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Repeated measurements

Repeated measurements - outcomes of a measurement of is...

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Repeated measurements If we measure once and we find as outcome we know that the system is in the th eigenstate of the Hamiltonian. That certainty means that if we measure the energy again we must find again. This is called the ``collapse of the wave function'': before the first measurement we couldn't predict the outcome of the experiment, but the first measurements prepares the wave function of the system in one particuliar state, and there is only one component left! Now what happens if we measure two different observables? Say, at 12 o'clock we measure the position of a particle, and a little later its momentum. How do these measurements relate? Measuring to be makes the wavefunction collapse to , whatever it was before . Now mathematically it can be shown that (8.31) Since is an eigenstate of the momentum operator, the coordinate eigen function is a superposition of all momentum eigen functions with equal weight. Thus the spread in possible
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Unformatted text preview: outcomes of a measurement of is infinite! incompatible operators The reason is that and are so-called incompatible operators, where (8.32) The way to show this is to calculate (8.33) for arbitrary . A little algebra shows that (8.34) In operatorial notation, (8.35) where the operator , which multiplies by 1, i.e., changes into itself, is usually not written. The reason these are now called ``incompatible operators'' is that an eigenfunction of one operator is not one of the other: if , then (8.36) If was also an eigenstate of with eigenvalue we find the contradiction . Now what happens if we initially measure with finite acuracy ? This means that the wave function collapses to a Gaussian form, (8.37) It can be shown that (8.38) from which we read off that , and thus we conclude that at best (8.39) which is the celeberated Heisenberg uncertainty relation....
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Repeated measurements - outcomes of a measurement of is...

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