Going from 1+1=2 to Quantum Mechanics

Let

Variable

Matrix
Matrix Addition

Ket (State)

Angular Frequency
Superposition
Wave-particle Duality

Element
In
Intersection
Partitions
Set
Standard Number System
String
Subset
Union

Function
Addition
Summation

Differentiation
Integration
Limit
Taylor Series

Absolute Value of Complex Number
Complex Number
Complex Number Decomposition
Conjugate
Euler Form

Basis State
Bra-ket
Cases
Completeness
Gram-Schmidt Procedure
Inner product
Kronecker Delta
Linear Independence
Magnitude
Matrix Multiplication
Normal
Operator
Orthogonality
Orthonormality
Square Matrix
Symmetric Matrix
Transpose
Vector
Vector Space

Hilbert Space
Observable
Particle Energy
Particle Phase
Planck-Einstein Relation
Reduced Planck constant

Interference
Quantum Dynamics
Quantum System
Schrödinger equation, Hamiltonian I
Spin-1 System
Spin-1/2 System

Adjoint
Anti-Hermitian Operator
Characteristic Polynomial
Commutator
Determinant
Diagonalization
Eigenbasis
Eigenspace
Eigenstate
Eigenvector/Eigenvalue
Hermitian Matrix
Hermitian Operator
Invariant Subspace
Matrix Exponential
Matrix Invertibility
Normal Operator
Positive Semi-Definite (PSD) Operator
Projection
Projector
Span
Spectral Decomposition
Subspace
Trace
Unitary Operator

Trigonometry
Complex Number Trigonometry

Qubit
Bloch Sphere
Degeneracy
Observable on a qubit
Pauli Matrices

Probability Theory
Cauchy-Schwarz Inequality
Conditional Probability
Definite
Density Matrix
Expected Value of an Observable
Gaussian Distribution
Joint Random Variables
Numerical Random Variable/Expected Value
Poisson Distribution
Spread
Variance

Born Rule
Heisenberg Uncertainty Relation
Measuring a Quantum State
Probability amplitude

Baker, Campbell, and Hausdorff (1897-1906)
Cross Product
Kronecker Product
Tensor
Tensor Product
Tensor Product Space

Basic Decoding Theory
Basic Distinguishability Theory
Holevo's Theorem (Holevo, 1973) - Quantum Information Capacity
No-Cloning Theorem

Bennet and Brassard (1984) Protocol
Classical Cryptography

Compatibility
Compatibility of Observables
Ehrenfest Theorem
Example of Finding Hamiltonian
Example of Quantum Dynamics
Hamiltonian Rotation Quirk
Heisenberg Picture (Heisenberg, 1925)
Isolated System
Schrödinger equation, Hamiltonian II
Time Energy Uncertainty
Uniform dynamics
Unitary Evolution

Bell States
Bell's Theorem
Clauser, Horne, Shimony, and Holt (1969)
Composite System
Einstein, Podolsky, and Rosen (1935)
Entanglement
Hamiltonian on a Composite System
Local Realism
Measuring a Composite System
Neutron Interferometry
No Communication Theorem
Observables on a Composite System
Singlet
States on a Composite System

Bar-Yossef, Jayram, and Kerenidis (2004)
Greenberger, Horne, and Zeilinger (1989)
Quantum Teleportation

Continuous Identity
Continuous Operator
Continuous Orthonormality
Continuous Time Fourier Transform
Convolution
Dirac Delta
Non Denumerable Basis

de Broglie
Degrees of Freedom
Gaussian Wave Packet
Momentum Operator
Physical Wave Functions
Position Operator
Probability Density
Time-Independent Schrödinger Equation
Wave function
Wave Packet
Wave Packet Momentum

Double Slit Experiment
Particle in 1d
Particle in Box
Tunnelling

Quantum Harmonic Oscillator
Classic Harmonic Oscillator
Ladder of States
Number Operator
QHO Coherent States
QHO Observables
QHO Quantum Dynamics
QHO States killed
QHO Wave Function

Probability Current
Continuity Equation
Evanescent
Even Odd Functions
Flow of Probability
Quantum Tunneling
Stationary States
Stationary States 1d
Transmission Reflection Coefficients
Variational Method

Probability Current Problem 1
Probability Current Problem 2
Probability Current Problem 3
Probability Current Problem 4

Classical Computer
Quantum Computer
Quantum Gate Cnot
Quantum Gate Controlled u
Quantum Gate Controlled z
Quantum Gate Hadamard
Quantum Gate Not
Quantum Gate Rotate
Quantum Gate Swap
Universality Theorem

Going from 1+1=2 to Quantum MechanicsChaimongkol, 2026

Intersection

The intersection of two Sets $A$ and $B$ is the set of all elements that are in $A$ and $B$ .

A\cap B=\{x\in A \text{ and } x\in B\}