&=-\frac{i}{\hbar}<[\hat{\mathbf{r}}\cdot\hat{\mathbf{p}},\hat{H}]>=-\frac{i}{2m\hbar}\langle[\hat{\mathbf{r}}\cdot\hat{\mathbf{p}},\hat{\mathbf{p}}^2]\rangle-\frac{i}{\hbar}\langle[\hat{\mathbf{r}}\cdot\hat{\mathbf{p}},\hat{V}(\hat{\mathbf{r}})]\rangle \\
+&=\frac{\langle\hat{\mathbf{p}}^2\rangle}{m}-\langle\hat{\mathbf{r}}\cdot\nabla\hat{V}(\hat{\mathbf{r}})\rangle=2\langle\hat{K}\rangle-\langle\hat{\mathbf{r}}\cdot\nabla\hat{V}(\hat{\mathbf{r}})\rangle.
+\end{align}
+</math>
-&\frac{d}{dt}<xp>\\
+For a stationary state, the expectation value of <math>\hat{\mathbf{r}}\cdot\hat{\mathbf{p}}</math> is constant in time.  This gives us the relation,
-&=\frac{1}{i\hbar}<[xp,H]> \\
-&=\frac{ 2<p^{2}> }{2m}+\frac{1}{i\hbar}<xpV-xVp>\\
-&=\frac{2<p^{2}>}{2m}+ \frac{1}{i\hbar}\int_{-\infty}^{\infty}dx[\psi^{*}x\frac{\hbar}{i}\frac{\partial}{\partial x} (V\psi)-xV\frac{\hbar}{i}\frac{\partial}{\partial x}\psi] \\
-&= \frac{2<p^{2}>}{2m}+<x\frac{\partial V}{\partial x}>
-\end{align}
+<math>2\langle\hat{K}\rangle=\langle\hat{\mathbf{r}}\cdot\nabla\hat{V}(\hat{\mathbf{r}})\rangle.</math>
-</math>
-Taking time average at both sides, we have
+This is the virial theorem.
-<math>
+As an example of its application, let us consider the isotropic three-dimensional harmonic oscillator,
-\frac{<xp>_{t=T}-<xp>_{t=0}}{T}=2\frac{1}{T}\int_{0}^{T}dt<\frac{p^{2}}{2m}>-\frac{1}{T} \int_{0}^{T}dt  <x\frac{\partial V}{\partial x}>
+<math>\hat{V}(\hat{\mathbf{r}})=\tfrac{1}{2}m\omega^2\hat{r}^2.</math>
-</math>
-For <math>T \rightarrow \infty , LHS  \rightarrow 0 </math>.
+The right-hand side of the virial theorem is given by
-For stationary state, the expectation values are constant in time, so we arrive
+<math>\hat{\mathbf{r}}\cdot\nabla\hat{V}(\hat{\mathbf{r}})=m\omega^2\hat{r}^2=2\hat{V}(\hat{\mathbf{r}}).</math>
-<math>2<KE>=<x\frac{\partial V}{\partial x}></math>, which is known as the Virial Theorem.
-In 3D, it is modified to
+Therefore,
-<math>2<KE>=<\mathbf{r}  \cdot \nabla V> =  -<\mathbf{r}  \cdot  \mathbf{F} > </math>.
+<math>\langle\hat{K}\rangle=\langle\hat{V}\rangle.</math>

The Virial Theorem: Difference between revisions

Latest revision as of 10:43, 16 August 2013

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