Advanced Quantum Physics Summary
Wednesday, Feb 5, 2025 | 3 minutes read | Update at Friday, Jun 27, 2025
Chapter-by-chapter summaries of Advanced Quantum Mechanics by Pieter Kok from the University of Sheffield
Recently I have become bored, and in order to find something interesting to do I started browsing some science textbooks (i know im such a nerd).
Quantum physics really interests me so I decided to start reading and summarising the chapters for myself and anyone else (although im not sure why you are here).
Please note that I can be and probably will be wrong in my summaries, because this is from my own understanding, or lack thereof.
I will attempt to summarise each chapter, although I will probably only summarise the chapters that I am able to or the chapters that interest me.
Link: https://phys.libretexts.org/Bookshelves/Quantum_Mechanics
Notes : This textbook and everything contained is licensed under CC BY-NC-SA 4.0
1: Linear Vector Spaces and Hilbert Space
Chapter Overview
The modern version of quantum mechanics was formulated in 1932 by John von Neumann in his famous book Mathematical Foundations of Quantum Mechanics, and it unifies Schrödingers wave theory with the matrix mechanics of Heisenberg, Born, and Jordan. The theory is framed in terms of linear vector spaces, so the first couple of lectures we have to remind ourselves of the relevant mathematics.
The basis of quantum mechanics relies on vectors |ψ⟩, |ϕ⟩ and complex numbers a,b,c, etc.
Linear vector space V is a mathematical structure of vectors and numbers that obey the following rules.
- |ψ⟩ + |ϕ⟩ = |ϕ⟩ + |ψ⟩ (commutativity)
- |ψ⟩ + (|ϕ⟩ + |χ⟩) = (|ψ⟩ + |ϕ⟩) + |χ⟩ (associativity)
- a(|ψ⟩ + |ϕ⟩) = a|ψ⟩ + a|ϕ⟩ (linearity)
- (a + b)|ψ⟩ = a|ψ⟩ + b|ψ⟩ (linearity)
- a(b|ϕ⟩) = (ab)|ϕ⟩
Note: The symbol for linear vector space resembles a V but is not a V, I was unable to input the actual symbol.
There also exists a null vector 0, where |ψ⟩ + 0 = |ψ⟩, and a conjugate |ϕ⟩ vector for every vector |ψ⟩ where |ϕ⟩ + |ψ⟩ = 0.
For each |ψ⟩ vector there is a ⟨ϕ| vector, and the set of dual vectors forms the linear vector space V*.
Note: Im skipping a bit about the rules of the inner product of the dual vectors and the norm of linear vector space. The norm of linear vector space is called Hilbert space, and we can always assume that all linear vector spaces are Hilbert spaces.
For linear vector spaces with an inner product, we can derive the Cauchy-Schwarz Inequality:
⟨ϕ ∣ ψ⟩^2 ≤ ⟨ψ ∣ ψ⟩⟨ϕ ∣ ϕ⟩
