Stability of Matter

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Quantum Mechanics A
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\Psi\rangle evolves in time.
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
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
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
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
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

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 \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 r_0\!, the angular velocity ω, the charge of the particle e\!, and the speed of light, c\!. Therefore the rate of energy emission can be written by the function of those factors

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

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

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

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

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

Since we are considering er_0\! as one parameter, let's multiply by r_0^2\! and divide by \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

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

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

\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.

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