• 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

Probability Density

When measuring a wave function

Pψ(x)=∣ψ(x)∣2P_\psi(x)=|\psi(x)|^2Pψ​(x)=∣ψ(x)∣2
Time-Independent Schrödinger Equation