Phy5645/Heisenberg Uncertainty Relation 3

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Let us assume that a particle has the wavefunction,

We now wish to verify the Heisenberg Uncertanity Principle for this case. To do so, we need to find the uncertainties in position and momentum, and

We will calculate the expectation values one by one.

since the integrand is odd and thus the integral over all space is zero.

Since the integral is of a Gaussian times a power of , we are able to use the known results for such integrals.

Similarly to because the integrand will be an odd function as well.

Combining these results, we obtain and

Finally,

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