User:Able806/Quantum mechanics

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Quantum mechanics consists of the breakthrough in physics in the 1920s in understanding how particles behave inside atoms. Classical mechanics, as initially discovered by Isaac Newton, cannot explain atomic behavior. Erwin Schrödinger is generally credited with the formulation of the Schrödinger equation, around 1926. Other contributions were from Werner Heisenberg, Niels Bohr, John von Neumann, and Hermann Weyl.

Classical mechanics would predict that an electron orbits a proton just as planets orbit the sun. Classical electromagnetism would predict that the orbiting electron would emit a time-varying electrical field just as a radio station does. But the electron would lose energy as it emits this radiation, and would orbit closer and closer to the proton, until it collapses into the proton! Such a model cannot be correct.

Quantum mechanics posits that an electron (or any other sub-atomic particle) behaves as both a wave and a particle. As a result of the wave nature of the electron, the position of the electron can never be precisely known. Whenever it is attempted to be measured, knowledge of the electron's velocity is lost. Hence, there is an inherent uncertainty that prevents precisely measuring both the position and the momentum simultaneously. This is known as the Heisenberg Uncertainty Principle.

Quantum mechanics forms the basis for our understanding of chemical reactions, as well as all computers and electronic devices today.

An important aspect of Quantum Mechanics is the predictions it makes about the radioactive decay of isotopes. Radioactive decay processes, controlled by the wave equations, are random events. A radioactive atom has a certain probability of decaying per unit time. As a result, the decay results in an exponential decrease in the amount of isotope remaining in a given sample as a function of time. The characteristic time required for 1/2 of the original amount of isotope to decay is known as the "half-life" and can vary from quadrillionths of a second (9B) to quintillions of years (186W).

Contents

The Four Ideas that led to Quantum Theory

Blackbody Radiation

A blackbody is an object that is a perfect absorber of radiation. In the ideal case, it absorbs all of the light that falls on it. While such an object does not reflect any light, if it is heated is can radiate light. The study of this radiated light generated controversy in the late 19th century. Specifically, there was a problem explaining the spectrum of the thermal radiation emitted from a blackbody.

The Photoelectric Effect

In 1905, Einstein made the radical proposal that light consisted of particles called photons.

Bohr Theory of the Atom

de Broglie Wavelength of Material Particles

The Schrödinger Equation

The behavior of a particle of mass m subject to a potential V (x,t) is described by the following partial differential equation:

i\hbar\frac{\partial \psi(x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \psi(x,t)}{\partial x^2}+V(x,t)\psi(x,t)

where ψ(x,t) is called the wavefunction. The wavefunction contains information about where the particle is located, its square being a probability density. A wavefunction must be defined and continuous everywhere. In addition it must be square-integrable, meaning:

\int_{-\infty}^{\infty} |\psi(x,t)|^2 dx < \infty

Postulates of Quantum Mechanics

Postulate 1: States of physical systems are represented by vectors

The state of a physical system is described by a state vector |\psi\rangle that belongs to a complex Hilbert space.

The superposition principle holds, meaning that if | φ1\rangle, | φ2\rangle,..., | φn\rangle are kets belonging to the Hilbert space, the linear combination

|\chi\rangle = α1 | φ1\rangle + α2 | φ2\rangle +...+ αn | φn\rangle

is also a valid state that belongs to the Hilbert space. States are normalized to conform to the Born probability interpretation, meaning

\langle \psi|\psi\rangle = 1

If a state is formed from a superposition of other states, normalization implies that the squares of the expansion coefficients must add up to 1:

\langle \chi|\chi\rangle = |\alpha1|2 + | α2|2 +...+ | αn|2 = 1

Postulate 2: Physical observables are represented by operators

Physically measureable quantities like energy and momentum are known as observables. Mathematacally, an observable is a Hermitian operator that acts on state vectors in the Hilbert space. The eigenvectors of Hermitian operators from an orthonormal basis of the state space for the system.

Postulate 3: The possible results of a measurement are the eigenvalues of an operator

Postulate 4: The probability of obtaining a given measurement result

Postulate 5: The state of a system after measurement

Postulate 6: The time evolution of a quantum system is governed by the Schrödinger equation

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