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## Homework Statement

A wave packet is described by the momentum-space wave function A(p)=C when 0<p<p

_{0}, and A(p)=0 for all other values of p. Here C is a constant.

i) Normalize this wave function by solving for C in terms of p

_{0}.

ii) Calculate the expectation values <p> and <p

^{2}>. From these compute the standard deviation in terms of p

_{0}.

## Homework Equations

For normalization: h(integral from 0 to p

_{0})A*(p)dp=1, h being Planck's constant.

<p>=h(integral from 0 to p

_{0})pA*(p)dp

<p

^{2}>=h(integral from 0 to p

_{0})p

^{2}A*(p)dp

(I am not entirely sure if this equation for <p

^{2}> is correct, it may be <p

^{2}>=h(integral from 0 to p

_{0})p

^{2}A*(p)A(p)dp)

standard deviation=<p

^{2}> - <p>

^{2}

## The Attempt at a Solution

i) When I normalized the wave function from 0 to p

_{0}, I got C=1/sqrt(h*p

_{0}).

ii) This is the part I'm struggling with. Typically <p> would be equal to zero when integrated from -infinity to +infinity, but this is from 0 to p

_{0}. Doing this I am getting <p>=p

_{0}.

For <p

^{2}> I am integrating from 0 to p

_{0}and getting <p

^{2}>=(1/3)p

_{0}

^{2}.

One of these values (or both) can't be correct because when I try to calculate the standard deviation=<p

^{2}> - <p>

^{2}= (1/3)p

_{0}

^{2}- p

_{0}

^{2}= -(2/3)p

_{0}

^{2}. I'm pretty sure standard deviations can't be negative. So the part of the question I really can't figure out is what I'm doing wrong when I try to find the expectation values.