Redshift Academy

Wolfram Alpha:         

  Search by keyword:  

Astronomy

-
Astronomical Distance Units .
-
Celestial Coordinates .
-
Celestial Navigation .
-
Location of North and South Celestial Poles .

Chemistry

-
Avogadro's Number .
-
Balancing Chemical Equations
-
Stochiometry
-
The Periodic Table .

Classical Mechanics

-
Blackbody Radiation .

Classical Physics

-
Archimedes Principle
-
Bernoulli Principle
-
Center of Mass Frame
-
Comparison Between Gravitation and Electrostatics
-
Compton Effect .
-
Coriolis Effect
-
Cyclotron Resonance
-
Dispersion
-
Doppler Effect
-
Double Slit Experiment
-
Elastic and Inelastic Collisions .
-
Electric Fields
-
Error Analysis
-
Fick's Law
-
Fluid Pressure
-
Gauss's Law of Universal Gravity .
-
Gravity - Force and Acceleration
-
Hooke's law
-
Ideal and Non-Ideal Gas Laws (van der Waal)
-
Impulse Force
-
Inclined Plane
-
Inertia
-
Kepler's Laws
-
Kinematics
-
Kinetic Theory of Gases .
-
Kirchoff's Laws
-
Maxwell's Equations .
-
Moments and Torque
-
Nuclear Spin
-
One Dimensional Wave Equation .
-
Pascal's Principle
-
Phase and Group Velocity
-
Poiseuille's Law
-
Refractive Index
-
Rotational Dynamics
-
Simple Harmonic Motion
-
Specific Heat, Latent Heat and Calorimetry
-
The Gas Laws
-
The Laws of Thermodynamics
-
The Zeeman Effect .
-
Young's Modulus

Climate Change

-
Keeling Curve .

Cosmology

-
Baryogenesis
-
Cosmic Background Radiation and Decoupling .
-
CPT Symmetries
-
Dark Matter .
-
Friedmann-Robertson-Walker Equations .
-
Hubble's Law
-
Inflation Theory
-
Introduction to Black Holes .
-
Olbers' Paradox .
-
Planck Units .
-
Stephen Hawking's Last Paper .
-
Stephen Hawking's PhD Thesis .
-
The Big Bang Model
-
Vacuum Energy .

Finance and Accounting

-
Amortization
-
Annuities
-
Brownian Model of Financial Markets .
-
Capital Structure
-
Dividend Discount Formula
-
Lecture Notes on International Financial Management
-
NPV and IRR
-
Periodically and Continuously Compounded Interest
-
Repurchase versus Dividend Analysis

Game Theory

-
The Truel .

General Relativity

-
Basis One-forms .
-
Catalog of Spacetimes .
-
Curvature and Parallel Transport
-
Einstein's Field Equations
-
Geodesics
-
Gravitational Waves
-
Hyperbolic Motion and Rindler Coordinates .
-
Quantum Gravity
-
Ricci Decomposition .
-
Ricci Flow .
-
Stress-Energy-Momentum Tensor .
-
Tensors
-
The Area Metric
-
The Dirac Equation in Curved Spacetime .
-
The Equivalence Principal
-
The Essential Mathematics of General Relativity
-
The Induced Metric
-
The Light Cone .
-
The Metric Tensor .
-
The Principle of Least Action in Relativity .
-
Vierbein (Frame) Fields

Group Theory

-
Basic Group Theory .
-
Basic Representation Theory .
-
Building Groups From Other Groups .
-
Sets, Groups, Modules, Rings and Vector Spaces
-
Symmetric Groups .
-
The Integers Modulo n Under + and x .

Lagrangian and Hamiltonian Mechanics

-
Classical Field Theory .
-
Euler-Lagrange Equation .
-
Ex: Newtonian, Lagrangian and Hamiltonian Mechanics .
-
Hamiltonian Formulation .
-
Liouville's Theorem
-
Symmetry and Conservation Laws - Noether's Theorem .

