Finding the probability of 1s electron within a cubical volume

  • #1

Homework Statement


How to calculate the probability of finding an 1s electron within 1 picometer cubic region located 50pm from the nucleus.

Homework Equations


The probability of an 1s electron within a spherical volume of radius 'a' from nucleus can be find using the expression
$$\int_{0}^{a}\int_{0}^{\pi}\int_{0}^{2\pi}|\psi(r,,\theta,\phi)|^2r^2 dr \sin(\theta) d\theta d\phi$$
but how to find probability within a cubic region?

The Attempt at a Solution


I thought I could transform ${(r,\theta,\phi)}$ into (x,y,z) coordinates and integrate the above integral within proper limits, but the integral will become messy! and also I don't know whether it is the proper way of doing this problem, all I am asking is give me a hint how to do this problem?
 

Answers and Replies

  • #2
DrClaude
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You could approximate it by choosing a sphere of the same volume.
 
  • #3
TSny
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Considering the size of the cube, will the value of the wavefunction be approximately constant throughout the cube?
 
  • #4
Considering the size of the cube, will the value of the wavefunction be approximately constant throughout the cube?
Yes the value of wavefunction will be approximately constant throughout the cube.
 
  • #5
Yes the value of wavefunction will be approximately constant throughout the cube.
so it is enough to find the probability density at 50pm times the volume of cube right?
 
  • #6
TSny
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Yes the value of wavefunction will be approximately constant throughout the cube.
Yes

so it is enough to find the probability density at 50pm times the volume of cube right?
Yes
 
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