# Finding the probability of 1s electron within a cubical volume

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1. Nov 8, 2017

### Muthumanimaran

1. The problem statement, all variables and given/known data
How to calculate the probability of finding an 1s electron within 1 picometer cubic region located 50pm from the nucleus.
2. Relevant 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?

3. 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?

2. Nov 9, 2017

### Staff: Mentor

You could approximate it by choosing a sphere of the same volume.

3. Nov 9, 2017

### TSny

Considering the size of the cube, will the value of the wavefunction be approximately constant throughout the cube?

4. Nov 10, 2017

### Muthumanimaran

Yes the value of wavefunction will be approximately constant throughout the cube.

5. Nov 10, 2017

### Muthumanimaran

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

6. Nov 10, 2017

Yes

Yes