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Hydrogen atom ground state, uncertainty relation

Hydrogen atom ground state, uncertainty relation

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From the table, we find that the normalized 1s wave function for Z=1 is
$$
psi(vec r) = psi_{000} = {1oversqrt{pi a_o^3}} e^{-r/a_o}.
eqno(1.1)
$$

bf (a) rm

Since $psi(vec r)$ is symmetric about the origin, we have $la xra = 0$ and also
$$
la x^2ra = la y^2ra = la z^2ra = {1over 3} la x^2 +y^2+z^2ra
= {1over 3} la r^2ra = {1over 3}int d^3vec r r^2|psi|^2 =
$$
$$
= {1over 3pi a_o^3} int_0^pi sintheta dtheta int_0^{2pi} dphi
int_0^infty r^2dr r^2e^{-2r/a_o}
= {4piover 3pi a_o^3}int_0^infty r^4e^{-2r/a_o} dr,
eqno(1.2)
$$
where $(r,theta,phi)$ are spherical coordinates. ...

$2.19