Electron Probability Graphing

[Seiryuu]

From any wavefunction we can determine the probability of where an electron may be. I don't understand the rationale of taking the square of R( r ) instead of using its absolute value. Could someone explain this?

 

NOTE: I haven't taken surface area into account yet.

Edited November 15, 2011 by Seiryuu

[mississippichem]

The wavefunctions can in theory be complex valued. So the "psi squared" is really just the real equivalent of some complex valued psi times its complex conjugate.

[Seiryuu]

Complex conjugate?

[mississippichem]

Complex conjugate?

 

The complex conjugate of a complex number is another complex number with the same real part and an oppositely signed imaginary part. For example the complex conjugate of 2-3i is 2+3i.

 

Try to multiply any complex number with its complex conjugate. What do you notice about the result?

[Seiryuu]

So you end up with a positive value. But that's not squaring, is it? You're just multiplying a complex number by its conjugate.

[mississippichem]

Squaring also gives you real values but not necessarily positive.

 

The purpose of the squaring (or multiplying by the complex conjugate which is more general) is so you can always get a positive and real probability.

 

After normalizing, if we want to integrate with respect to position to get a probability we need a positive real value.

 

Even more formally the wavefunction would be in terms of vectors and we would be using an inner product with the hermetian conjugate IIRC instead of squaring or multiplying by the complex conjugate and integrating.

 

In the case of real valued wave functions, you can just square it because the imaginary parts are zero.

Edited November 16, 2011 by mississippichem