Macroeconomics

-
Lecture Notes on International Economics
-
Lecture Notes on Macroeconomics
-
Macroeconomic Policy

Mathematics

-
Amplitude, Period and Phase
-
Arithmetic and Geometric Sequences and Series .
-
Asymptotes
-
Augmented Matrices and Cramer's Rule
-
Binomial Theorem (Pascal's Triangle)
-
Completing the Square
-
Complex Numbers
-
Composite Functions
-
Conformal Transformations .
-
Conjugate Pair Theorem
-
Contravariant and Covariant Components of a Vector
-
Derivatives of Inverse Functions
-
Double Angle Formulas
-
Eigenvectors and Eigenvalues
-
Euler Formula for Polyhedrons
-
Factoring of a3 +/- b3
-
Fourier Series and Transforms .
-
Fractals
-
Gauss's Divergence Theorem
-
Grassmann and Clifford Algebras .
-
Heron's Formula
-
Index Notation (Tensors and Matrices) .
-
Inequalities
-
Integration By Parts
-
Introduction to Conformal Field Theory .
-
Inverse of a Function
-
Law of Sines and Cosines
-
Line Integrals, ∮
-
Logarithms and Logarithmic Equations
-
Matrices and Determinants
-
Matrix Exponential
-
Mean Value and Rolle's Theorem
-
Modulus Equations
-
Orthogonal Curvilinear Coordinates .
-
Parabolas, Ellipses and Hyperbolas
-
Piecewise Functions
-
Polar Coordinates
-
Polynomial Division
-
Quaternions 1 .
-
Quaternions 2 .
-
Regular Polygons
-
Related Rates
-
Similar Matrices and Diagonalization .
-
Spherical Trigonometry
-
Stirling's Approximation
-
Sum and Differences of Squares and Cubes
-
Symbolic Logic
-
Tangent and Normal Line
-
Taylor and Maclaurin Series .
-
The Essential Mathematics of Lie Groups
-
The Limit Definition of the Exponential Function
-
Tic-Tac-Toe Factoring
-
Trapezoidal Rule
-
Unit Vectors
-
Volume Integrals

Microeconomics

-
Marginal Revenue and Cost

Nuclear Physics

-
-
Radioactive Decay

Particle Physics

-
Feynman Diagrams and Loops
-
Field Dimensions
-
Helicity, Chirality and Weyl Spinors .
-
Klein-Gordon and Dirac Equations .
-
Regularization and Renormalization
-
Scattering - Mandelstam Variables
-
Spin 1 Eigenvectors .

Probability and Statistics

-
Box and Whisker Plots
-
Buffon's Needle .
-
Categorical Data - Crosstabs
-
Chebyshev's Theorem
-
Chi Squared Goodness of Fit
-
Conditional Probability
-
Confidence Intervals
-
Data Types
-
Expected Value
-
Factor Analysis
-
Hypothesis Testing
-
Linear Regression
-
Monte Carlo Methods
-
Non Parametric Tests
-
One-Way ANOVA
-
Pearson Correlation
-
Permutations and Combinations
-
Pooled Variance and Standard Error
-
Probability Distributions
-
Probability Rules
-
Sample Size Determination
-
Sampling Distributions
-
Set Theory - Venn Diagrams
-
Stacked and Unstacked Data
-
Stem Plots, Histograms and Ogives
-
Survey Data - Likert Item and Scale
-
Tukey's Test
-
Two-Way ANOVA

Programming and Computer Science

-
Hashing
-
How this site works ...
-
More Programming Topics
-
MVC Architecture
-
Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
-
Public Key Encryption

Quantum Computing

-
Density Operators and Mixed States .
-
Entangled States .
-
The Qubit .

Quantum Field Theory

-
Creation and Annihilation Operators
-
Field Operators for Bosons and Fermions
-
Lagrangians in Quantum Field Theory
-
Path Integral Formulation
-
Relativistic Quantum Field Theory

Quantum Mechanics

-
Bohr Atom
-
Clebsch-Gordan Coefficients .
-
Commutators
-
Dyson Series
-
Electron Orbital Angular Momentum and Spin
-
Heisenberg Uncertainty Principle
-
Ladder Operators .
-
Multi Electron Wavefunctions .
-
Pauli Spin Matrices
-
Photoelectric Effect .
-
Position and Momentum States .
-
Probability Current
-
Schrodinger Equation for Hydrogen Atom .
-
Schrodinger Wave Equation
-
Spin 1/2 Eigenvectors
-
The Differential Operator
-
The Essential Mathematics of Quantum Mechanics
-
The Quantum Harmonic Oscillator .
-
The Schrodinger, Heisenberg and Dirac Pictures
-
The WKB Approximation
-
Time Dependent Perturbation Theory
-
Time Evolution and Symmetry Operations
-
Time Independent Perturbation Theory
-
Wavepackets

Semiconductor Reliability

-
The Weibull Distribution

Solid State Electronics

-
Band Theory of Solids .
-
Fermi-Dirac Statistics .
-
Intrinsic and Extrinsic Semiconductors .
-
The MOSFET
-
The P-N Junction

Special Relativity

-
4-vectors .
-
Electromagnetic (Faraday) Tensor .
-
Energy and Momentum in Special Relativity, E = mc2 .
-
Invariance of the Velocity of Light .
-
Lorentz Invariance .
-
Lorentz Transform .
-
Lorentz Transformation of the EM Field .
-
Newton versus Einstein
-
Spinors - Part 1 .
-
Spinors - Part 2 .
-
The Continuity Equation .
-
The Lorentz Group .

