Question

1. In this problem, we are going to look at a three-level system. A spin-1 particld is placed in a constant magnetic field along the a-direction with strength B,. The spin-1 particle İs initialized in a z-eigenstate with positive eigenvalue h, ie, the i 1,m 1) state. What is the probability to find the negative eigenvalue the spin along the z axis as a function of time? Assume that the spin-1 particle has inagnetic moment 2 × μιι, i.e. that of two electrons whose spins are locked together to form a spin-1 system. Hints: In order to write down the Hamilton operator you may want to choose a basis different from the standard basis. In addition, it may be useful to use results from the midterm. 3 points)

Answer 1

Solution Given thab ihe spin pavticle is placed in a costonk magnetic field alcng the x -dixection th st rength B, it is initialized in ΐ eigen state cith positive agen value h igen valuc h the(j-1, m.'> state the spin-1 part icle has magnetic rnC meat 2xμ Let us first calculate the tonian of the Syshem 3 た the three matyices o spin A particle 2 Hariitonian let c-

The eigen value of H "(-){(*.NC ») ●y.e {e(-지-o了-o NOCs let us find the eigen vector of the Hamilonian at:0 So the eigen vector for λ:0 2 norcnalized

e2e e 2 2 r> ag the cigon vector 고 -

2e o CS The eigen vector N2 oxalited vecto 2 the gane alized eigoo state ahich enve lopth time n be coher c 3、2시β8xk.

The parti cles tially of z eigen state esith eigen vee f So, for t o 2 fomalization constant eqcuation t ek all the Coficieoks t - k(142) ae scaltod by Q

So, Final No,ce have to find the probnbiits of finding the particle tn cigen value state oith i<e Fixs cse csill find No 豆. O o So

Probability 욀[014을 exp (.ite')-ĩ exp (it»)(-14-le.fte') tap (.ite)) ine :な、/2 +2 sino cohe the probability as afunction of time