Lecture 21

# Lecture 21 - EEE 434/591Quantum Mechanics L21:1 David K....

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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L21:1 EEE 434/591—Quantum Mechanics David K. Ferry Regents ! Professor Arizona State University

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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L21:2 PERTURBATION THEORY ! Sometimes it is difficult to get the real answer, since the number of problems which can be solved in quantum mechanics is countable (and a small number). ! In general, then, we can start with a problem that is close to the actual one, and one which we can solve exactly. ! We want to develop an approximation technique that gives the answers to the unknown problem in terms of those of the known problem. ! This assumes that the difference between the two problems may be considered to be small .
EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L21:3 http://www.clausewitz.com/Complex/ChaosDemos.htm One of the properties of chaotic systems is that two initial conditions, very close to each other, can lead to trajectories that diverge exponentially. Obviously such systems are not suitable for perturbation theory, since they do not remain close to the known system. Thus, we must be very careful to assure ourselves that the conditions we require for perturbation theory to work, are in fact satisfied in the system under investigation. But, let us proceed with the assumption that this caveat has been satisfied.

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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L21:4 We begin with the assumption that the total Hamiltonian can be split into two terms for which we assume that H 0 is solvable exactly: Then, we assume that we may rewrite the Hamiltonian and the solution to the total problem as: 1 0 H H H + = i E i H i = 0 0 ) 0 ( 0 0 ) 2 ( 2 ) 1 ( ) 0 ( 2 2 1 0 0 1 0 ... ... i E i H i E i H E E E E i i i i V H H H H i i i i i i = = + + + = + + + = + ! + = " Key starting points
EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L21:5 H i = E i i H 0 + " V ( ) i 0 + i 1 + 2 i 2 + ... ( ) = = E i (0) + E i (1) + 2 E i (2) + ... ( ) i 0 + i 1 + 2 i 2 + ... ( ) We now equate terms which are of the same power of ! : ... : : : 0 ) 2 ( 1 ) 1 ( 2 ) 0 ( 1 2 0 2 0 ) 1 ( 1 ) 0 ( 0 1 0 1 0 ) 0 ( 0 0 0 i E i E i E i V i H i E i E i V i H i E i H i i i i i i + + = + + = + =

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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L21:6 Clearly, our ! 0 term is just the known solutions of the reference problem.
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## This note was uploaded on 12/10/2011 for the course MAT 443 taught by Professor F during the Spring '11 term at Alaska Bible.

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Lecture 21 - EEE 434/591Quantum Mechanics L21:1 David K....

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