Statistical Mechanics

-
Entropy and the Partition Function
-
The Harmonic Oscillator
-
The Ideal Gas

String Theory

-
Bosonic Strings
-
Extra Dimensions
-
Introduction to String Theory
-
Kaluza-Klein Compactification of Closed Strings
-
Strings in Curved Spacetime
-
Toroidal Compactification

Superconductivity

-
Bardeen–Cooper–Schrieffer Theory
-
BCS Theory
-
Cooper Pairs
-
Introduction to Superconductivity .
-
Superconductivity (Lectures 1 - 10)
-
Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

-
Chiral Superfields
-
Generators of a Supergroup
-
Grassmann Numbers
-
Introduction to Supersymmetry
-
The Gauge Hierarchy Problem

The Standard Model

-
Electroweak Unification (Glashow-Weinberg-Salam)
-
Gauge Theories (Yang-Mills)
-
Gravitational Force and the Planck Scale
-
Introduction to the Standard Model
-
Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
-
Quantum Flavordynamics and Quantum Chromodynamics
-
Special Unitary Groups and the Standard Model - Part 1 .
-
Special Unitary Groups and the Standard Model - Part 2
-
Special Unitary Groups and the Standard Model - Part 3 .
-
Standard Model Lagrangian
-
The Higgs Mechanism
-
The Nature of the Weak Interaction

