• 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
  • Strings
  • 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
  • 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 Function
  • 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)
  • 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 (CTFT)
  • 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
  • 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
Going from 1+1=2 to Quantum MechanicsChaimongkol, 2026

Quantum System

A quantum system is defined by:

  1. A Hilbert space HHH

  2. A Hamiltonian H^\hat{H}H^ generating time evolution via the Schrödinger equation.

The Hilbert space’s dimensions defines the fixes the spin of the system. It evolution through time is described by quantum dynamics

Schrödinger equation, Hamiltonian I