Lect34_DensityOperator

Lect34_DensityOperator - Lecture 34: The `Density Operator...

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Lecture 34: The `Density Operator Phy851 Fall 2009
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The QM `density operator HAS NOTHING TO DO WITH MASS PER UNIT VOLUME The density operator formalism is a generalization of the Pure State QM we have used so far. New concept: Mixed state Used for: – Describing open quantum systems – Incorporating our ignorance into our quantum theory Main idea: – We need to distinguish between a `statistical mixture’ and a `coherent superposition’ Statistical mixture: it is either a or b, but we don’t know which one • No interference effects Coherent superposition: it is both a and b at the same time • Quantum interference effects appear
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Pure State quantum Mechanics The goal of quantum mechanics is to make predictions regarding the outcomes of measurements Using the formalism we have developed so far, the procedure is as follows: – Take an initial state vector – Evolve it according to Schrödinger's equation until the time the measurement takes place – Use the projector onto eigenstates of the observable to predict the probabilities for different results – To confirm the prediction, one would prepare a system in a known initial state, make the measurement, then re-prepare the same initial state and make the same measurement after the same evolution time. With enough repetitions, the results should show statistical agreement with the results of quantum theory
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Expectation Value The expectation value of an operator is defined (with respect to state | ψ〉 ) as: The interpretation is the average of the results of many measurements of the observable A on a system prepared in state | . – Proof: ψ A A A A A a a n n n = n n n n a a a = n n n n a a a = n n n a a = 2 n n n a a p = ) ( This is clearly the weighted average of all possible outcomes
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Statistical mixture of states What if we cannot know the exact initial quantum state of our system? – For example, suppose we only know the temperature, T, of our system? Suppose I know that with probability P 1 , the system is in state | ψ 1 , while with probability P 2 , the system is in state | 2 .
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This note was uploaded on 12/21/2011 for the course PHY 851 taught by Professor Moore,m during the Fall '08 term at Michigan State University.

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Lect34_DensityOperator - Lecture 34: The `Density Operator...

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