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Classical Physics

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Special Relativity

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Statistical Mechanics

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String Theory

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Superconductivity

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Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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test

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test

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Special Unitary Groups and the Standard Model - Part 1
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
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Formulas
Last modified: January 26, 2018

Entangled States ---------------- Consider 2 electrons 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 1 we can write: σx|u> = |d> σx|d> = |u> σy|u> = i|d> σy|d> = -i|u> σz|u> = |u> σz|d> = -|d> - - - - Where, |u> = | 1 | and |d> = | 0 | | 0 | | 1 | - - - - Similarly, for electron 2 we can write: τx|u> = |d> τx|d> = |u> τy|u> = i|d> τy|d> = -i|u> τz|u> = |u> τz|d> = -|d> Examples: σx|ud> = |dd> σx|dd> = |ud> σz|ud> = |ud> σz|dd> = -|dd> Likewise τy|ud> = -i|uu> We can define a general state for 2 electrons as a linear superposition as follows: α|uu> + β|ud> + γ|du> + δ|dd> such that: αα* + ββ* + γγ* + δδ* = 1 At first sight it might seem that only 4 are required to describe such a state. However, this is not true. Consider: α, β, γ, δ are complex => 8 variables. α2 + β2 + γ2 + δ2 = 1 => 7 variables Phase doesn't matter => 6 variables So in fact 6 variables are needed to describe the state. This is our first insight into the concept of ENTANGLED STATES. To explore this further consider 2 electrons that are physically located at different points in space. Because they are at different places, the Pauli exclusion principle does not come into play. For each electron we can write: state 1 = α1|u> + β1|d> state 2 = α2|u> + β2|d> The joint state of the 2 electrons can be expressed as the product of the respective individual electron states. Thus, (α1|u> + β1|d>)(α2|u> + β2|d>) = α1α2|uu> + α1β2|ud> + β1α2|du> + β1β2|dd> We only need four real numbers (α1α2, α1β2 etc) to describe a product state, as opposed to six for the general state, hence this product state is not entangled. This is because the two single-electron states are both phase-independent and unit length, hence we can reduce the total of real numbers needed by two for each state. Product states correspond to the situation in which each electron is prepared independently, and can be measured independently. This means that, for a product state, there is always a direction along which you will 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. This is equivalent to saying that each electron can be measured independently. Now construct the following 2 electron states: 1/√2{|ud> + |du>} ... TRIPLET STATE 1/√2{|ud> - |du>} ... SINGLET STATE By definition, these states are systems where the second electron is in the opposite configuration to the first. A pair of electrons in this state are said to be ENTANGLED. It turns out to be energetically favorable for the electrons to end up in the singlet state. The singlet state results by bringing 2 electrons close enough together so that spins interact. Again, as long as the electrons are not on top of each other, the PE principle is not violated. Each electron is like a little magnet and after a while they will orient themselves to form the lowest energy state of the 2 electron system. A photon may or may not be emitted in the process depending on the initial orientation of the electrons. For any direction we can show that the averages (expectation values) of the associated spins are zero, meaning that measurements along this direction are equally likely to be +1 or -1. Recall, the expectation value is given by <a|O|a>. {<ud| ± <du|}σx{|ud> ± |du>} => {<ud| ± <du|}{|dd> ± |uu>} = 0 {<ud| ± <du|}σy{|ud> ± |du>} => {<ud| ± <du|}{i|dd> ±' i|uu>} = 0 {<ud| ± <du|}σz{|ud> ± |du>} => {<ud| ± <du|}{|ud> ±' |du>} = 0 ±' ≡ -/+ Likewise, if we did the samething for τ we would get the same result. Clearly, this is a different situation from the single electron system where there is always an axis where <σ.n> = 1. In contrast, in the entangled case, any axis you measure has a 50/50 probability of being up or down (in other words, the expectation value is 0). This is quite different from the product state described above. There is a significant difference between the two states: 1/√2{|ud> + |du>} and 1/√2{|ud> - |du>} One way to distinguish them is to compute the result of the sum of the two spin operators, σ and τ, for each direction on each state (+ and -) This is analagous to the vector sum of a pair of directions, n and m. Thus, (σn + τn){|ud> ± |du>} neglecting 1/√2 coefficient for now which is equivalent to, σn{|ud> ± |du>} + τn{|ud> ± |du>} Consider {|ud> + |du>}, we get σz{|ud> + |du>} = { |ud> - |du>} τz{|ud> + |du>} = {-|ud> + |du>} ∴ σz{|ud> + |du>} + τz{|ud> + |du>} = 0 σx{|ud> + |du>} = {|dd> + |uu>} τx{|ud> + |du>} = {|uu> + |dd>} ∴ σx{|ud> + |du>} + τx{|ud> + |du>} = 2{|dd> + |uu>} σy{|ud> + |du>} = { i|dd> - i|uu>} τy{|ud> + |du>} = {-i|uu> + i|dd>} ∴ σy{|ud> + |du>} + τy{|ud> + |du>} = 2i{|dd> - |uu>} Now consider {|ud> - |du>}, we get σz{|ud> - |du>} = { |ud> + |du>} τz{|ud> - |du>} = {-|ud> - |du>} ∴ σz{|ud> - |du>} + τz{|ud> - |du>} = 0 σx{|ud> - |du>} = {|dd> - |uu>} τx{|ud> - |du>} = {|uu> - |dd>} ∴ σx{|ud> - |du>} + τx{|ud> - |du>} = 0 σy{|ud> - |du>} = { i|dd> + i|uu>} τy{|ud> - |du>} = {-i|uu> - i|dd>} ∴ σy{|ud> - |du>} + τy{|ud> - |du>} = 0 Thus, the expectation value of (σn + τn) is 0*. * From <ud| - <du|}(σn + τn){|ud> - |du>} Now, we can prove that fact is true for any arbitrary direction, n (n is a unit vector with components nx, ny, nz) by taking the dot product as follows: (σ + τ).n = (σx + τx)nx + (σy + τy)ny + (σz + τz)nz Therefore, since we have previously proved that all components of (σ + τ) are 0, (σ + τ) in the n direction must also equal 0. Einstein - Podolsky - Rosen Correlation --------------------------------------- As mentioned previously, a pair of electrons will ultimately produce a singlet (lowest energy) state if they are brought together so that their magnetic fields interact. Since the sum of the spins is always 0, it has the special property that if you measure a component of spin for one of the electrons, you instantly know the same component of spin for the other electron. This is known as the EINSTEIN - PODOLSKY - ROSEN CORRELATION. The curious fact is that the 2 electrons will remain entangled regardless of the separation between them unless either is 'disturbed' by a measurement. A measurement results in the collapse of the state of the of the original entangled system. This is equivalently described as the entanglement of the electron with the measurement apparatus. If you measure electron 1's spin, electron 2's gets set by the measurement ... but somehow electron 2 also instantly "knows" what spin it is supposed to take on. This seemingly involves communication between the two particles at speeds greater than the speed of light, which is in conflict with Einstein's theory of relativity (violation of locality). Einstein himself supported an alternative approach called 'hidden variables theory' which suggested that quantum mechanics was incomplete. However, this theory has the problems that violates Bell's theorem. This is discussed here. Monogamy of Entanglement ------------------------ If 2 electrons, are maximally entangled with each other they cannot be entangled at all with a third electron. In general, there will be a trade off between the amount of entanglement between 1 and 2 and 1 and 3 or 2 and 3. This is referred to as the MONOGAMY OF ENTANGLEMENT. In general, the more parties, the less entanglement