Introductory Quantum Mechanics — A Study Guide
“A physicist is just an atom’s way of looking at itself.” — Niels Bohr
“If you think you understand quantum mechanics, you don’t understand quantum mechanics.” — Richard Feynman
“If quantum mechanics hasn’t profoundly shocked you, you haven’t understood it yet.” — Niels Bohr
“Anyone who claims to understand quantum theory is either lying or crazy” — Richard Feynman
Opening
What is light?
Is light a particle or a wave?
After studying electrodynamics and optics and even just observing nature you might conclude light is a wave. And historically it became clear that “light must be a wave” when Maxwell showed that light really is an electromagnetic wave.
Physics seemed finished. The foundations were laid. Classical Physics, Thermodynamics, Statistical Physics, Optics, and Electrodynamics. These theories explained everything in the natural world. And light was understood.
“There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.” — Lord Kelvin
There were however some problems like the “Ultraviolet Catastrophe” and the “Photoelectric Effect.” Which the theories at the time could not explain.
Then came Max Planck. He solved the ultraviolet catastrophe by assuming the existence of “light quanta,” a smallest discrete packet of light with particle-like properties, which would later be called a photon.
Albert Einstein then used this concept to explain the photoelectric effect (which later earned him a Nobel Prize).
“According to the assumption to be contemplated here, when a light ray is spreading from a point, the energy is not distributed continuously over ever-increasing spaces, but consists of a finite number of ‘energy quanta’ that are localized in points in space, move without dividing, and can be absorbed or generated only as a whole.” — Albert Einstein
So light is a particle, then?
No. And yes, sort of. It’s complicated.
So it seems Kelvin was wrong. But he wasn’t just wrong. He was very wrong. Until now we have only dicussed what is considered “The Old Quantum Theory”
Things were about to get a whole lot weirder.
The Double Slit Experiment showed that particles such as electrons produced interference patterns you would expect from waves.
Strange. But physicists got clever and decided to shoot the electrons through the slits one at a time. This way they could not interact with eachother and produce “wave-like behavior” by some strange mechanism.
The result: The same pattern. The same exact wave interference pattern.
When learning this for the first time the only proper response is to fall out of your chair.
How could these individual electrons possibly interfere with eachother when shot through the slits one by one?
But still physicists tried to be clever and decided to use a detector to observe which slit each electron went through. They placed a detector at one of the slits.
The result: The detector ‘beeped’ or detected an electron 50% of the time (as expected).
But there was no wave inteference pattern.
It turns out the act of measurement interfered with the outcome of the experiment. The act of observation or measurement resulted in the electron behaving like a particle again and not like a wave.
Again, if you were to learn this for the first time and did not fall out of your chair yet, surely now you must be on the floor.
Or else you haven’t grasped the significance of what is being told yet.
I highly recommend you watch the folllowing video on the double slit experiment:
This was a long introduction. But the point was to explain a bit about the history of Quantum Mechanics, and, more importantly, expose you to the fact that this theory is quite radical, strange, and counter-intuitive.
This theory has huge physical as well as metaphysical implications. It’s no mystery as to why it took so long for this theory to be accepted as real physics.
But today Quantum Mechanics is one of the most succesful theories in physics. And our modern world: nanotechnology, semiconductors, transistors, modern computers, is all and only made possible because of this theory.
Introductory Quantum Mechanics
A subject such as “Introductory Quantum Mechanics” or “Quantum 1” will typically be given in the 1st or 2nd year of undergrad physics.
This subject will mostly just introduce you to the concepts in Quantum Mechanics and some of the formalisms used.
Scope of Subject
Quantum Mechanics is a physical theory that describes the “microscopic world.” This microscopic world is nature at the scale of atoms and subatomic particles (systems with dimensions smaller than ~ 0.1 Å).
The major ways in which Quantum Mechanics differ from Classical Mechanics are that physical quantities (energy, momentum, etc.) are quantized (restricted to discrete values), Wave-Particle Duality, and the Uncertainty Principle.
In modern QM, a fundamental mathematical function called “The Wave Function” contains all the information of a system in the form of probabiltiy amplitudes of what a measurement of physical quantities will yield.
The theory is very succesful. But it fails at speeds approaching the speed of light (~ 3×10⁸ m/s)
See the image below:
Important topics for this subject (I have divided it into two parts):
Part 1:
- The Wave Function
- The Schrödinger Equation
- The Hydrogen Atom
- Wave-Particle Duality, De Broglie Relation, Quantum Numbers
- Heisenberg Uncertainty Principle
Part 2:
- Operators
- Dirac Notation
- Generalized Uncertainty Principle
- Perturbation Theory
- Varitional Methods
Important maths:
- (Partial) Differential Equations
- Complex Numbers
- Linear Algebra
Where to begin?
I recommend you start by reading the following Wikipedia article:
Next it’s important to get really familiar with two new concepts:
- The Wave Function. The following is a must-watch:
- Schrödingers Wave Equation: Another must-watch (1-min long):
Make sure you also read about these concepts and think about what they really are.
It’s also a good idea to learn about The Infinite Potential Well Problem and The Finite Potential Well Problem.
Another great video:
Step Plan
- If you have an introductory physics books (like Giancoli): Read the chapters on QM and do the exercises
- The following lecture notes are a great resource when first learning QM: https://oer.physics.manchester.ac.uk/QM/Notes/Notes.pdf
- I also recommend an introductory QM book (like Griffith’s Introduction to Quantum Mechanics)
- Basically every chapter in Griffith’s book is important
Thanks for reading!
