By: Tao Steven Zheng (鄭濤)

This article was originally posted on?Brilliant.org?and the Math & Physics Wikia by the same author.

Introduction

Quantum mechanics?is a very peculiar physical theory. Its discovery was not derived from simple laws that are undeniably valid through everyday experience. Unlike classical mechanics, with Newton’s laws of motion describing the actions of macroscopic objects, quantum mechanics would never have taken form without the explanation of?blackbody radiation?(Planck 1900), the structure of the?atom?(Bohr 1913), and countless experimental successes. In this lecture, we will focus on the mathematical description of modern quantum mechanics, starting with the hypotheses of Max Planck, Albert Einstein, and Louis de Broglie. From these hypotheses, we will derive the equations and concepts that fundamentally describe the microscopic world.

The Pre-Quantum Wars

Stephen Hawking described quantum theory as “The dreams that stuff is made of”. Well, such an insubstantial statement is actually not far from truth! In fact, the physical formalism of quantum mechanics was made possible by bold guesses that happened to have triumphed over earlier knowledge (which we now call classical physics). Yet the reason why this “New Physics” claimed victorious is not at all mysterious. You see, physicists prior to Planck, and especially Einstein, were completely sure of themselves that the study of physics was nearly complete. They knew how planets and stars moved, how electromagnetic fields work, and how all other natural sciences followed from just two pillars: MECHANICS and ELECTROMAGNETISM.

However, despite this common arrogance most physicists shared during the late 19th century, there were ongoing feuds over which theories were to die, and which scientists were on the fringe. There was the longest debate about whether light behaved like a wave (Huygens) or a particle (Newton), which temporarily ended in favour of waves through the?double-slit experiment?(Young 1803). In fact, the wave-particle debate was not only confined to light, but ordinary matter as well. Boltzmann’s entropic theory of thermodynamics will be remembered as immortal; however, the majority of scientists in Europe at the time rejected his physical intuition of matter being composed of atoms and molecules. Sadly, Boltzmann took his life before every other physicist in Europe accepted his theory of the existence of atoms. Boltzmann’s picture of the world being composed of discrete particles will pave the way to a deeper understanding of nature for the future of mankind.

Planck’s Brave New World

Although Boltzmann’s contributions were immeasurable, the person who truly kickstarted quantum mechanics would be Max Planck (even though he was somewhat dubious of his own theory). In 1900, Max Planck described the relationship between the temperature of an object and its emitted frequency using?Boltzmann statistics. This was made possible by imposing a postulate unthinkable at the time: objects radiate and absorb in discrete packets of energy (which he called?‘quanta’). In doing so, he gave the relation??where??is the energy,??is Planck’s constant, and??is the frequency of light. Although Planck was lucky to have solved the blackbody radiation problem, he did not understand why his “magic postulate” fundamentally worked, nor would he expect to have ignited a revolution in physics.

Five years later, Albert Einstein would make the next step in clarifying Planck’s claim. First, Einstein recognized that Planck’s ‘quantized packet of energy’ as a fundamental unit of energy. By concretizing Planck’s hypothesis, Einstein immortalized the idea by renaming Planck's ‘quanta’ the?photon. There are two new features to Einstein’s photon:

(1) The photon is a massless particle with energy

(2) Photons travel by the propagation of electromagnetic radiation (light waves)

Thus, it is made clear that light is both a particle and a wave.

In 1924, Louis de Broglie furthered the development of quantum mechanics by making another bold claim: all particles possess?wave-particle duality, where the momentum and energy of a particle are given by two relations:

where ?and .

Louis de Broglie's hypothesis radically changed our understanding of matter. His 'matter-waves' generalizes Einstein's photon being both a particle and a wave to every particle we observe in the universe (so fundamentally, all particles possess a wavelength).

With Planck’s hypothesis of the ‘quanta’, Einstein’s interpretation of the photon, and de Broglie’s wave-particle duality, physicists were more comfortable discussing quantum mechanics.

References

[1] ''Modern Physics for Scientists and Engineers'' by John R. Taylor, Chris D. Zafiratos, Michael A. Dubson

[2] ''Introduction to Quantum Mechanics'' by David J. Griffith

[3] ''Quantum Mechanics Demystified'' by David MacMahon

[4] ''Essential Quantum Mechanics'' by Gary E. Bowman

[5] ''Understanding Physics'' by Michael Mansfield, Colm O'Sullivan