# The Azimuth Project Quantum techniques for stochastic mechanics (course) lecture 3

## Quantum Techniques for Stochastic Mechanics

• Lecture 3 of 4

• 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

• $\psi : \rightarrow C$

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

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

• $U : L^2(x) \rightarrow L^2(x)$

• $\langle U\psi , U \phi \rangle = \langle \psi, \phi\rangle$

• Reversible, unitary

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

• $\psi(t) = e^{-i t H} \psi(0)$

• $H = H^\dagger$

Stochastic mechanics

• $\psi : x \rightarrow R^+$

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

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

• $U: L^1(x) \rightarrow L^1(x)$

• $\integral U \psi = \integral \psi$

• $\psi \geq 0$, $\Rightarrow$ $U \psi \geq 0$

• $\frac{d}{d t} \psi(t) = H \psi(t)$

• $\psi(t) = e^{t H}\psi(0)$

• $H_{i\neq j} \geq 0$

• $\sum_i H_{ij} = 0$