Topology

-

Units, Constants and Useful Formulas

-
Constants
Last modified: January 19, 2022 ✓

Entangled States ---------------- Consider 2 electrons that are physically located at different points in space. Because they are at different places, the Pauli exclusion principle is not violated. For each electron we can write their spin wavefunctions as: |ψA> = α|uA> + β|dB> |ψB> = γ|uA> + δ|dB> Each of these states describes a qubit. The states |u> and |d> are called PURE states. The superposition of these states is also a pure state. Also, a superposition is said to be coherent if there is an operator that, if applied to one state, can turn it into another that is also present in the superposition. In the above case σx(1/√2)|u> = (1/√2)|d> and vice versa so |ψ> is a coherent superposition. If a system is a superposition of states it exists simultaneously in each of these states until a measurement is made. The joint state of the 2 qbits can be expressed as the tensor product of the respective individual qubits. Thus, |ψAB> = (α|uA> + β|dA>) ⊗ (γ|uB> + δ|dB>) = αγ|uAuB> + αδ|uAdB> + βγ|dAuB> + βδ|dAdB> ... (1) Where α2γ2 + α2δ2 + β2γ2 + β2δ2 = 1 States belonging to a composite space that can be factored into individual states belonging to separate subspaces are also referred to as SEPARABLE states. Separable states correspond to the situation in which each electron is prepared independently, and can be measured independently. This means that, for a separable state, there is always a direction along which you can measure the spin of the first electron to be ±1 with 100% certainty, and there is always a direction along which you will measure the spin of the second electron to be ±1 with 100% certainty. We can define a general state for 2 qubits as follows: |ψAB> = a|uAuB> + b|uAdB> + e|dAuB> + f|dAdB> ... (2) As it stands, this state cannot be factored into (1) unless a = αγ, b = αδ, e = βγ, f = βδ and the condition af - be = 0 is enforced. In general, however, this is not the case. A state that cannot be factored (is not separable) is referred to as an ENTANGLED state. Entanglement requires coherence, but coherence does not imply entanglement. Note that a qubit represents the basic state of a single spin 1/2 particle and is therefore neither separable nor entangled. One other point to note is that (1) can be written as αγ(|uu> + (δ/γ)|ud> + (β/α)|du> + (β/α)(δ/γ)|dd> indicating 3 degrees of freedom (αγ, β/α, δ/γ) to describe this state. Whereas (2) has 4 degrees of freedom. A consequence of this is that there are states where the 2 individual particles are not independent of one another. Now focus on the following state (the singlet): |ψ> = 1/√2{|uAdB> - |dAuB>} This state is not separable. It is easy to see this since it would imply that αγ = 0, αδ = 1/√2, βγ = 1/√2 and βδ = 0, which is not possible. For any direction we can show that the expectation values(averages) of the associated spins are zero, meaning that measurements along this direction are equally likely to be +1 or -1. First we need to discuss basis states. Basis States ------------ The eigenvectors of the σ matrices are: σx: - - - - - - | 0 1 || 1 | = +1| 1 | | 1 0 || 1 | | 1 | - - - - - - - - - - - - | 0 1 || 1 | = -1| 1 | | 1 0 || -1 | | -1 | - - - - - - σy: - - - - - - | 0 -i || -i | = +1| -i | | i 0 || 1 | | 1 | - - - - - - - - - - - - | 0 -i || i | = +1| i | | i 0 || 1 | | 1 | - - - - - - σz: - - - - - - | 1 0 || 1 | = +1| 1 | | 0 -1 || 0 | | 0 | - - - - - - - - - - - - | 1 0 || 0 | = -1| 0 | | 0 -1 || 1 | | 1 | - - - - - - The basis states |u> and |d> are chosen to be the eigenvectors of σz. - - - - |u> = | 1 | and |d> = | 0 | | 0 | | 1 | - - - - - - - - - - - - σx| 1 | = | 0 | σx| 0 | = | 1 |   | 0 | | 1 |   | 1 | | 0 | - - - - - - - - - - - - - - - - σy| 1 | = | 0 | σy| 0 | = | -i |   | 0 | | i |   | 1 | | 0 | - - - - - - - - - - - - - - - - σz| 1 | = | 1 | σz| 0 | = | 0 |   | 0 | | 0 |   | 1 | | -1 | - - - - - - - - Consider 2 qubits and different spin matrices, σ and τ. σ acts on the first electron and does nothing to the second electron. τ does the opposite. For electron A we can write: σx|uA> = |dA> σx|dA> = |uA> σy|uA> = i|dA> σy|dA> = -i|uA> σz|uA> = |uA> σz|dA> = -|dA> - - - - Where, |u> = | 1 | and |d> = | 0 | | 0 | | 1 | - - - - Similarly, for electron B we can write: τx|uB> = |dB> τx|dB> = |uB> τy|uB> = i|dB> τy|dB> = -i|uB> τz|uB> = |uB> τz|dB> = -|dB> Recall, that the expectation value, <>, of an operator is given by <ψAB|σ|ψAB>. For the singlet: <σx> = <(<uAdB| - <dAuB|)(1/√2)|σx|(1/√2)(|uAdB - |dAuB>)> = (1/2)(<uAdB| - <dAuB|)(|dAdB> - |uAuB>) = 0 <σy> = <(<uAdB| - <dAuB|)(1/√2)|σy|(1/√2)(|uAdb - |dAuB>)> = (1/2)(<uAdB| - <dAuB|)(i|dAdB> + i|uAuB>) = 0 <σz> = <(<uAdB| - <dAuB|)(1/√2)|σz|(1/√2)(|uAdb - |dAuB>)> = (1/2)(<uAdB| - <dAuB|)(|uAdB> + |dAuB>) = 0 Likewise, if we did the same analysis for τ we would get the same result. Therefore, in the entangled situation there is no direction of definite spin for either A's or B's electron (i.e. <σx> = <τx> = <σy> = <τy> = <σz> = <τz> = 0). For example, if the system is prepared in the +z direction, A will never be able measure the spin of her electron in the +z direction and obtain +1 with 100% certainty. Instead, she will measure +1 50% of the time and -1 50% of the time. B will encounter exactly the same situation with his electron. Because of this they both conclude that their electrons must be in an entangled state. A’s measurement will cause the wave function to collapse. If A finds her electron to be spin-up then B will find his electron to be spin down i.e. For the singlet state prior to measurement, P(|uA>) = P(|uB>) = (1/√2)2 = 1/2 After measurement the system collapses into the normalized eigenstate: 1|uAdB> + 0|dAuB> With P(|uA>) = 12 = 1 and P(|uB>) = 02 = 0. We can compare this result to the one obtained for the separable state (1/2(|uA> + |dA>) ⊗ (|uB> + |dB>). We get: <σx> = <(1/2)(|uAuB> + |uAdB> + |dAuB> + |dAdB>) |σx|(1/2)(|uAuB> + |uAdB> + |dAuB> + |dAdB>) = (1/2)(|uAuB> + |uAdB> + |dAuB> + |dAdB>) (1/2)(|dAuB> + |dAdB> + |uAuB> + |uAdB>) = (1/4)(4) = 1 <σy> = <(1/2)(|uAuB> + |uAdB> + |dAuB> + |dAdB>) |σy|(1/2)(|uAuB> + |uAdB> + |dAuB> + |dAdB>) = (1/2)(|uAuB> + |uAdB> + |dAuB> + |dAdB>) (1/2)(i|dAuB> + i|dAdB> - i|uAuB> - i|uAdB>) = 0 <σz> = <(1/2)(|uAuB> + |uAdB> + |dAuB> + |dAdB>) |σz|(1/2)(|uAuB> + |uAdB> + |dAuB> + |dAdB>) = (1/2)(|uAuB> + |uAdB> + |dAuB> + |dAdB>) (1/2)(|uAuB> + |uAdB> - |dAuB> - |dAdB>) = 0 Likewise, if we did the same analysis for τ we would get the same result. Therefore, if a system of qubits is prepared in the +z direction and A's electron is independent, she would measure the spin in the +z direction to be +1 (i.e. <σx> = 1, <σy> = <σz> = 0). Likewise for Bob's electron (i.e. <τx> = 1, <τy> = <τz> = 0). We could also show that if the system of qubits is prepared and measured in the ±x direction we get: <σx> = (1/2)((<ψxxx) = +1 <σx> = (1/2)((<ψ-xx-x) = -1 <σy> = <σz> = 0 and so on. For the separable state: |ψ> = (1/√2)(|uA> + |dA>) ⊗ (1/√2)(|uB> + |dB>) = (1/2)|uAuB> + (1/2)|uAdB> + (1/2)|dAuB> + (1/2)|dAdB> (Normalization: (1/2)2 + (1/2)2 + (1/2)2 + (1/2)2 = 1) For the separable state prior to measurement, P(|uA>) = P(|uB>) = (1/2)2 + (1/2)2 = 1/2 After measurement by A the system collapses into the normalized eigenstate: (1/√2)|uAuB> + (1/√2)|uAdB> + 0|dAuB> + 0|dAdB> ∴ P(|uA>) = (1/√2)2 = 1/2 After measurement by B the system collapses into the normalized eigenstate: (1/√2)|uAuB> + 0|uAdB> + (1/√2)|dAuB> + 0|dAdB> ∴ P(|uB>) = (1/√2)2 = 1/2 We have focussed on the singlet state in our analysis but there are 3 other entangled states of particular importance. These 3 states together with the singlet are called the BELL STATES: |Φ+> = 1/√2{|uAuB> + |dAdB>} |Φ-> = 1/√2{|uAuB> - |dAdB>} |Ψ+> = 1/√2{|uAdB> + |dAuB>} |Ψ-> = 1/√2{|uAdB> - |dAuB>} - singlet It should be