This material focuses not on pure linear algebra, but rather on its applications in engineering and quantum mechanics.
Geometrically, the meaning of an eigenvalue signifies the scaling (expansion or contraction) and occasionally the inversion of an invariant coordinate axis.
The parity operator (P hat) stands as one of the most fundamental operators in quantum mechanics. For any physical system governed by a symmetric potential, the Hamiltonian (H hat) and the parity operator strictly commute, yielding the relation [H hat, P hat] = 0. This mathematically guarantees the existence of a complete set of simultaneous eigenstates. It physically implies that both the energy and the spatial symmetry of the system can be determined simultaneously with absolute precision, bypassing the constraints of Heisenberg's uncertainty principle. The attached material explicitly demonstrates the algebraic and analytic properties of P hat.
We solve the eigenvalue problem to determine the eigenvalues and eigenvectors, and understand their geometric and physical significance.
This material is not the pure linear algebra or functional analysis covered in mathematics departments. It is applied material utilized in engineering and quantum mechanics.
We will also cover the inverse Fourier transform, which localizes position by extending a finite-dimensional vector space to an infinite-dimensional vector space and summing an infinite number of momentum bases.
In quantum mechanics, the Fourier transform is not merely an abstract mathematical tool for space transformation, but the strict mathematical representation of wave superposition.
To spatially localize a particle, an infinite spectrum of plane waves possessing different momenta must be continuously added (superposed).
The integral of the inverse Fourier transform precisely represents this continuous summation.
I hope this is helpful to those learning this for the first time.
This investigates the geometric meaning of the eigenvalue problem for a 3 times 3 operator. Geometrically, the eigenvectors represent invariant directions that undergo no change in orientation under a given linear transformation.
Under what conditions does interference fail to occur, even within a linear combination of pure states?
I could not answer this confidently during my undergraduate years.
I hope this material is helpful to beginners in quantum mechanics.
This material is not pure linear algebra intended for a mathematics department.
It consists of linear algebra applied to fields such as engineering and physics.
I hope this is helpful.
This material covers the mathematical calculation and physical meaning of fundamental expectation values and standard deviations through introductory examples.
Hope this helps with your studies.
This document presents the physical meaning of the expectation value, representing the statistical average of quantum mechanical measurements, and the standard deviation, representing the dispersion of the probability distribution.
Hope this helps with your studies.
This post provides reference material regarding the types of operators, specifically linear and non-linear operators, that frequently appear in undergraduate quantum mechanics.
It is hoped that this helps.
Please note that this material isn't intended as a pure mathematics guide to linear algebra or functional analysis.
Instead, it's geared towards applied fields like quantum information, quantum computing, physical AI, and quantum mechanics.
Hope you find it helpful!
Since 2025, linear algebra-based quantum mechanics has been increasingly and rigorously demanded by cutting-edge industries.
However, existing literature on linear algebra-based quantum mechanics is highly inaccessible, as it predominantly targets graduate-level studies and lacks a cohesive connection to fundamental knowledge.
By constructing a framework with linear algebra and naturally integrating the Schrödinger equation based on tunneling wavefunctions applied in advanced semiconductor physics within it, this approach will significantly assist students in any scenario, whether pursuing graduate education or industrial employment.
Furthermore, this foundation directly extends to fields such as quantum computing, quantum information science, physical AI, and quantum field theory.
Consequently, this is the imperative reason why undergraduate students must learn linear algebra-based quantum mechanics at an early stage.
There are four major branches of mechanics in physics.
As of 2026, global technological advancements, including quantum computing, physical AI, quantum information science, and semiconductors, are all fundamentally driven by quantum mechanics.
However, this modern advancement relies on a linear algebraic foundation to treat wavefunctions, rather than the traditional wave mechanics approach.
Those who first master this methodology will secure highly promising prospects, whether seeking industry employment or advancing to graduate school.
Today's post marks the introduction to quantum mechanics based on linear algebra.
Basic quantum mechanics utilizing the algebraic approach is recently being most closely applied to physical AI, quantum information, and quantum computing.
I hope this material provides assistance in learning operators through examples.
This material is not a conventional linear algebra text for a mathematics department.
Since linear algebra is a crucial discipline that enables rapid access to quantum mechanics, quantum computing, physical AI, and quantum information science, maximal effort has been exerted to ensure the comprehension of certain elements of functional analysis as well.
I hope this provides substantial assistance.
As the first proof of de Broglie's matter waves—demonstrating that particles also possess wave properties—this experiment is invariably covered at least once at the beginning of quantum mechanics.
Undergraduate quantum mechanics using operators is often too basic, while graduate-level books (like Sakurai) have a very high barrier to entry. To help everyone easily learn advanced topics like angular momentum, spin, quantum information, quantum computing, and physical AI, I have carefully updated and improved the existing lecture materials. I hope you find this very helpful for your studies.
This material focuses on quantum mechanics within the framework of operator-based linear algebra.
As of 2026, linear algebra-based quantum mechanics is rapidly becoming an essential requirement across both industry and academia—driven by its critical roles in quantum computing, quantum information science, Physical AI, and the development of 2-nanometer semiconductor processes.
I have made every effort to break down the complex concepts of functional analysis and explain them as simply as possible.
I hope this material proves highly useful to your studies.