Question

while describing the electron we use the different quantum numbers: n, l, m, s.

where l is known as orbital angular momentum quantum number.

when l=0, means its orbital angular momentum is zero. It means there is no rotation, but how is it possible?

Answer 1

I've read up on this. A book "Chemical Principles" by Peter Atkins suggests that the electron in an s-orbital has such a simple(spherical) path to follow that it does so with a high velocity. So you shouldn't imagine it moving around the nucleus, like Earth revolves around the sun. Rather, as it is the quantum world, we should imagine the electron as moving so fast that it is evenly distributed through the spherical path. (Just like pi-electrons are evenly distributed all over a benzene ring, but you shouldn't imagine them as "circling" the ring).
The p and d orbitals have much more complicated paths, so the electron's motion in these paths can be imagined to be a regular motion of a 'ball moving in a dumb-bell shaped path.
Was that ok?

Answer 2

whatever you write is ok. But, I am thinking about the "ANGULAR MOMENTUM". Angular momentum has non-zero value of an object if and only if it is rotating. Since for l=0 case angular momentum is zero then how can I imagine that an object is rotating.

In books it is written that for l=0 case electron has spherical path but it means electron is rotating.

so, these two ideas are contradicting to me.........

Answer 3

The total angular momentum:
\[j = |ℓ ± s|\]

Answer 4

That's what I am saying. We shouldn't imagine the s-orbital electron to be rotating. We should picture it statically. Just a photo of the whole electron being distributed evenly around the nucleus. It is most difficult to see the motion of a s-electron, because, being attracted to the e=nucleus most strongly, it has the highest speed. So, it's motion is too fast to be predicted. Let us not forget that we're dealing with the quantum world, the dimensions of the sphere are extremely extremely small.
An extremely extremely small path to cover and a very speed to cover it -What will happen? The electron moves so fast that it is uniformly distributed all around the nucleus at any instant.
Don't think in terms of Newtonian Mechanics here. This is quantum mechanics. In Newtonian mechanics, if there was no orbital angular momentum of a planet, it would imply that the planet wasn't rotating. But in quantum mechanics, a zero angular momentum doesn't necessarily mean so. It could also mean what I have explain.

Answer 5

However, I am no expert on quantum mechanics.What UncleRhaukus said might be correct.

Answer 6

All l = 0 means is that the electron spends just as much time going clockwise around the nucleus as it does counter-clockwise. To have positive angular momentum, you need to be advancing in only one direction in your orbit, as seen from its pole. So, for example, if you get in a jet plan and fly to London, then fly back again, then your angular momentum around the Earth, measured over your complete journey, is zero. You spent just as much time going east as you did west.

You might well ask: how is it possible for the electron to reverse its direction of travel without an acceleration and hence emission of a photon? To some extent, this is the mystery of a stationary state. But the answer is probably that it doesn't: it reverse direction, absorbs a photon, then reverses direction again and absorbs it, and so forth. Relativity tells us that particles are always in equilibrium with a sea of virtual photons like this, that allow them to change their energy and angular momentum on a moment-to-moment basis all the time. That is, on the smallest time scales, and when you are not taking measurements, there is no need for energy or angular momentum to be conserved. Part of the magic of quantum mechanics -- still unexplained -- is why this is true, and yet energy and momentum must and are conserved whenever you make a measurement.

Answer 7

Oops, I meant the electron reverse direction, absorbing a virtual photon, then reverse direction again and emits it back.