MIT5_74s09_lec12

MIT5_74s09_lec12 - MIT OpenCourseWare http:/ocw.mit.edu...

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MIT OpenCourseWare 5.74 Introductory Quantum Mechanics II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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p. 1 THE DENSITY MATRIX The density matrix or density operator is an alternate representation of the state of a quantum system for which we have previously used the wavefunction. Although describing a quantum system with the density matrix is equivalent to using the wavefunction, one gains significant practical advantages using the density matrix for certain time-dependent problems – particularly relaxation and nonlinear spectroscopy in the condensed phase. The density matrix is formally defined as the outer product of the wavefunction and its conjugate. ρ ( t ) ψ ( t ) ( t ) . (1.1) This implies that if you specify a state χ , the integral χρχ gives the probability of finding a particle in the state . Its name derives from the observation that it plays the quantum role of a probability density. If you think of the statistical description of a classical observable obtained from moments of a probability distribution P , then ρ plays the role of P in the quantum case: A = A P A d A ( ) (1.2) A = A = [ ] . (1.3) T r where Tr[…] refers to tracing over the diagonal elements of the matrix. The last expression is obtained as follows. For a system described by a wavefunction ( t ) = c n ( t ) n , (1.4) n the expectation value of an operator is * ˆ A ˆ c tc n () = t m n (1.5) At m , nm Also, from eq. (1.1) we obtain the elements of the density matrix as ( t ) = t ( ) * ( t ) n m n m , (1.6) nm tnm ,
p. 2 We see that ρ nm , the density matrix elements, are made up of the time-evolving expansion coefficients. Substituting into eq. (1.5) we see that ˆ = A mn nm () t At , (1.7) = Tr A t ˆ In practice this makes evaluating expectation values as simple as tracing over a product of matrices. So why would we need the density matrix? It is a practical tool when dealing with mixed states. Pure states are those that are characterized by a single wavefunction. Mixed states refer to statistical mixtures in which we have imperfect information about the system, for which me must perform statistical averages in order to describe quantum observables. A mixed state refers to any case in which we subdivide a microscopic or macroscopic system into an ensemble, for which there is initially no phase relationship between the elements of the mixture. Examples include an ensemble at thermal equilibrium, and independently prepared states.

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MIT5_74s09_lec12 - MIT OpenCourseWare http:/ocw.mit.edu...

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