Entanglement

BIRTH OF QUANTUM MECHANICS

The era of quantum mechanics was ushered by Warner Heisenberg in 1925, and together with Max Born and Pascual Jordan, he introduced the matrix-algebra formulation in the same year. One year later (1926), the Austrian physicist Erwin Schrödinger independently introduced his wave mechanics and proved the equivalence of the two methods.

Matrix mechanics is analogous to classical mechanics, the central idea being that all physical quantities must be represented by infinite self-adjoint matrices4. If the matrices q and p represent the position and momentum variables of a particle, then the canonical commutation rule

\mathbf{qp} - \mathbf{pq} = i\hbar

is satisfied, where

\displaystyle\hbar=\frac{h}{2\pi}=1.054572\times 10^{-34}\mbox{J.s.}

h denotes Planck’s constant; the boldface type is used to represent matrices. The new theory spectacularly accounted for nearly all spectroscopic data known at the time, especially after the concept of the electron spin was included into the theoretical framework.

On the other hand, Schrödinger equation describes the quantum mechanical behavior objects such as atoms and subatomic particles. Its form is iconic:

\displaystyle H(t)\mid\psi (t)\rangle=i\hbar\frac{\partial}{\partial t}\mid\psi (t)\rangle

The Schrödinger equation is a partial differential equation whose solution gives the wave function which is a representation of the state of the system. The Schrödinger equation depicts how the wave function evolves over time. The state itself describes the probability that a system has a particular position, momentum, spin, etc.

The Schrödinger equation plays the analogous role that Newton’s laws of motion play in classical mechanics. The evolution conserves information, and is completely reversible.

Following Schrödinger’s demonstration of the equivalence of the matrix and wave versions of quantum mechanics, Max Born presented a statistical interpretation of the wave function (see box). Shortly after, Pascual Jordan in Göttingen and Paul Dirac in Cambridge, England, created a set of unified equations known as “transformation theory” that gave a logical picture of quantum mechanics using linear vector spaces and clarified the roles of both wave mechanics and matrix mechanics.

All of these developments formed the basis of what is now regarded as quantum mechanics.


4. later identified with operators on a Hilbert space

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