# Magnetic moment precession around magnetic field

I have a question regarding the magnetic moment of an atom. If you just have one atom, and you apply an external magnetic field in lets say the z-direction. Then the magnetic moment will precess around the direction of the magnetic field applied, and depending on the quantum numbers it can precess in different quantized directions.

The question is rather simple: WHY does it precess around the magnetic field ? My book only states that it do indeed precess if a magnetic field is applied, but as I'm asking, I'm not sure WHY it does ? Why can't it just point in one direction all the time ?

• Your book probably expects you to recognize that when you have a body with angular momentum subjected to a torque that is not colinear with the angular momentum you get precession. This is exactly like a toy top in that regard. – dmckee Aug 11 '13 at 17:31

The reason is simple. With the existence of external magnetic field, the coupling involve the magnetic moment quantum number $m$ and the z-component of angular momentum $L_z$: $$\mathcal{H}\propto \vec{B}\cdot \vec{L_z}$$ However, in quantum mechanics, the magnitude of possible $L_z$ are strictly less than magnitude of total angular momentum $L$: $$|L|=\sqrt{\ell(\ell+1)}\hbar$$ $$|L_z|=m\hbar$$ where $m\le \ell$.

So, how is it possible in the classical sense? The only possibility is that the angular momentum vector $\vec{L}$ is not align with the external magnetic field. It cant move toward z-axis either, otherwise, it will violate the above principle.

Mathematically, you can solve the Hamiltonian and find the expectation value of all $\langle L_x\rangle$, $\langle L_y\rangle$, $\langle L_z\rangle$ and you will see that there is sinusoidal oscillation in x- and y- direction. And there is a angle between $L_z$ and $L$ that is exactly equal to $\cos^{-1}(L_z/L)$, which is exactly the meaning of preccession.

corresponding to magnetic moment of atom there is magnetic field so when a atom is placed in external magnetic field there is interaction in magnetic moment and external magnetic field and energy depends upon the angle btw these two so mag moment try to align in B direc.but theta can not be changed because if it will change energy of atom will change which cant happen so mag momemt start to precise around B with same interaction energy

## protected by Qmechanic♦Nov 29 '13 at 12:34

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