lecture12_sup2

lecture12_sup2 - 5.61 Fall 2007 Lecture Summary 12-15,...

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5.61 Fall 2007 Lecture Summary 12-15, Supplement page 1 Ehrenfest’s Theroem In the lecture notes for the harmonic oscillator we derived the expressions for x ˆ ( t ) and ( t ) using standard approaches – integrals involving Hermite polynomials (see pages 17 and 18, Lecture Summary 12-15). The calculations are algebraically intensive, but showed that p ˆ x x ˆ ( t ) and ( t ) oscillate at the vibrational frequency. The results were as follows: p ˆ x x ( t ) = ( 2 α ) 1 cos ( ω vib t ) = 2 µ ! 1 2 cos ( vib t ) and 1 p ( t ) = 2 i ! 2 ( e i vib t e i vib t ) = ! 2 1 sin ( vib t ) The issue considered here is an approach to calculate x ( t ) and p ( t ) in a more straightforward manner. Classically, (we use m instead of since we are dealing with a free particle) dx p = mv = m dt So, quantum mechanically we might expect p ( t ) = m d x ( t ) dt . But, is this expression valid ? We can show that in fact it is with the following argument.
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This note was uploaded on 09/24/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.

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lecture12_sup2 - 5.61 Fall 2007 Lecture Summary 12-15,...

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