- #1

gfd43tg

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## Homework Statement

## Homework Equations

$$ \psi_{100} = \frac {1}{\sqrt{\pi a^{3}}} e^{-r/a} $$

## The Attempt at a Solution

a)

$$\langle r \rangle = \frac {1}{\pi a^{3}} \int_0^{2 \pi} d \phi \int_{0}^\pi d \theta \int_0^{\infty} r^{3} e^{-2r/a} dr$$

This comes out to be ##\frac {3}{2}a##

$$\langle r^{2} \rangle = \frac {1}{\pi a^{3}} \int_0^{2 \pi} d \phi \int_{0}^\pi d \theta \int_0^{\infty} r^{4} e^{-2r/a} dr$$

Which comes out as ##3a^{2}##

b)

I know ##r^{2} = 3x^{2}##, so the answer for the expectation value of ##x^{2}## is one third the expectation value of ##r^{2}##, therefore ##\langle x^{2} \rangle = a^{2}##

However, I am confused how to find ##\langle x \rangle##. Do I just say ##x = r sin \theta##, therefore ##dx = sin \theta dr + r cos \theta d \theta##?

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