How Are the Components of Angular Momentum Derived Mathematically?

[jg370]

Homework Statement

In my study of Quantum Mechanics, I am using Introduction to Quantum Mechanics by David J. Griffiths. So far I have done quite well. However, as I come to the section on Angular Momentum, I need help to get further.

Homework Equations

Classically, the angular momentum of a particle is given by:

$$\mathbf{L = r\times p }$$

This is all good. But this is followed by component form of the above equation as:

$$L_x = yp_z-zp_y, L_y = zp_x-xp_z, L_z = xp_y-yp_z$$

I am curious how $$L_x, L_y , L_x$$ are mathematically derived

The Attempt at a Solution

I have look at other textbooks and various posting on internet but I have not been able to find anything to help me with. Hopefully, someone will suggest some thing.

Thanks

 

[D H]

Let $\mathbf r = x\hat x + y\hat y + z \hat z$ and [ltex]\mathbf p = p_x \hat x + p_y \hat y + p_z \hat z[/itex]. What ls $\mathbf L = \mathbf r \times \mathbf p$?

 

[ideasrule]

By the way, that definition is just part of classical mechanics. It's not unique to quantum mechanics.