The discovery of the photon was one of the major initiators of quantum mechanics.
Light was very well known to be a wave through diffraction experiments. So how could it also be a particle???
This process "started" in 1900 with Planck's law which was based on discrete energy packets being exchanged as exposed at On the Theory of the Energy Distribution Law of the Normal Spectrum by Max Planck (1900).
This ideas was reinforced by Einstein's explanation of the photoelectric effect in 1905 in terms of photon.
In the next big development was the Bohr model in 1913, which supposed non-classical physics new quantization rules for the electron which explained the hydrogen emission spectrum. The quantization rule used made use of the Planck constant, and so served an initial link between the emerging quantized nature of light, and that of the electron.
The final phase started in 1923, when Louis de Broglie proposed that in analogy to photons, electrons might also be waves, a statement made more precise through the de Broglie relations.
This event opened the floodgates, and soon matrix mechanics was published in quantum mechanical re-interpretation of kinematic and mechanical relations by Heisenberg (1925), as the first coherent formulation of quantum mechanics.
It was followed by the Schrödinger equation in 1926, which proposed an equivalent partial differential equation formulation to matrix mechanics, a mathematical formulation that was more familiar to physicists than the matrix ideas of Heisenberg.
Inward Bound by Abraham Pais (1988) summarizes his views of the main developments of the subjectit:
- Planck's on the discovery of the quantum theory (1900);
- Einstein's on the light-quantum (1905);
- Bohr's on the hydrogen atom (1913);
- Bose's on what came to be called quantum statistics (1924);
- Heisenberg's on what came to be known as matrix mechanics (1925);
- and Schroedinger's on wave mechanics (1926).
Bibliography:
MIT 8.06 Quantum Physics III, Spring 2018 by Barton Zwiebach by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Instructor: Barton Zwiebach.
Free material from university courses:
- physics.weber.edu/schroeder/quantum/QuantumBook.pdf (archive) "Notes on Quantum Mechanics" pusbliehd by Daniel V. Schroeder (2019) The author is from from Weber State University.
Looks very impressive! Last update marked 2011 as of 2020.
Goes up to "A.15 quantum field theory in a Nanoshell", Ciro have to review it to see if there's anything worthwhile in that section.
Personal page says he retired as of 2020: www.eng.fsu.edu/~dommelen/ But hopefully he has more time for these notes!
And he appears to have his own lightweight markup language that transpiles to LaTeX called l2h: www.eng.fsu.edu/~dommelen/l2h/
One single universal wavefunction, and every possible outcomes happens in some alternate universe. Does feel a bit sad and superfluous, but also does give some sense to perceived "wave function collapse".
Solving the Schrodinger equation with the time-independent Schrödinger equation by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Before reading any further, you must understand heat equation solution with Fourier series, which uses separation of variables.
Once that example is clear, we see that the exact same separation of variables can be done to the Schrödinger equation. If we name the constant of the separation of variables for energy, we get:
- a time-only part that does not depend on space and does not depend on the Hamiltonian at all. The solution for this part is therefore always the same exponentials for any problem, and this part is therefore "boring":
- a space-only part that does not depend on time, bud does depend on the Hamiltonian:Since this is the only non-trivial part, unlike the time part which is trivial, this spacial part is just called "the time-independent Schrodinger equation".Note that the here is not the same as the in the time-dependent Schrodinger equation of course, as that psi is the result of the multiplication of the time and space parts. This is a bit of imprecise terminology, but hey, physics.
Because the time part of the equation is always the same and always trivial to solve, all we have to do to actually solve the Schrodinger equation is to solve the time independent one, and then we can construct the full solution trivially.
Once we've solved the time-independent part for each possible , we can construct a solution exactly as we did in heat equation solution with Fourier series: we make a weighted sum over all possible to match the initial condition, which is analogous to the Fourier series in the case of the heat equation to reach a final full solution:
- if there are only discretely many possible values of , each possible energy . we proceed and this is a solution by selecting such that at time we match the initial condition:A finite spectrum shows up in many incredibly important cases:Equation 3.Solution of the Schrodinger equation in terms of the time-independent and time dependent parts.
- if there are infinitely many values of E, we do something analogous but with an integral instead of a sum. This is called the continuous spectrum. One notable
The fact that this approximation of the initial condition is always possible from is mathematically proven by some version of the spectral theorem based on the fact that The Schrodinger equation Hamiltonian has to be Hermitian and therefore behaves nicely.
It is interesting to note that solving the time-independent Schrodinger equation can also be seen exactly as an eigenvalue equation where:The only difference from usual matrix eigenvectors is that we are now dealing with an infinite dimensional vector space.
- the Hamiltonian is a linear operator
- the value of the energy
Eis an eigenvalue
Furthermore:
- we immediately see from the equation that the time-independent solutions are states of deterministic energy because the energy is an eigenvalue of the Hamiltonian operator
- by looking at Equation 3. "Solution of the Schrodinger equation in terms of the time-independent and time dependent parts", it is obvious that if we take an energy measurement, the probability of each result never changes with time, because it is only multiplied by a constant
Schrödinger equation for a one dimensional particle with . The first step is to calculate the time-independent Schrödinger equation for a free one dimensional particle
Then, for each energy , from the discussion at Section "Solving the Schrodinger equation with the time-independent Schrödinger equation", the solution is:Therefore, we see that the solution is made up of infinitely many plane wave functions.
TODO are there experiments, or just theoretical?
Not the same as Hermite polynomials.
www.physics.udel.edu/~jim/PHYS424_17F/Class%20Notes/Class_5.pdf by James MacDonald shows it well.
And then we can prove the ladder properties easily.
The commutator appear in the middle of this analysis.
Quantum numbers appear directly in the Schrödinger equation solution for the hydrogen atom.
However, it very cool that they are actually discovered before the Schrödinger equation, and are present in the Bohr model (principal quantum number) and the Bohr-Sommerfeld model (azimuthal quantum number and magnetic quantum number) of the atom. This must be because they observed direct effects of those numbers in some experiments. TODO which experiments.
E.g. The Quantum Story by Jim Baggott (2011) page 34 mentions:This refers to forbidden mechanism. TODO concrete example, ideally the first one to be noticed. How can you notice this if the energy depends only on the principal quantum number?
As the various lines in the spectrum were identified with different quantum jumps between different orbits, it was soon discovered that not all the possible jumps were appearing. Some lines were missing. For some reason certain jumps were forbidden. An elaborate scheme of ‘selection rules’ was established by Bohr and Sommerfeld to account for those jumps that were allowed and those that were forbidden.
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