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linear algebra so i can understand qm

Welcome! I'm excited to help you bridge the gap between linear algebra and the stunning, probabilistic world of quantum mechanics. Before we construct our personalized learning path and dive into how vectors represent physical states, let's make sure we tailor this perfectly to your background.

To tailor this, could you tell me a bit about your current background in math (e.g., have you taken calculus or basic linear algebra?) and how much quantum mechanics you've encountered so far?

I am comfortable with both calculus and basic linear algebra, but I want to see how they connect to QM.

It is great to have you here! Since you already have a solid handle on calculus and basic linear algebra, we can skip the dry introductory computational drills and focus entirely on the deep conceptual machinery that connects these mathematics directly to physical quantum phenomena.

Your path
Linear Operators and Matrices
Eigenvalues, Eigenvectors, and Spectral Theory
Advanced Linear Algebra Topics for QM

The State Vector and Hilbert Space

In classical mechanics, you specify the state of a particle by listing its exact position and momentum as numbers. In quantum mechanics, things are fundamentally different: the state of a physical system is represented by a vector in a complex known as a Hilbert space.
In quantum mechanics, we write these state vectors using Dirac's bra-ket notation. A state vector is called a ket and is written with a vertical bar and a right angle bracket, like this: ∣ψ⟩\lvert \psi \rangle. Think of this ket as an abstract column vector that contains all the physical information about the system.
To make this concrete, imagine a simple quantum system: a single photon's polarization. It could be horizontally polarized, vertically polarized, or any linear combination of both. We can represent the pure horizontal state as the ket ∣H⟩\lvert H \rangle and the pure vertical state as the ket ∣V⟩\lvert V \rangle. In a two-dimensional complex vector space, we can write these as simple basis vectors:
∣H⟩=(10),∣V⟩=(01)\lvert H \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \lvert V \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}
Any other polarization state is just a superposition—a linear combination—of these two states, written as:
∣ψ⟩=α∣H⟩+β∣V⟩=(αβ)\lvert \psi \rangle = \alpha \lvert H \rangle + \beta \lvert V \rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}

Continue to the definition of Hilbert spaces

To understand quantum mechanics deeply, we have to transition from basic vector spaces to a more specialized playground: the Hilbert space.

What Makes a Hilbert Space?

A is not just any vector space. It is a specific type of complex vector space equipped with an inner product, and it must possess a mathematical property called completeness.

The inner product is what allows us to compute