Lectures on Quantum Mechanics 1: Origins

Introduction

Quantum mechanics (QM) is a very peculiar physical theory in that 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 (more like the absence of failure). 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 are to die, and which scientists were on the fringe. There was the longest debate of whether light behaved as a wave (Huygens) or particle (Newton), which temporarily ended in favour of waves by the double-slit experiment (Young 1803). In fact, the wave-particle debate was not only confined to light, but ordinary matter as well. This debate resulted in the biggest tragedy in physics –– the death of Ludwig Boltzmann. 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 submitted to his theory on 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 QM would be 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 $E= h \nu$ where $E$ is the energy, $h$ is Planck’s constant, and $\nu$ 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.

Blackbody curves

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 $E= h \nu$

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.

Photoelectric Effect

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

$p= \hbar k = \frac{h}{\lambda}$ and $E = \hbar \omega = \frac{{p}^{2}}{2m}$

• $\hbar = \frac{h}{2 \pi}$

• $k = \frac{2 \pi}{\lambda}$

• $\omega = 2 \pi \nu$

Wave-Particle Duality

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 wavelength).

With Planck’s hypothesis of the ‘quanta’, Einstein’s interpretation of the photon, and de Broglie’s wave-particle duality, we are well equipped (from a physical basis) to consider the rest of quantum mechanics. In the next lecture, we will derive the Schrödinger equation out of de Broglie’s relations and the wave-particle duality.

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

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