pointed out that for the Φ states A and B's measurements will result in identical spins whereas in the Ψ states, measurements will result in opposite spins. Einstein - Podolsky - Rosen Correlation --------------------------------------- The curious fact is that A and B's electrons will remain entangled regardless of the separation between them unless either is 'disturbed' by a measurement. Once a measurement has been made, the state of the original entangled system collapses and the entanglement is destroyed. The fact that the spin of B's electron instantly "knows" what spin it is supposed to take on (and vice versa), seemingly involves communication between the two particles at speeds greater than the speed of light, which is in conflict with Einstein's special theory of relativity (violation of locality). Einstein referred to the phenomenon as 'spooky; action at a distance and supported an alternative idea called 'hidden variables theory' which suggested that quantum mechanics was incomplete. However, later on Bell's theorem would suggest that the existance of local hidden variables is impossible. Hidden Variables and Bell's Theorem ----------------------------------- Bell's theorem showed that if local hidden variables existed, certain experiments could be performed where the result would satisfy a Bell inequality. If, on the other hand, if Bell's inequality was violated local hidden variables could not exist. Bell's Theorem -------------- Let N be the number of objects. N(A not B) + N(B not C) ≥ N(A not C) Therefore, N(1 + 2) + N(4 + 7) ≥ N(1 + 4) Therefore, N(1 + 4) + N(2 + 7) ≥ N(1 + 4) So that N(2 + 7) ≥ 0 This is generaly written as: N(A,~B) + N(B,~C) ≥ N(A,~C) Let ↑↗→ be possible measurement directions for A and B's electrons. Interpret P(↑,↗) as the probability A finds her electron in the ↑ direction and B finds his electron in the ↗ direction which, for the singlet, is equivalent to the ~↗ = ↙ direction for A. Returning to the above Venn diagram, we can define A = ↑, B = ↗ and C = →. Substituting into Bell's theorem gives: N(↑,↗) + N(↗,→) ≥ N(↑,→) We can equate N with a probability (i.e. P = n/NTotal). To compute these probabilities it is helpful to use the projection operators for spin. These can be written in terms of the σ matrices as follows: - - - - - - ℙ = (I + σz)/2 = | 1 0 | + | 1 0 | = | 1 0 |   | 0 1 | | 0 -1 | | 0 0 | - - - - - - ℙ = (I/2 + (σx + σ2)/2√2 - - - - - - ℙ = (I + σx)/2 = | 1 0 | + | 0 1 | = | 1/2 1/2 |    | 0 1 | | 1 0 | | 1/2 1/2 | - - - - - - These act on electron 1. There is an equivalent set of operators that act on electron 2: For ℙ: - - - - - - - - - - - - | 1 0 || 1 | = | 1 | and | 1 0 || 0 | = | 0 | | 0 0 || 0 | | 0 | | 0 0 || 1 | | 0 | - - - - - - - - - - - - Therefore, ℙ|u> = |u> and ℙ|d> = |0> For ℙ: - - - - - - - - - - - - | 1/2 1/2 || 1 | = | 1/2 | and | 1/2 1/2 || 0 | = | 1/2 | | 1/2 1/2 || 0 | | 1/2 | | 1/2 1/2 || 1 | | 1/2 | - - - - - - - - - - - - Therefore, ℙ|u> = |u> and ℙ|d> = |0> For ℙ: - - - - - - (1/2√2)| √2+1 1 || 1 | = (1/2√2)| √2+1 | | 1 √2-1 || 0 | | 1 | - - - - - - and, - - - - - - (1/2√2)| √2+1 1 || 0 | = | 1 | | 1 √2-1 || 1 | | √2-1 | - - - - - - The probabilities can then be computed as: P = <ψ|ℙ12|ψ> Return to the singlet state and compute the probability P(↑,↗). <ψ|(I/2 + σz/2)(I/2 + (τx + τz)/2√2)|ψ> Breaking it down piece by piece: ℙ|(1/√2)(|ud> - |du>) = (1/√2)|ud> ℙ|(1/√2)|ud>)> = (1/√2){|u> ⊗ ℙ|d>} = (1/√2){|u> ⊗ {1|u> + (√2-1)/2√2|d>} = (1/√2){|uu> ⊗ (√2-1)/2√2|ud>} P(↑,↗) = <(<ud| - <du|)(1/√2)|(1/√2){|uu> ⊗ (√2-1)/2√2|ud>} = <ud|(1/√2)(1/√2)(√2-1)/2√2|ud> = (1/2)(√2-1)/2√2 = 0.075 Likewise, compute the probability P(↗,→). <ψ|(I/2 + (σx + σz)/2√2)(I/2 + τx/2)|ψ> P(↗,→) = 0.075 Finally, for P(↑,→). <ψ|(I/2 + σz/2)(I/2 + τx/2)|ψ> <(<ud| - <du|)(1/√2)|ℙ|(1/√2)(|ud> - |du>) P(↑,→) = 0.25 P(↑,↗) + P(↗,→) ≱ P(↑,→) There is another way to look at this. A B ↑↗→ ↑↗→ === === N1 +++ --- + = aligned, - = counter