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Unformatted text preview: Introductory comments • This course is PHYS811, the second semester of the Department’s Quantum Mechanics sequence. • I am Dr. Pittel and my office is in Sharp Lab Rm 202. • The text for the course is Quantum Mechanics Third Edition, by Eugen Merzbacher. • We will cover the material from chapters 14 through 19 and then a little from chapter 23 this semester. • I will be assigning weekly reading from the book, but at the same time will be preparing my own set of lecture notes, which will be made available on my web site prior to the associated lecture. Feel free to bring the lecture material with you, so that you do not have to spend the entire lecture taking notes. The lecture will sometimes, but not always, parallel the text. • I will also be assigning weekly problem assignments, sometimes from the text and sometimes not. All such assignments should be handed in a week after assignment and will be graded. • I will not give a midterm examination, but will give a final examination. • At the end of the semester your grade will be obtained by an equal weighting of the homework assignments and the final examination. 1 CHAPTER 14 During week one, I would like you to read Chapter 14 of Merzbacher, pages 315342. 1. The Time development of a Quantum System A quantum system isolated from external influences evolves in time in a completely predictable manner. In the approach you have learned up to now, the time evolution of the system is governed by the timedependent Schr¨ odinger equation, which in Dirac notation takes the form i ¯ h d dt  Ψ( t ) > = H ( t )  Ψ( t ) > (1) This is an equation that defines how the state vector  Ψ( t ) > evolves in time. Now let us assume that we know the state vector  Ψ( t ) > at some time t and we wish to evaluate it at a later time t . Let’s define the time evolution operator T ( t,t ) by  Ψ( t ) > = T ( t,t )  Ψ( t ) > (2) If we insert (1) into (2) we find that i ¯ h ∂ ∂t T ( t,t )  Ψ( t ) > = HT ( t,t )  Ψ( t ) > so that i ¯ h ∂ ∂t T ( t,t ) = HT ( t,t ) (3) To this point, we have made no specific assumptions regarding the hamiltonian H . If, however, H is independent of time then we can directly integrate (3) and (making use of the obvious initial condition T ( t ,t ) = 1) obtain T ( t,t ) = exp i H ¯ h ( t t ) ¶ (4) Thus, in this formulation of Quantum Mechanics state vectors evolve in time in the presence of a timeindependent hamiltonian according to  Ψ( t ) > = e i ¯ h H ( t t )  Ψ( t ) > (5) What about the time evolution of the expectation value of some operator A . This operator may or may not itself vary in time. 2 The time derivative of its expectation value for a given state vector Ψ( t ) > is given by i ¯ h d dt < Ψ( t )  A  Ψ( t ) > = < Ψ( t )  HA  Ψ( t ) > + < Ψ( t )  AH  Ψ( t ) > + i ¯ h < Ψ( t )  ∂ ∂t  Ψ( t ) > When differentiating < Ψ( t )  and Ψ( t ) > , I used the timedependent Schrodinger equation (1) and the related hermitan adjoint equation to treat the associated derivatives.(1) and the related hermitan adjoint equation to treat the associated derivatives....
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This note was uploaded on 10/28/2009 for the course PHYS 306 taught by Professor Laverty during the Spring '09 term at University of Delaware.
 Spring '09
 LAVERTY
 mechanics

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