Introductory Course
Fundamental Experiments of Quantum Mechanics
The Schrödinger equation derived from four principles and one experimental measurement.
Introduction
A beam of electrons passes through a thin crystalline layer and produces a series of rings on a screen: a diffraction pattern, characteristic of waves. The classical description of electron motion does not explain this pattern. Associating a wave with the electrons allows us to interpret it, but raises another question: which equation describes that wave, and how can we derive it?
For a non-relativistic particle in a static electric potential, we derive the Schrödinger equation without taking it as our starting point. The work is concentrated in cards 3, 4 and 5: following the steps, we see which properties of the law depend on the principles and where a measurement is needed.
In cards 6 and 7 we use the equation to calculate the distribution of particles scattered in Rutherford’s experiment and compare it with measurements. Later we study the allowed energy values of the hydrogen atom and their relation to the frequencies of the light it emits.
The four principles
We assume four principles:
- 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.
- 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.
- 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.
- Agreement with Newton. Under the conditions in which Newton’s mechanics is confirmed by experience, the theory must give the same predictions (F = ma).
The first three fix the general form of the evolution equation. The fourth guides the search for a law describing the effect of forces on the particle. The resulting law contains a constant that the principles do not determine: we obtain its value from the measurement of electron diffraction and identify it with Planck’s constant.
How to read this site
The thirteen cards require knowledge of Newtonian mechanics, electromagnetism, and differential and integral calculus, but not Analytical Mechanics. The course is intended for third-year Engineering students.
The first two cards present electron experiments and diffraction; the third introduces the complex numbers and vectors needed for the derivation. The theory should be read in order. We can first follow its assumptions and conclusions, then return to the calculations and proofs collected in the notes. Readers who start with the experimental cards can return to the third to understand how the predictions are obtained.
The experiments
The experiments with electrons, diffraction, Rutherford scattering, Franck–Hertz, the photoelectric effect and emission spectra were carried out in the laboratory. For Stern–Gerlach, the apparatus was designed at LAFIDIN but not built. The cascaded experiments are thought experiments: they illustrate the first three principles without proving them.
The interactive virtual laboratories allow us to vary the parameters and observe the models’ predictions. They accompany the cards without replacing experimental measurements.
The origin of the work
The material originated in a 1999 degree thesis. The experiments were performed using LEYBOLD equipment at the LAFIDIN teaching laboratory and in the “demonstration hall” of LEYBOLD DIDACTIC in Milan. The treatment of cascaded Stern–Gerlach experiments draws on R. P. Feynman (The Feynman Lectures on Physics, vol. III), with a different formalism and graphical representations.
The web edition reorders the nine original cards, separates some experimental and theoretical parts, and adds the card on complex numbers and state vectors. The changes are described in the note What has changed since 1999.
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”).