Lecture 07

Lecture 07 - EEE 434/591Quantum Mechanics L7:1 New physics...

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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L7:1 EEE 434/591—Quantum Mechanics David K. Ferry Regents’ Professor Arizona State University New physics building Göttingen University Max Born, Werner Heisenberg, Pascual Jordan
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L7:2 In this lecture, we want to return to the time-dependent Schrödinger equation, and work out some details about averages, time variation of operators, and such details to give us further insight into this view of quantum mechanics. It will also let us examine how measurements might yield insight into the world of quantum mechanics. This becomes important, since the Copenhagen interpretation does not admit to reality being there before a measurement is made.
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L7:3 Time Variation in the Schrödinger Equation Now, we can separate the variables by assuming a form as so that
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L7:4 Now, dividing by Ψ gives Hence Function of x alone Function of t alone
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L7:5 This may be solved to give In general, the spatial variation produces a set of eigenfunctions each of which has a different eigenvalue (the energy) and frequency
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L7:6 Hence, we can quite generally write the total wave function as Normalizing the total wave function gives Ψ ( x , t ) = c i ψ i ( x ) e iE i t / i i ( x ) 2 dx R = 1 Ψ ( x , t ) 2 dx R = c j * i , j j * ( x ) R i ( x ) e i ( E i E j ) t / dx = c j * c i δ ij e i ( E i E j ) t / i , j = c i 2 i = 1
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L7:7 The average (expectation) value of x is given by x ( t ) = Ψ , x Ψ ( ) Ψ * ( x , t ) x −∞ Ψ ( x , t ) dx = c j * c i ψ j * ( x ) x i ( x ) e i ( E i E j ) t / dx −∞ i , j = c j * c i x ji e i ( E i E j ) t / i , j How does this average change with time?
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L7:8 d x ( t ) dt = d dt Ψ * ( x , t ) x −∞ Ψ ( x , t ) dx = ∂Ψ * ( x , t ) t
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Lecture 07 - EEE 434/591Quantum Mechanics L7:1 New physics...

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