Stability of Matter: 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 13:35, 8 August 2013

One of the most important problems to inspire the creation of quantum mechanics was the model of the Hydrogen atom. After Thompson discovered the electron, and Rutherford the nucleus (or Kern, as he called it), the model of the Hydrogen atom was refined to one of the lighter electron of unit negative elementary charge orbiting the larger proton, of unit positive elementary charge. However, it was well known that classical electrodynamics required that accelerated charges must radiate electromagnetic waves, and therefore lose energy. One question of interest is what determines the rate 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 \rho\!} of this radiation and how high is this rate?

The parameters that would describe this radiation for an electron in Bohr's model, in which the electron is assumed to travel in quantized circular orbits around the nucleus, are the radius of the orbit 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 r_0\!} , the angular velocity 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 \omega} , the charge of the particle 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 e\!} , and the speed of light, 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 c\!} . Therefore the rate of energy emission can be written by the function of those factors

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 \rho=\rho(r_0,\omega,e,c)\!} .

However, far away from the atom, 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 \rho\! } can only depend on the the dipole moment, so we can express the rate as

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 \rho(er_0, \omega, c) \!}

Since light is energy, we are looking for how much energy is emitted per unit time: 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 [\rho]=\frac{\text{energy}}{\text{time}}.} Knowing this much already imposes constraints on the possible dependence 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 \rho\!} on 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 er_0\!} , 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 \omega\!} , and 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 c\!} . We now use dimensional analysis to construct a quantity with units of energy.

From potential energy for coulombic electrostatic attractions: 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 \text{energy}=\frac{e^2}{\mbox{length}}\!}

Since we are considering 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 er_0\!} as one parameter, let's multiply 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 r_0^2\!} and divide 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 \text{length}^2\!} . Also, the angular velocity has units of frequency, so to obtain a quantity with units of energy per unit time, we simply multiply our result by the angular velocity, obtaining

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 \frac{\text{energy}}{\text{time}}=\frac{e^2 }{\text{length}}\frac{r_0^2}{\text{length}^2}*\omega.}

In addition, 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 c/\omega \!} has units of length, so we can write

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 \frac{\text{energy}}{\text{time}} \sim \frac{e^2r_0^2 }{(c/\omega)^3} \omega = \frac{e^2r_0^2 }{c^3}\omega^4\sim\frac{1}{r_0^4 }.}

Therefore, as the dipole loses energy by radiating, the radius of the electron's orbit decrease. That is, the rate of emission increases as the radius decreases. As a result, classical physics predicts that atoms should collapse because the electron will radiate away all of its energy.

