Solutions of Heisenberg Equations of Motion for Angular Momenta

[Dixanadu]

Hey guys,

Imma type this up in word so its nice and clear!

http://imageshack.com/a/img32/2013/3q8s.jpg

 

[MuIotaTau]

Your equations look right to me, although I don't have a lot of time to check thoroughly right now. So your equations are coupled differential equations, right? They are of the form

$$y_1' = a y_2$$
$$y_2' = -a y_1$$
$$y_3' = 0$$

where the prime mark denotes differentiation. The last equation should be simplest to solve. What does it mean if a quantity's time derivative is zero? Does it change?

For the other two, I'm a bit rusty on some parts of differential equations, but I would start by taking the time derivative of your first two equations. See if doing that makes a solution clear.

 

[Dixanadu]

damn, sorry posted by accident. Let me type that again

 

[Dixanadu]

So our first equation is:

$\frac{d<L_{x}>}{dt}=\omega_L <L_{y}>$

If we differentiate this again w.r.t time:

$\frac{d^2 <L_{x}>}{dt^2}=\omega_L \frac{d<L_{y}>}{dt}$

But we know from our second equation that

$\frac{d<L_{y}>}{dt}=-\omega_L <L_{x}>$

Substituting gives

$\frac{d^2 <L_{x}>}{dt^2}=-\omega^{2}_{L} <L_{x}>$

Which has the solution $<L_{x}>=A cos(\omega_L t) + B sin(\omega_L t)$

But the weird thing is: if we repeat this same thing for the second differential equation, we get the same thing - that is

$\frac{d^2 <L_{y}>}{dt^2}=-\omega^{2}_{L} <L_{y}>$

So does this basically mean that $<L_{x}>=<L_{y}>$ and that $<L_{z}>$ is constant? I don't see how it can make sense that $<L_{x}>=<L_{y}>$...

 

[paranormal]

Hello,

Thank you for sharing this image and your work on the solutions of Heisenberg Equations of Motion for Angular Momenta. It appears that you have derived the equations for the time evolution of the angular momenta operators in quantum mechanics, which is an important aspect of understanding the behavior of particles at the atomic and subatomic level. Your work provides a valuable contribution to the field and I appreciate your dedication to advancing our understanding of the fundamental laws of physics. Keep up the good work!