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Lect18_ClassicalLimit

# Lect18_ClassicalLimit - Quiz#7 Expectation values obey the...

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Lecture 18: Classical limit II PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Quiz #7 Expectation values obey the Heisenberg equation of motion 1. For the Hamiltonian: Find the equations for Do they form a closed set? 2. What is the solution when at t=0 we have [ ] A H i A dt d , h = 2 2 2 2 1 2 X m m P H ! + = ? = X dt d ? = P dt d 0 x X = 0 = P [ ] ) ( ), ( X V i P X V ! = h

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Review Last lecture we derived the equation of motion for expectation values: We also discussed how Classical Mechanics describes the motion of narrow wavepackets ! X " gives the center of the wavepacket ! P " gives the velocity of the center times the mass, i.e. the classical momentum We then evaluated the commutator Valid for any function V ( x ) From which we found the QM eqs of motion: Here F ( x )=- V_ ( x ) is the classical force From this we see that the requirement for classical mechanics to provide a good approximation is: d dt A = i h H , A [ ] V ( X ), P [ ] = i h " V ( X ) m P X dt d = ) ( X F P dt d = F ( X ) " F X ( ) Force Expectation Value The expectation value of the force is:
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Lect18_ClassicalLimit - Quiz#7 Expectation values obey the...

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