aligned N2 ++- --+ N3 +-+ -+- N4 +-- -++ N5 -++ +-- N6 -+- +-+ N7 --+ ++- N8 --- +++ Experimentally, we can compute the probabilities as follows: P(↑,↗) = (N3 + N4)/NT P(↗,→) = (N2 + N6)/NT P(↑,→) = (N2 + N4)/NT P(↑,↗) + P(↗,→) ≥ P(↑,→) ∴ (N3 + N4)/NT + (N2 + N6)/NT ≥ (N2 + N4)/NT ∴ (N3 + N4) + (N2 + N6) ≥ (N2 + N4) ∴ (N2 + N4) + (N3 + N6) ≥ (N2 + N4) Components of Spin ------------------ The eigenvectors of the Pauli matrices represents an observable describing the spin of a spin 1/2 particle in the three spatial directions. - - - - - - σz: | 1 0 | => | 1 |, | 0 |   | 0 -1 | | 0 | | 1 | - - - - - - - - - - - - σx: | 0 1 | => (1/√2)| 1 |, (1/√2)| 1 |   | 1 0 | | 1 | | -1 | - - - - - - - -      - - - - | 1 | = (1/√2)(χ+x + χ-x) = (1/2)| 1 | + (1/2)| 1 | | 0 |     | 1 | | -1 | - -      - - - - - - - - - - σy: | 0 -i | => (1/√2)| 1 |, (1/√2)| 1 |   | i 0 | | i | | -i | - - - - - - - -      - - - - | 1 | = (1/√2)(χ+y + χ-y) = (1/2)| 1 | + (1/2)| 1 | | 0 |     | i | | -i | - -      - - - - Therefore, P(|χ+x>) = P(|χ+y>) = 1/2 Alternatively, we can look at this as the overlap of χ+x with χ+z. Then: P(χ+x) = <u+z|u+x><u+z|u+x>* = |<u+z|u+x>|2      - - - - <u+z|u+x> = | 1 0 || 1/√2 | = 1/√2      - - | 1/√2 |      - - So, P(χ+x) = (1/√2)(1/√2) = 1/2 Therefore, if you line up spin in any direction and measure an orthogonal component, it will have an equal probability of being up or down. We can generalize this to find the components of spin in any direction when the electron is line up along the +z axis. Let n be a unit vector with components nx, ny, nz that represents the measurements axis. Therefore, σ.n represents the component of σ along n, i.e. σ.n = σxnx + σyny + σznz - - - - - - = | 0 nx | + | 0 -iny | + | nz 0 | | nx 0 | | iny 0 | | 0 -nz | - - - - - - - - = |nz nx-iny| |nx+iny -nz | - - - - = |nz n-| where n- = nx - iny and n+ = nx + iny |n+ -nz| - - This is Hermitian with eigenvalues of +/-1. Now, (σ.n)2 = (σxnx + σyny + σznz)(σxnx + σyny + σznz)   = nx2 + ny2 + nz2 + nxnyxσy + σyσx)   + nxnzxσz + σzσx) + nynzyσz + σzσy)   = nx2 + ny2 + nz2   = 1 (since σiσj = -σjσi) Check: If the electron spin is lined up along the +z axis and measured in the x direction we get nz = 1 and nx = ny = 0 which leaves us with the σz matrix and a positive eigenvector equal to: - - | 1 | | 0 | - - Then the probability, Px = <u+z|u+x><u+z|u+x>* = |<u+z|u+x>|2      - - - - <u+z|u+x> = | 1 0 || 1/√2 | = 1/√2      - - | 1/√2 |      - - So, P+x = (1/√2)(1/√2) = 1/2 as before. The General Case ---------------- Now, what about the situation where we prepare the spin in direction n and measure it in direction m where the angle between m and n is θ. m n \ / \ / \ θ / \ / o Let |σ.n=1> be an eigenvector of σ.n with eigenvalue +1 that satisfies: σ.n|σ.n=1> = +1|σ.n=1> This is interpreted as preparing the electron spin in the up state along n and then measuring the component of the spin along the same axis. Likewise, for direction, m: σ.m|σ.m=1> = 1|σ.m=1> Note: For the spin down case we would write: |σ.n=-1> and σ.n|σ.n=-1> = -1|σ.n=-1> The eigenvectors of σ.n=1 can be found from: - - - - - - |nz n-|| α | = +1| α | |n+ -nz|| β | | β | - - - - - - From which we get nzα + n-β = α n+α - nzβ = β These equations are equivalent. Solving for β gives: n+α/(1 + nz) = α(1 - nz)/n- ∴ n+/(1 + nz) = (1 - nz)/n- ∴ n+n- = 1 - nz2 ∴ nx2 + ny2 + nz2 = 1 Choose β = α(1 - nz)/n- Normalize: -   - - - | α α(1 - nz/n+)|| α    | -   - | α(1 - nz)/n- |   - - = α2 + α2(1 - nz)(1 - nz)/n+n- = α2((n+n- + 1 - 2nz + nz2)/n+n-) = α2((1 - nz2 + 1 - 2nz + nz2)/n+n-) = α2(2 - 2nz)/n+n-) = α22(1 - nz)/(1 - nz)2 = α22(1 - nz)/(1 - nz)(1 + nz) = 2α2/(1 + nz) So, to get the normalization factor, N, we need to find the inverse √ of this. Thus, we get: N = √{(1 + nz)/2α2} = (1/α)√{(1 + nz)2} So the eigenvector of σ.n=1 is: - - |σ.n=1> = (1/α)√{(1 + nz)/2}| α    |  | α(1 - nz)/n- | - - - - = | √{(1 + nz)/2}    | | √{(1 + nz)/2}{(1 - nz)/n-} | - - Now