between them. The Density Matrix ------------------ A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states. This should be contrasted with a single state vector that describes a quantum system in a pure state. The density matrix is written for a 4 state system as: - - | ρ1 0 0 0 | ρ = | 0 ρ2 0 0 | | 0 0 ρ3 0 | | 0 0 0 ρ4 | - - where ρi is the probability for being in the ith state. As all probabilities must sum to 1 the trace of ρ, Tr(ρ), must therefore = 1. The density matrix is an Hermitian operator. Entanglement Entropy -------------------- How do we quantify the degree (amount) of entanglement? One way to do this to compute the QUANTUM ENTROPY defined as: S = -Tr(ρlogρ) In classical mechanics this is equivalent to: S = -ΣiPilogPi The entropy is measure of our degree of knowledge about the state of the system. In the case where there is only one state we know exactly what state the system is in and the entropy is 0. There will be only one non-zero entry in the density matrix and the system is in a PURE STATE. Conversely, if the number of states is large, then we have very little knowledge about the precise state of the system and the entropy is large. The system is in a MIXED STATE and there will be more than non-zero entry in the density matrix. If we have absolutely no knowledge about how the sustem was prepared, all the elements in the density matrix will be equal. Consider the observable, O. O is an Hermitian operator. <O> = Tr(Oρ) which is similar to the classical, <O> = ΣiOiPi Tr(Oρ) = Σi<i|Oρ|i> by definition = Σij<i|O|j><j|ρ|i> = Σij<i|O|i>ρij since <j|ρ|i> = ρij This is the expectation value of O for state i weighted by the probability that the state is i Consider a state consisting of 2 systems a and b. If system a has N possible states and system b has n states, then the total number of states - n x N. For example, in the case of 2 spins we could have uu, dd, ud and du. The general state of the combined system can be written as: Σabψ(a,b)|ab> where ψ(a,b) are complex coefficients such that Σabψ*ψ = 1 Now consider a measurement only on system 1 only with no change to system b. The observable is represented by the operator, O. Now compute the expectation value of O. <O> = Σaa'bb'ψ(a',b')*<a'b'|O|ab>ψ(a,b) Since b is completely unchanged, b' = b, and we can write: <O> = Σaa'bψ(a',b)<a',b)|O|ab>ψ*(a',b)ψ(a,b) = Σaa'b<a'b|O|ab>ψ*(a',b)ψ(a,b) = Σaa'<a'b|O|ab>Σbψ*(a',b)ψ(a,b) Now b is completely passive in <a'b|O|ab>. Therefore: = Σaa'<a'|O|a>Σbψ*(a',b)ψ(a,b) = Σaa'Oa'aΣbψ*(a',b)ψ(a,b) = Σaa'Oa'aρaa' where ρaa' ≡ Σbψ*(a',b)ψ(a,b) <O> = Tr(Oρ) Proof: Tr(Oρ) = <a'|Oρ|a'> = Σaa'<a'|O|a><a|ρ|a'> = Σaa'Oa'aρa'a What this means is that if you have a combined system in a pure state, ψ(a,b), and you measure either sub-system (in this case a) alone they will, in general, be described by a mixed state with an associated density matrix. Consider the singlet 1/√2{|ud> - |du>}. This is an entangled state. ψ(u,u) = 0 ψ(u,d) = 1/√2 ψ(d,u) = -1/√2 ψ(d,d) = 0 Now ρaa' ≡ Σbψ*(a',b)ψ(a,b). This leads to: ρuu = ψuuψ*uu + ψudψ*ud = 1/2 ρdd = ψduψ*du + ψddψ*dd = 1/2 ρud = ψuuψ*du + ψudψ*dd = 0 ρdu = ψduψ*uu + ψddψ*ud = 0 The desity matrix becomes: - - ρ = | 1/2 0 | | 0 1/2 | - - and for the entropy we get: S = -Tr(ρlogρ) = -{(1/2)log(1/2) + (1/2)log(1/2)} = log(2) It is fairly easy to show that 2 is the maximum entropy ever allowed. This represents a density matrix where all of the elements are equal. Therefore, we have a combined system in a pure state but we have complete ignorance about either of the sub-systems (we have focussed on a here but the same is true for b). There a specific case where this is not true. Consider a product state. This is a non-entangled state. ψ(a,b) = ψ(a)χ(b) ρaa' = Σbφ(a)χ(b)φ*(a')χ*(b)     = φ(a')φ*(a)Σbχ(b)χ*(b)     = φ(a)φ*(a') <O> = Oaa' = <φ|O|φ> In this case, each sub-sub system is in pure state, the density matrix is 0 and the entropy is 0. We know everthing about the combined system and its sub-systems.