Stability of Matter

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.

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

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:

Where $r\!$ is the orbital radius, and we neglect the motion of the proton by assuming it is much much more massive than the electron.

The question is: What determines the rate $\rho \,\!$ of this radiation and how fast is this rate?

The electron in the Bohr's model involves factors of radius $r_{0}\!$, angular velocity $\omega \!$, charge of the particle $e\!$, and the speed of light, $c\!$. Therefore the rate of the energy dissipation 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 follows

$\rho (er_{0},\omega ,c)\!$

What is the dimension of $\rho \,\!$?

Essentially, since light is energy, we are looking for how much energy is passed in a given time: $[\rho ]={\frac {\text{energy}}{\text{time}}}\!$

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: ${\text{energy}}={\frac {e^{2}}{\mbox{length}}}\!$

Since we are considering $er_{0}\!$ as one parameter, let's multiply by $r_{0}^{2}\!$ and divide ${\text{length}}^{2}\!$. 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

${\frac {\text{energy}}{\text{time}}}={\frac {e^{2}}{\text{length}}}{\frac {r_{0}^{2}}{{\text{length}}^{2}}}*\omega$

In addition, since $c/\omega \!$ gives the dimension of length, we can write

${\frac {\text{energy}}{\text{time}}}\sim {\frac {e^{2}r_{0}^{2}}{(c/\omega )^{3}}}\omega ={\frac {e^{2}r_{0}^{2}}{c^{3}}}\omega ^{4}\sim {\frac {1}{r_{0}^{4}}}\!$

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 decrease. As a result, classically the matter should collapse.