How Do I Create the State Vector χ for Larmor Precession in a 2D B-field?

[Benhur]

Homework Statement

I have a situation with a spin 1/2 particle in a couple of magnetic fields (one - Bo - in the Z direction, and another - Γo - in the X direction). I found the energy eigenvalues for the system, but I don't know how to mount the time-dependent state vector, χ(t). I need this to determine the expected values for Sx, Sy and Sz. (It's a problem inspired in the Example 4.3 of the Griffiths book: Introduction to Quantum Mechanics - Chapter 4 -)

Relevant Equations

The Hamiltonian in the matrix form (I guess it is made by the relation H = -γ.Bo.Sz - γ.Γo.Sx); Pauli spin matrices; some other relations in the Chapter 4 of the book.

I just tried to find the eigenvalues (for the energy), obtaining E = ±(γħ.√(Bo² + Γo²))/2 and the corresponding eigenvectors for the H matrix. But I don't know what to do to create de state vector χ.

 

[kuruman]

Do you know $\chi (0)$? If so, write it is as a linear combination of the eigenvectors (I assume you found them) and tack on the time dependence in the usual way, $e^{-iE_1 t/\hbar}$ and $e^{-iE_2 t/\hbar}$.

 

[Benhur]

So, I don't have a specific initial state. Instead, I have the general expression χ(0) = (a b). I was thinking in use the consideration of the book (a = cos(α/2) and b = sin(α/2)). But... in the end, precession seems like it's going to get weird (am I right to say that the precession it's going to be with a angle between Z and X axis?)

 

[kuruman]

The precession will be around the resultant of the two components of the external magnetic field. All you are doing is redefining the coordinate system relative to the coordinate system in Griffiths in which the field $B_0$ is taken to be along the $z$-axis. The Larmor frequency in your case will be $\omega_L=\gamma \sqrt{B_0^2+\Gamma_0^2}$. Also the expectation value of $S_z$ in your system of coordinates will end up having a time dependence. It looks like a lot of algebra that will obfuscate rather than clarify one's understanding of the physical situation. In this problem we put the axis of quantization $z$ along the field direction (without loss of generality) for a good reason, it makes the analysis simple to do and understand.

Compare this with projectile motion. Sure, you could devise a coordinate system in which the acceleration of gravity is at arbitrary spherical angles $\theta$ and $\phi$ relative to the cartesian axes, but what will you gain from that other than a more complex description of the motion?

 

[Benhur]

Yes, I got it. Thank you for the explanation, I was very confused but now I can see a little bit better. I will explore more to see how far I get.