Why is probability defined in space rather than at a point in quantum mechanics?

[Naman Jain Kota]

I have started quantum mechanics on my own using online lectures. So i have very basic doubts:

1) probability of electron is defined in space rather than a point. My question is why don't we comment about probability at a point.
I thought two possible explanation that:
1.1)ψ²dx the dx term goes zero so probability at point is zero
1.2)electron is itself not a point object(which i don't know is true or not) 2)if we are given real valued wavefunction graphs can we can we comment on probability density without knowing the complex part.(i hope not)

 

[BvU]

Hello Naman,

1) probability density is defined. As you indicate, probability itself can only be given for a volume, not for a point.

2) no we can not. But 'real valued wavefunction graphs' are rare: most of the time we get presented amplitude graphs.

 

[Naman Jain Kota]

Ok, so can we say probability at a point is zero always.
And is electron point object??

Hello Naman,

1) probability density is defined. As you indicate, probability itself can only be given for a volume, not for a point.

2) no we can not. But 'real valued wavefunction graphs' are rare: most of the time we get presented amplitude graphs.

 

[BvU]

For all we know it is, yes. http://gabrielse.physics.harvard.edu/gabrielse/overviews/ElectronSubstructure/ElectronSubstructure.html are of the order of $10^{-20}$ m. Just try to imagine the charge density !