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Wave packets - (10.15 Actually I shall show below that...

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Wave packets (states of minimal uncertainty) One of the questions of some physical interest is ``how can we create a qunatum-mechanical state that behaves as much as a classical particle as possible?'' From the uncertainty principle, (10.13) this must be a state where and are both as small as possible. Such a state is known as a ``wavepacket''. We shall see below (and by using a computer demo) that its behavior depends on the Hamiltonian governing the system that we are studying! Let us start with the uncertainty in . A state with width should probably be a Gaussian, of the form (10.14) In order for to be normalised, we need to require
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Unformatted text preview: (10.15) Actually, I shall show below that with (10.16) we have (10.17) The algebra behind this is relatively straightforward, but I shall just assume the first two, and only do the last two in all gory details. (10.18) Thus (10.19) Let act twice, (10.20) Doing all the integrals we conclude that (10.21) Thus, finally, (10.22) This is just the initial state, which clearly has minimal uncertainty. We shall now investigate how the state evolves in time by usin a numerical simulation. What we need to do is to decompose our state of minimal uncertainty in a sum over eigenstates of the Hamiltonian which describes our system!...
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