Linear Algebra in Quantum Mechanics

Linear algebra is a branch of mathematics, but the truth of it is that linear algebra is the mathematics of data. Matrices/operators and vectors are the language of data. In quantum mechanics application, it is used to construct a base to express everything of a subatomic particle in a mathematical way in order to analyse.

Linear algebra is about linear combinations. That is, using arithmetic on columns of numbers called vectors and arrays of numbers called matrices, to create new columns and arrays of numbers. Linear algebra is the study of vector spaces and mappings that are required for linear transforms.

A vector space is a set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space R^n, where every element is represented by a list of real numbers. For a general vector space, the scalars are members of a field, in which case V is called a vector space over F.

Euclidean n-space R^n is called a real vector space, and C^n is called a complex vector space.

In order for defining a vector space, there are eight axioms, which I put in my notes at the bottom of this article. Before talking more precisely about some definitions and calculations in vector space, let’s clarify what does quantum physics have to do with this:

Quantum mechanics "lives" in a Hilbert space, (boring definition: Every finite-dimensional, complex, inner product vector space is a Hilbert space), and the vectors inside it are actually what known as wave functions which describe probability distribution (you know that everything is uncertain in quantum world). Then the mathematics of quantum mechanics is pretty much about the linear operators in the Hilbert space.

Below are the basic glossaries of the mathematics in quantum mechanics and in normal linear algebra. You can see them as the same thing just with different names ( just like Yirui and Steph;) ). I definitely am going to talk about these ‘fancy’ words in the later articles.

Quantum mechanics Linear algebra

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wave function vector

linear operator matrix

eigenstates eigenvectors

Hilbert space physical system

Hermitian operator physical observable

A meme about Hilbert space that actually highlights some of the essential stuff (and a Hilbert)