@@ Line 1: / Line 1: @@
-One of the most important problems to inspire the creation of Quantum Mechanics was the model of the Hydrogen Atom. After Thompson discovered the electron, and Rutherford, the nucleus (or Kern, as he called it), the model of the Hydrogen atom was refined to one of the lighter electron of unit negative elementary charge orbiting the larger proton, of unit positive elementary charge. However, it was well known that classical electrodynamics required that accelerated charges must radiate electromagnetic waves, and therefore lose energy. For an electron that moves in circular orbit about the more massive nucleus under the influence of the Coulomb attractive force, here is a simple non-relativistic model of this classical system:
+{{Quantum Mechanics A}}
+One of the most important problems to inspire the creation of quantum mechanics was the model of the [[Hydrogen atom]]. After Thompson discovered the electron, and Rutherford the nucleus (or Kern, as he called it), the model of the Hydrogen atom was refined to one of the lighter electron of unit negative elementary charge orbiting the larger proton, of unit positive elementary charge. However, it was well known that classical electrodynamics required that accelerated charges must radiate electromagnetic waves, and therefore lose energy. One question of interest is what determines the rate <math>\rho\!</math> of this radiation and how high is this rate?
-Where <math>r\!</math> is the orbital radius, and we neglect the motion of the proton by assuming it is much much more massive than the electron.
-'''So the question is:  What determines the rate <math>\rho</math> of this radiation?  and how fast is this rate?'''
+The parameters that would describe this radiation for an electron in Bohr's model, in which the electron is assumed to travel in quantized circular orbits around the nucleus, are the radius of the orbit <math>r_0\!</math>, the angular velocity <math>\omega</math>, the charge of the particle <math>e\!</math>, and the speed of light, <math>c\!</math>. Therefore the rate of energy emission can be written by the function of those factors
-The electron in the Bohr's model involves factors of radius  <math>r_0\!</math>, angular velocity <math>\omega\!</math>, charge of the particle <math>e\!</math>, and the speed of light, <math>c\!</math>. Therefore the rate of the energy dissipation can be written by the function of those factors
 :<math>\rho=\rho(r_0,\omega,e,c)\!</math>.
-However, far away from the atom, <math> \rho \!</math> can only depend on the the dipole moment, so we can express the rate as follows
+However, far away from the atom, <math> \rho\! </math> can only depend on the the dipole moment, so we can express the rate as
 :<math> \rho(er_0, \omega, c) \!</math>
+Since light is energy, we are looking for how much energy is emitted per unit time: <math>[\rho]=\frac{\text{energy}}{\text{time}}.</math>  Knowing this much already imposes constraints on the possible dependence of <math>\rho\!</math> on <math>er_0\!</math>, <math>\omega\!</math>, and <math>c\!</math>.  We now use dimensional analysis to construct a quantity with units of energy.
-'''What is the dimension of <math>\rho\,\!</math>?'''
+From potential energy for coulombic electrostatic attractions: <math>\text{energy}=\frac{e^2}{\mbox{length}}\!</math>
-Essentially, since light is energy, we are looking for how much energy is passed in a given time: <math>[\rho]=\frac{\text{energy}}{\text{time}} \!</math>
-Knowing this much already imposes certain constraints on the possible dimensions. By using dimensional analysis, let's construct something with units of energy.
-From potential energy for coulombic electrostatic attractions: <math>\text{energy}=\frac{e^2 }{\mbox{length}} \!</math>
+Since we are considering <math>er_0\!</math> as one parameter, let's multiply by <math>r_0^2\!</math> and divide by <math>\text{length}^2\!</math>.  Also, the angular velocity has units of frequency, so to obtain a quantity with units of energy per unit time, we simply multiply our result by the angular velocity, obtaining
+:<math>\frac{\text{energy}}{\text{time}}=\frac{e^2 }{\text{length}}\frac{r_0^2}{\text{length}^2}*\omega.</math>
-Since we are considering <math>er_0\!</math> as one parameter, let's multiply by <math>r_0^2\!</math> and divide <math>\text{length}^2\!</math>. Also, the angular velocity is in frequency, so to get the above equations in energy/time, just multiply it with the angular velocity, then we have
+In addition, <math> c/\omega \!</math> has units of length, so we can write
-:<math>\frac{\text{energy}}{\text{time}}=\frac{e^2 }{\text{length}}\frac{r_0^2}{\text{length}^2}*\omega </math>
+:<math>\frac{\text{energy}}{\text{time}} \sim \frac{e^2r_0^2 }{(c/\omega)^3} \omega = \frac{e^2r_0^2 }{c^3}\omega^4\sim\frac{1}{r_0^4 }.</math>
-In addition, since <math> c/\omega \!</math> gives the dimension of length, we can write
-:<math>\frac{\text{energy}}{\text{time}} \sim \frac{e^2r_0^2 }{(c/\omega)^3} \omega = \frac{e^2r_0^2 }{c^3}\omega^4\sim\frac{1}{r_0^4 } \!</math>
-Therefore, as the dipole loses energy by radiating, the radius of the electron's orbit decrease. That is, the rate of emission increases as the radius decrease. As a result, classically the matter should collapse.
+Therefore, as the dipole loses energy by radiating, the radius of the electron's orbit decrease. That is, the rate of emission increases as the radius decreases.  As a result, classical physics predicts that atoms should collapse because the electron will radiate away all of its energy.

Stability of Matter: Difference between revisions

Latest revision as of 13:35, 8 August 2013

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