Master's Thesis

Fundamental Experiments of Quantum Mechanics

The experiments that made the theory inevitable — and the Schrödinger equation derived from four principles, without postulating it.

Introduction

Why does a beam of silver atoms, passing through a magnet, split into exactly two? Why do electrons, fired at a crystal, form diffraction patterns as if they were waves? In the early twentieth century a series of experiments broke classical physics and forced the construction of a new theory: Quantum Mechanics.

This site retraces those experiments — actually performed in the laboratory, not merely recounted — and shows how, starting from them, the theory can be built one principle at a time, until the Schrödinger equation is derived rather than postulated. The only prerequisites are Newtonian mechanics and electromagnetism.

The origin of this work

This material began as a degree thesis: the aim was to carry out the fundamental experiments of Quantum Mechanics and to prepare, starting from their analysis, a teaching path introducing the conceptual foundations of the theory. The work was presented at the 85th National Congress of the Italian Physical Society (SIF), held in Pavia in September 1999 (communication: Prof. C. Luponio, Prof. G. Mastrocinque, F. Palma [author], “The formulation of quantum physics through experiments performed and theoretically interpreted”).

It dates from 1999, but it deals with fundamental experiments and principles of physics, unchanged since then: the content is fully current.

The experiments

The experiments — those with electrons, electron diffraction, Rutherford scattering, Franck–Hertz, the photoelectric effect, atomic emission spectra — were carried out with LEYBOLD equipment, partly at the LAFIDIN teaching laboratory and partly in Milan, in the “demonstration hall” of LEYBOLD DIDACTIC. In this web edition, each experiment is also accompanied by an interactive simulated laboratory, where you can vary the parameters and watch the results change.

The path opens with the Stern–Gerlach experiment, the conceptually cleanest entry point into quantum behaviour. For this one, an apparatus was designed from scratch within the LAFIDIN laboratory — and building it on a student budget turned out to be a story in itself, involving a 1000 °C furnace, an impossible vacuum, and, in the original 1922 version, a cheque from one of the founders of Goldman Sachs. That story is told in the behind-the-scenes note of Chapter 1; the card presents the full design, analyses its operation theoretically, and its simulated laboratory lets you explore the design parameters yourself — and see what breaks.

The theory: four principles and one measurement

The theoretical part constitutes an introduction to Quantum Mechanics, treated from the outset with Dirac notation. The main result is the derivation of the Schrödinger equation: it is not postulated, but obtained from a set of elementary principles and from the requirement of consistency with Newton’s equation within the latter’s range of applicability. The derivation can be stated intuitively, building it one principle at a time and in the particular case of a particle moving in vacuum under electromagnetic forces:

1. Complex function. The state of a particle is described by a function that assigns a complex number to each position. The probabilities of finding the particle in the various regions of space are proportional to the sums, over each region, of the squared moduli of the function.
2. Linear superposition. Two states can be combined, each with a complex coefficient, and the combination is still a state. The state evolves continuously in time and, under evolution, each term evolves as it would on its own, with the same coefficient, and the subsequent state is the sum of the evolved terms.
3. Conservation of the total. As long as the system is not observed, the sum of the squared moduli over all positions does not change during the evolution.→ These three principles, together, are enough to fix the general form of the evolution equation. They do not yet say how the forces acting on the particle enter it.
4. Agreement with Newton. Under the conditions in which Newton’s mechanics is confirmed by experience — bodies sufficiently large and heavy — the theory must give the same predictions (F = ma).→ Imposing this requirement, the Schrödinger equation appears in its entirety, and with it a new constant, foreign to classical physics. Its value is supplied by an entirely quantum experiment, the diffraction of an electron beam through a thin crystalline layer: the constant turns out to coincide with the celebrated Planck constant.

From these four principles, and one measurement, the Schrödinger equation is derived, without postulating it and without resorting to Analytical Mechanics.

The introduction to complex numbers and to the principles of Quantum Mechanics through cascaded Stern–Gerlach experiments draws on R. P. Feynman’s treatment (The Feynman Lectures on Physics, vol. III); here, however, it has been recast with a somewhat different formalism and with original graphical representations, designed to be more intuitive.

The whole treatment is independent of Analytical Mechanics: it relies only on Newton’s theory and on electromagnetism, and is therefore — while remaining a far from trivial subject — accessible to third-year Engineering students.

How to read this site

The work is divided into nine “cards”; each card contains an experimental part accompanied by a theoretical part. The cards follow a progressive path: the first two (Stern–Gerlach and cascaded Stern–Gerlach) introduce the fundamental concepts and complex numbers through original graphical representations, and are accessible even without advanced mathematics. The cascade is a thought experiment: its laboratory realisation came only recently, and a note in card two tells that story. In the later cards the formalism grows gradually, up to the derivation of the Schrödinger equation. Reading in order is recommended, but each card’s experimental part stands on its own.

Credits

Degree Thesis in Electrical Engineering — University of Naples “Federico II”, Faculty of Engineering. LAFIDIN teaching laboratory (Laboratory, Physics, Teaching, Engineering). Academic year 1998/1999.

Supervisors: Prof. Scipione Bobbio, Prof. Carlo Luponio · Candidate: Faustino Palma (student no. 44/956).

Contact: Faustino Palma on LinkedIn ↗

Work presented at the 85th National Congress of the Italian Physical Society (SIF), Pavia, September 1999 (Prof. C. Luponio, Prof. G. Mastrocinque, F. Palma [author], “The formulation of quantum physics through experiments performed and theoretically interpreted”).