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Quantum techniques for stochastic mechanics (course) lecture 3 (changes)

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Quantum Techniques for Stochastic Mechanics

  • Lecture 3 of 4

  • Link to course homepage

  • Quantum Techniques for Stochastic Mechanics, by Jacob Biamonte, QIC 890/891 Selected Advanced Topics in Quantum Information, The University of Waterloo, Waterloo Ontario, Canada, (Spring term 2012).

  • Given Aug 14th, 2012 in Waterloo Canada

Lecture Content

  • Review of the master equation vs the rate equation

  • Amoeba field theory

  • Stationary states

  • Stochastic mechanics vs quantum mechanics

  • The lecture began with a review of lecture 1

Comparing quantum and stochastic mechanics

Quantum mechanics Stochastic mechanics
$latex \psi: X \rightarrow C$ $ \psi:X \rightarrow R^+$
Content Cell Content Cell

Quantum mechanics

  • ψ:C \psi : \rightarrow C

  • L 2(x)={ψ:C; x|ψ(x)| 2dx} L^2(x) = \{\psi : \rightarrow C; \integral_x |\psi(x)|^2 dx \rangle \infinity\}

  • ,:L 2(x)×L 2(x)C \langle -, -\rangle : L^2(x) \times L^2(x) \rightarrow C

  • U:L 2(x)L 2(x) U : L^2(x) \rightarrow L^2(x)

  • Uψ,Uϕ=ψ,ϕ \langle U\psi , U \phi \rangle = \langle \psi, \phi\rangle

  • Reversible, unitary

  • iddtψ(t)=Hψ(t) i \frac{d}{d t} \psi(t) = H \psi(t)

  • ψ(t)=e itHψ(0) \psi(t) = e^{-i t H} \psi(0)

  • H=H H = H^\dagger

Stochastic mechanics

  • ψ:xR + \psi : x \rightarrow R^+

  • L 1(x)={ψ:xR; x|ψ(x)|dx} L^1(x) = \{\psi : x \rightarrow R; \integral_x |\psi(x)| d x \rangle \infinity\}

  • ψ:=:L 1(x)R \langle \psi \rangle := \integral: L^1(x) \rightarrow R

  • U:L 1(x)L 1(x) U: L^1(x) \rightarrow L^1(x)

  • Uψ=ψ \integral U \psi = \integral \psi

  • ψ0 \psi \geq 0, \Rightarrow Uψ0 U \psi \geq 0

  • ddtψ(t)=Hψ(t) \frac{d}{d t} \psi(t) = H \psi(t)

  • ψ(t)=e tHψ(0) \psi(t) = e^{t H}\psi(0)

  • H ij0 H_{i\neq j} \geq 0

  • iH ij=0 \sum_i H_{ij} = 0

References