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The Heisenberg uncertainty relation for the position and momentum of a particle that moves only in one dimension is
In this expression Dx and Dp are the uncertainties in position and momentum. They are the root-mean-square deviations from the expectation values: i.e,
| (DA)2 = < (A- < A > )2 > = < A2 > - < A > 2, |
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for an arbitrary operator A. The expectation value < A > is defined as
| < A > = |
ó
õ |
dx y(x)*Ay(x) |
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(and a similar expression for A2) with the system being described by the normalized wavefunction y.
a) Calculate < x > , < x2 > , and Dx for the wavefunctions.
with n = 1,2,3,¼.
b) Calculate < p > , < p2 > , and Dp for the same wavefunctions.
c) Show that the Heisenberg uncertainty relation indeed holds for these wavefunctions.
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