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    Hydrogen atom ground state, uncertainty relation

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    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}.

    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,
    where $(r,theta,phi)$ are spherical coordinates. ...

    Solution Summary

    The solution discusses the hydrogen atom ground state, and uncertainty relation.