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In applying conservation of angular momentum for an electron we use the relation $mvr=\frac {nh}{2 \pi}$ however the angular momentum has terms of velocity and distance from the line of action. According to Heisenberg's principle of Uncertainty we can only measure one of them with absolute precision and increasing precision in either increases our error in the other. Also as we know we can only get a probability region (orbitals) where the electrons are more likely to be find. So how do we compute the angular momentum precisely with these constraints?

An example would be an electron in the 1s orbital of the hydrogen atom. The orbital itself is spherically symmetrical however the electron can be anywhere in the region and neither it's velocity or position can be computed with absolute certainty without compromising on the other

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    $\begingroup$ can you give an example of what type of interaction you are talking about? To conserve there must be an interaction and I cannot off hand visualize one where consrvation of angular momentum is applied to a sngle particle. $\endgroup$
    – anna v
    Nov 29 '19 at 5:10
  • $\begingroup$ @annav I've updated. Kindly have a look $\endgroup$ Nov 29 '19 at 5:24
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Conservation laws , either of quantum numbers or of energy, momentum,angular momentum need a "before and after",usually an interaction is allowed or not according to the conservation laws.

The example:

an electron in the 1s orbital of the hydrogen atom.

There is no before and after in this case. The atom is eternal, unless it interacts.

The orbital itself is spherically symmetrical however the electron can be anywhere in the region and neither it's velocity or position can be computed with absolute certainty without compromising on the other.

The electrons velocity and position cannot be computed at all( let alone with certainty). It is a probabilistic case. If the atom interacts, and the electron is detected , one can start on conservation laws.

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So how do we compute the angular momentum precisely with these constraints?

We do not compute the angular momentum precisely with these constraints.

Conservation of momentum is like an accounting principle.

The money in your bank account consists of money that is yours plus money that you owe. Maybe you don't know exactly how much money is in the account, or how much you owe. But still it's exactly true that the unknown amount of money there is equal to the unknown amount that's yours plus the unknown amount that you owe.

And it will still be true after your bank account has a state change, and money is deposited or money is paid from it. Banks create money out of nothing all the time, but whenever they create more money they create the same amount of debt so it always balances out. When the loan is paid off the money disappears. There is never any money that doesn't belong to anybody, though sometimes it's totally unclear who it belongs to.

Momentum is like that. It's conserved by definition.

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