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Bml
Bml
07:11
@JohnRennie Hi :-)
John Rennie
John Rennie
Hi :-)
Bml
Bml
What do you think about my previous doubt? :-)
John Rennie
John Rennie
If you have an electron with some wavefunction ψ then |ψ|² is the probability density
That is, if you take some infinitesimal volume dV then the probability the electron is in that small volume is:
dP = |ψ|² dV
Bml
Bml
Yes
John Rennie
John Rennie
And to get the probability the electron is in some non-infinitesimal volume you integrate over the volume so:
P(V) = ∫ |ψ|² dV
But this isn't really anything to do with the uncertainty principle.
Bml
Bml
07:19
@JohnRennie I suspected it...What is the reason ?
John Rennie
John Rennie
Do you mean what is the explanation for the uncertainty principle?
Bml
Bml
@JohnRennie First of all, why are you confusing the uncertainty principle with the Born rule...And what is the reason why the former has nothing to do with the latter. Thank you very much...
John Rennie
John Rennie
I have another question to asnwer so I'll have to pause for a bit. I'll ping you when I'm free.
Bml
Bml
@JohnRennie OK, thanks.
John Rennie
John Rennie
07:46
@Bml Hi :-)
Bml
Bml
Hi :-)
John Rennie
John Rennie
The HUP doesn't say you can't know the position of the electron. It says you cannot simultaneously know the position and the momentum.
 
3 hours later…
Bml
Bml
10:33
@JohnRennie I know it differently. It is possible to know both velocity and position simultaneously with some precision, but only up to a certain point. We cannot know both with arbitrary precision. Yes?
John Rennie
John Rennie
Yes. I guess I should say you cannot simultaneously know the position and the momentum precisely.
Bml
Bml
After that is not correct to see the square modulus as a probability distribution for where the electron is. It tells how how likely it is to measure an electron in a given position. We can't say that the electron was definitely there just before we measured it. Yes? Right?
John Rennie
John Rennie
It's important to understand that the electron does not have a position. It exists in a superposition of all possible positions as described by the wavefunction.
Suppose the wavefunction that describes the electron as being exactly at the position xᵢ is written as δ(xᵢ)
Then our wavefunction ψ can be written as:
ψ = Σᵢ aᵢ δ(xᵢ)
where aᵢ is just a constant.
The aᵢs tell us how much of each δ(xᵢ) is in our wavefunction ψ.
Does this make sense so far?
Bml
Bml
@JohnRennie Yes
Electron cloud, IMO, is one of these ideas that stems from a general reluctance or inability not to think classically.If the electron does not have a well-defined position, then it could still admit a classical model by being a cloud (or distribution) of electric charge. Whereas, it is not a cloud of electric charge. In the QM model of the atom bound energy states are fundamental, not the position of the particles.
Thinking of a proton and an electron (in the case of Hydrogen) as an energy eigenstate is a fundamentally non classical concept.
Right?
John Rennie
John Rennie
So:
ψ² = Σᵢ,ⱼ aᵢ δ(xᵢ) aⱼ δ(xⱼ)
But all the cross products where i ≠ j will be zero so this reduces to:
ψ² = Σᵢ aᵢ² δ²(xᵢ)
So you can see that the ψ² is telling us about the constants aᵢ² that tell us how much of each position is in our wavefunction.
So it's a bit misleading to say ψ² tells us the probability of the electron being at that position. It's more like telling us how much of the electron is at that position.
Bml
Bml
10:54
@JohnRennie How much you are likely to find the electron in this position, right?
John Rennie
John Rennie
Yes
Bml
Bml
What about my thinking of electron cloud?
John Rennie
John Rennie
Well you're correct it's not a cloud in the classical sense, though I often use this analogy with students so it's not a terrible analogy.
Bml
Bml
Yes, but it is not a real cloud, right? IMO, the cloud picture is close to be right, but we have to interpret the cloud correctly. It's not to be understood as if the electron is a continuous entity smeared as a "cloud" over the whole space.

The cloud rather represents the probability distribution for the position of the electron around the nucleus. This is the basics of the Born Rule, which lies at the heart of the Minimal Statistical Interpretation
Yes? What do you think?
John Rennie
John Rennie
Yes, it's not a real cloud.
In fact you need to be careful about taking the wavefunction physically as it's far from clear if the wavefunction is a physical object at all. Most of the physicists I know would take the view that it's a mathematical function that describes the particle.
Whether the wavefunction is actually a real object is unknown.
Bml
Bml
11:17
So, where is the electron when we measure it?
John Rennie
John Rennie
Any measurement process interacts with the wavefunction and that interaction changes the wavefunction. So when we measure the position we end up with a different wavefunction.
The original wavefunction is gone.
Bml
Bml
@JohnRennie So we cannot say anything about the position of the electron even after the measurement... We can only say that it is around the nucleus (although there is a slight chance of finding it inside the nucleus), right?

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