we can apply the same reasoning to get the eigenvector of σ.m=1: - - |σ.m=1> = | √{(1 + mz)/2}    | | √{(1 + mz)/2}{(1 - mz)/m-} | - - The probability of measuring the state σ.m=1 given that the electron has been prepared in the state σ.n=1 is: P(σ.n=1 σ.m=1) = |<σ.n=1|σ.m=1>|2 Which, after a lot of tedious math leads to, P(σ.n=1 σ.m=1) = (1 + nxmx + nymy + nzmz)/2 = (1 + n.m)/2 = (1 + cosθmn)/2 since |n| = |m| = 1 = cos2mn/2) (using trig identity) Alternative derivation using polar coordinates: nx = cosφsinθ ny = sinφsinθ nz = cosθ - - σ.n = | cosθ exp(-iφ)sinθ | λ = ±1 | exp(iφ)sinθ -cosθ | - - det(σ.n - λI)v = 0. For λ = +1 we get: - - - - | cosθ - 1 exp(-iφ)sinθ || α | = 0 | exp(iφ)sinθ -cosθ - 1 || β | - - - - ∴ β = exp(iφ)(1 - cosθ)α/sinθ = αexp(iφ)sin(θ/2)/cos(θ/2) and, β* = αexp(-iφ)sin(θ/2)/cos(θ/2) Normalization: α2 + β2 = α2 + β*β = α2(1 + sin2(θ/2)/cos2(θ/2)) = 1 ∴ α2 = 1/(1 + sin2(θ/2)/cos2(θ/2)) Using the identity sinθ = 2sin(θ/2)cos(θ/2) this becomes: α2 = cos2(θ/2) Which gives: α = cos(θ/2) and β = sin(θ/2) |+> = cos(θ/2)|u> + exp(iφ)sin(θ/2)|d> This represents the qubit in terms of the Bloch sphere. Likewise: |-> = sin(θ/2)|u> - exp(iφ)cos(θ/2)|d> Density Operator Formulation: ℙn = (I + σ.n)/2 - - - - = (1/2)| 1 0 | + (1/2)| cosθ sinθ | | 0 1 | | sinθ -cosθ | - - - - - - - - - - = (1/2)| 1 0 | + (1/2)| 0 sinθ | + (1/2)| cosθ 0 | | 0 1 | | sinθ 0 | | 0 -cosθ | - - - - - - - - = | cos2(θ/2) (1/2)sinθ | | (1/2)sinθ sin2(θ/2) | - - This is the probability to measure |σ.n=1> if the initial state is |σ.m=1> if the state is prepared in a direction θ ... |ψ> = (1/√2)|u> + (1/√)|d> - - ρ = (1/2)|u><u| = | 1/2 0 | | 0 0 | - - ∴ P(ℙn) = Tr(ρℙn) = (1/2)cos2(θ/2) - - ρ' = (1/2)|d><d| = | 0 0 | | 0 1/2 | - - ∴ P(ℙn) = Tr(ρ'ℙn) = (1/2)sin2(θ'/2) σ.n=1 | / |θ / | / |/ o |\ | \ |θ'\ | \ σ.n=-1 If an electron is prepared in a certain direction and measured in another, the result of the measurement will be either spin up or spin down. The interesting thing is that if the exact same experiment is repeated a second time, the result may or may not be the same. If the experiment is repeated multiple times, a probability distribution of the measured spins will be obtained. The above equation shows that the smaller the angle between the prepared state and the measured state, the greater the probability of spin up occurrences. For example if the prepared and measured state are in the same direction the probability will be 1 as expected. If the prepared state and the measured state are in opposite directions, then the probability will be 0 (θ = 180). If the prepared state and the measured state are orthogonal then the probability will be 1/2 (θ = 90) as previously found. The above is the result for a single electron. In the case of the singlet we need to modify this to account for the normalization factors of (1/√2). We get: P(σ.n=1 σ.m=1) = (1/2)|<σ.m=1|σ.n=1>|2 = (1/2)(1 + cosθ)/2 ≡ (1/2)cos2(θ/2) (trig identity) We can now find the probabilities as follows: P(↑↗) case: ↑| /↗ | / θ = π/4 |θ/ |/ o / / /~↗ Counterclockwise angle between ↑ and ~↗ = 5π/4 P(↑↗) = (1/2)cos2((5π/4)/2) = 0.075 P(↑→) case: ↑| | | θ = π/2 |θ ------o------ ~→ Counterclockwise angle between ↑ and ~→ = 3π/2 P(↑→) = (1/2)cos2((3π/2))/2 = 0.25 P(↗→) case: ↑| /↗ | / θ = π/4 |θ/ |/ ------o ~→ Counterclockwise angle between ~→ and ↗ = 5π/4 P(↗→) = (1/2)cos2((5π/4))/2 = 0.075 Clearly, the last inequality cannot be true and Bell's theorem is violated. The implication of this is that the hidden variables that says each particle carries with it all of the required information at the time of separation, and nothing needs to be transmitted from one particle to the other at the time of measurement, cannot be true. Understanding 'spooky' action at a distance is one of the major unsolved mysteries of Physics. The study of String Theory and the idea of special extra spatial dimensions may offer some hope in this regard.