Chapter 1: The Birth of Quantum Concepts

At the dawn of the 20th century, groundbreaking experimental findings began to challenge the longstanding belief that light was merely a manifestation of electromagnetic waves, as described by Maxwell's equations. Max Planck's work on black-body radiation, which addressed the 'ultraviolet catastrophe' and earned him a Nobel Prize in 1918, hinted at a discrete, particle-like nature of light. This notion was reinforced by Albert Einstein's explanation of the photoelectric effect, which garnered him the Nobel Prize in 1921. Both Planck and Einstein introduced the idea that light, while exhibiting wave-like properties, is also emitted and absorbed in discrete packets of energy called photons. The energy of these photons is articulated through the Planck-Einstein relationship:

E = ℏω

where 𝜔 signifies the angular frequency of the electromagnetic wave, E denotes the energy of the corresponding photon, and ℏ is Planck's constant.

The particle-like nature of light was compellingly demonstrated by Arthur Compton in 1923. In his experiments, Compton scattered X-rays off free electrons and observed a notable decrease in frequency, a phenomenon now termed the Compton effect. This outcome could not be reconciled with the classical wave description of light, which predicted no frequency shift during scattering. Instead, the results aligned perfectly with the photon theory proposed by Einstein and Planck.

To further illustrate this, let's look at the Compton scattering effect:

As the 1920s progressed, even more astonishing evidence emerged that suggested not only the particle-like behavior of light but also the wave-like behavior of matter itself. Electron diffraction experiments, such as the 1927 Davisson-Germer experiment, revealed that electrons scattered off crystalline surfaces produced distinct diffraction patterns—an indication of wave interference.

Animation of the emergence of an electron diffraction pattern, by Thierry Dugnolle:

To explain this wave-like nature of electrons, physicist Louis de Broglie proposed in his 1924 doctoral dissertation titled "On the Theory of Quanta" that every material particle is associated with a wave, now known as a matter wave or de Broglie wave, characterized by a specific de Broglie wavelength. This concept not only captured the wave-like behavior of particles but also accounted for the discrete frequency spectra observed in atoms, reflecting the presence of quantized electron orbits.

Louis De Broglie eloquently stated in his 1929 Nobel lecture, "The electron can no longer be conceived as a single, small granule of electricity; it must be associated with a wave, and this wave is no myth; its wavelength can be measured and its interferences predicted."

To connect the wave and particle perspectives of matter, de Broglie established a relationship linking the properties of a particle to its corresponding matter wave. This relationship can be expressed as follows:

p = ℏk

where p represents the particle's momentum, 𝝀 is the de Broglie wavelength, and k is the wave-vector associated with the matter wave. In three spatial dimensions, this relationship can be generalized using vector notation.

Together, the de Broglie relation and the Einstein-Planck relation form the cornerstone of early quantum theory by linking a particle's energy and momentum to the frequency and wavelength of its associated wave.

Chapter 2: The Arrival of the Schrödinger Equation

Now, let's consider a simple one-dimensional de Broglie "plane wave," characterized by an amplitude A, a wave-vector k, and an angular frequency 𝜔. This wave, which we will refer to as a wave function, can be expressed mathematically as:

Ψ(x, t) = A e^{i(kx - ωt)}

Using Euler's identity, we can represent this wave in its trigonometric form. The actual quantity represented by Ψ(x, t) would later be interpreted as a probability amplitude through Born's probability amplitude interpretation.

Next, we can utilize the de Broglie and Einstein-Planck relations to relate the properties of the wave function to those of the associated particle. This leads us to define the total energy of the particle as a combination of kinetic and potential energy:

E = T + V

where m is the mass of the particle, p its momentum, and V(x) is a spatially-dependent potential field.

Erwin Schrödinger's landmark paper in 1926, titled "An undulatory theory of the mechanics of atoms and molecules," introduced the time-dependent Schrödinger equation, a differential equation that delineates how the wave function evolves over time. By taking the time derivative of the wave function Ψ(x, t), we can derive the equation:

iħ ∂Ψ/∂t = HΨ

where H represents the Hamiltonian operator. The time-dependent Schrödinger equation is the most comprehensive form from which all other versions can be derived. By exploring a time-independent potential field, one can derive the time-independent Schrödinger equation, although this discussion exceeds the scope of this article.

To further explore the implications of de Broglie's hypothesis, we can build upon the concept of matter waves to derive Heisenberg's quantum uncertainty principle. This derivation is the focus of another article, where we analyze how localized matter waves can be represented using the Fourier transform, leading to the emergence of the uncertainty principle.