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Math Help - Multiple Integrals Shrödinger Problem

  1. #1
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    Multiple Integrals Shrödinger Problem

    I've been set the following problem and have managed to complete parts i) and ii). However, I'm struggling to grasp the concept for part iii). What I would like to be able to do is come up with an expression for the probability of the electron being at radius r and then use differentiation to maximise it. But all I have is the probability of it being in a certain volume dV. Could someone please give me a hint?

    Many Thanks.

    Q9. According to the Schr¨odinger equation of quantum mechanics, an electron in the
    ground state of a hydrogen atom has a spherically-symmetric spatial probability density
    distribution P(r) = Ke−2r/a, i.e. the probability of locating the electron in a volume
    dV at radius r is P(r)dV . Here, r is the distance of the electron from the proton, a is
    a fixed constant with dimension of length and K is a (dimensional) constant chosen to
    make the probability of finding the electron anywhere equal to unity. Calculate:
    i. the constant K;
    4
    ii. the average value of r; and
    iii. the most probable value of r.
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  2. #2
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    Mar 2010
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    <br />
\displaystyle {<br />
Pr(r) \; dr=P(r) \; dV=K e^{-2r/a} \; 4 \pi r^2 dr<br />
}<br />

    The most probable value of r is when

    <br />
Pr(r) \; is \; max.<br />

    Or

    <br />
\displaystyle {<br />
\frac{d \; Pr(r)}{dr}=0.<br />
}<br />
    Last edited by zzzoak; December 30th 2010 at 04:24 PM.
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