About precessing directon of angular motion in quantum mechanics

[KFC]

Hi all,
I got a feeling that learning quantum mechanics is easy and hard. Most of time, it is easy to "accept" all the concepts given in the book by simply looking at their mathematical interpretation. But it is hard if you really take it serious to try to understand everything from the physical knowledge learned in classical point of view.

One problem I am learning is about the (quantized) angular momentum. The classical picture of angular momentum is pretty straightforward. Though the quantization of the angular momentum in quantum physics is not a super hard concept to accept (at least from math used to solve the PDE of Schrödinger), I still got stuck on some aspect especially on the z component of the angular momentum. My textbook emphasizes that angular momentum does not have a prefer direction. If that's the case, what's the point to select a special axis z for problem analysis. Or I ask which direction is really the z axis pointing to? I was told that the electron could have many different angular momentum pointing different direction and all those angular momenta are precessing about a same axis called z axis. It is confusing why they all precessing about the same direction?

 

[Simon Bridge]

The z axis for the particle with angular momentum may point in any direction in the lab.
A gyroscope may be set spinning in any orientation, but it definitely has an axis of rotation, and maybe of precession, what would you use to refer to it?

You can also establish a reference axis using lab equipment, as in the stern-gerlach experiment.
An electron in an atom experiences a direction imposed by the nuclear magnetic moment.

In another sense... it can be like how everything has x,y, and z components to linear velocity.
How does everything have the same set of axes?

 

[KFC]

The z axis for the particle with angular momentum may point in any direction in the lab.
A gyroscope may be set spinning in any orientation, but it definitely has an axis of rotation, and maybe of precession, what would you use to refer to it?

You can also establish a reference axis using lab equipment, as in the stern-gerlach experiment.
An electron in an atom experiences a direction imposed by the nuclear magnetic moment.

In another sense... it can be like how everything has x,y, and z components to linear velocity.
How does everything have the same set of axes?

Thanks. I am still digesting this information. But as I can understand form above so far, since the angular momentum does not have prefer direction, so you could always call a specific direction as z so you could project whatever angular momentum onto that z and the associated x and y, is that what you mean?

We know form the text that the z components of the angular momentum is quantized and has the value of $m_l\hbar$, so does it mean no matter what z axis is, you always have the z component as $m_l\hbar$? This is quite confusing. Let's assume there is a magnetic field pointing upward but I choose the z axis of our system along the direction making 45 degree to the magnetic field, will the z component of the angular momentum still be $m_l\hbar$?

 

[Simon Bridge]

Thanks. I am still digesting this information. But as I can understand form above so far, since the angular momentum does not have prefer direction, so you could always call a specific direction as z so you could project whatever angular momentum onto that z and the associated x and y, is that what you mean?

Close: see below.

We know from the text that the z components of the angular momentum is quantized and has the value of $m_l\hbar$, so does it mean no matter what z axis is, you always have the z component as $m_l\hbar$?

It means that whenever you measure the amount of angular momentum in some direction, that direction becomes the direction used in the general form of the mathematics and labelled "the z-direction" in the equations.

Let's assume there is a magnetic field pointing upward but I choose the z axis of our system along the direction making 45 degree to the magnetic field, will the z component of the angular momentum still be $m_l\hbar$?

No - the angular momentum is quantized in the direction you measured it, all you did was use a different label.

Did you look up the Stern-Gerlach experiment?

 

[KFC]

Close: see below.

It means that whenever you measure the amount of angular momentum in some direction, that direction becomes the direction used in the general form of the mathematics and labelled "the z-direction" in the equations.

No - the angular momentum is quantized in the direction you measured it, all you did was use a different label.

Did you look up the Stern-Gerlach experiment?

[/QUOTE]

Thanks a lot. I think I get some idea now.