Approximating Probability for a Wave Function

  • #1
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Homework Statement


The wavefunction at t = 0 is given by
$$\Psi = N*e^{-\frac{r}{a_0}}$$
where ##r = |\mathbf{x}|##. ##a_0## is a constant with units of length. The electron is in 3 dimensions.
Find the approximate probability that the electron is found inside a tiny sphere centered at the origin with ##b_0 << a_0## [Hint: the exact calculation is much harder than the approximate estimate]

Homework Equations


N was found in the previous part of the problem:
$$N = \frac{1}{\sqrt{\pi a_0^3}}$$

For probability:
$$P = \int_0^{b_0} |\Psi|^2 d^3x$$

The Attempt at a Solution


I first calculated ##|\Psi|^2##.
$$|\Psi|^2 = N^2e^{-\frac{2r}{a_0}}$$

Using spherical coordinates to write the integral:
$$P = \int_0^{b_0} |\Psi|^2 d^3x = \int_0^{2\pi}\int_0^{\pi}\int_0^{b_0} N^2e^{-\frac{2r}{a_0}} r^2sin(\phi)drd\phi d\theta$$

Using ##u = \frac{2r}{a_0}## and simplifying the integral by plugging in N gives me:
$$P = \frac{1}{2}\int_0^{\frac{2b_0}{a_0}} e^{-u}u^2du$$

My issue is how to approximate it give ##b_0 << a_0##.
All the answer choices are proportion to ##\frac{b_0^2}{a_0^2}## or ##\frac{b_0^3}{a_0^3}##.
 

Answers and Replies

  • #2
vela
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If ##b_0 \ll a_0##, then in the integral ##u## will be close to 0 so you can say ##e^u \approx 1##.
 
  • #3
166
18
If ##b_0 \ll a_0##, then in the integral ##u## will be close to 0 so you can say ##e^u \approx 1##.
Would I integrate first then approximate it, or use ##e^{-u} = 1## inside of the integal such that the integral just becomes the integral of ##u^2##?
 

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