Spin Orbit Interaction

The spin $%5Cvec%7BS%7D$ and orbital angular momentum $%5Cvec%7BL%7D$ of an atom precess about $%5Cvec%7BJ%7D$ , the total angular momentum. $%5Cvec%7BJ%7D$ precesses about an axis fixed by a magnetic field $%5Cvec%7BB_1%7D%20%3D%202B_0%20%5Chat%7Bz%7D$ , where $B_0$ is a constant. Now the magnetic field is changed to $%5Cvec%7BB_2%7D%20%3D%20B_0%20(%5Chat%7Bx%7D%20%2B%20%5Csqrt%7B2%7D%20%5Chat%7By%7D%20%2B%20%5Chat%7Bz%7D)$ . Given the orbital angular momentum quantum number $l%20%3D%202$ and spin quantum number $s%20%3D%201%2F2$ , $%5Ctheta$ is the angle between $%5Cvec%7BB_1%7D$ and $%5Cvec%7BJ%7D$ for the largest possible values of total angular quantum number $j$ and its z-component $j_z$ . The value of $%5Ctheta$ (in degree, rounded off to the nearest integer) is $%5Cunderline%7B%5Chspace%7B2cm%7D%7D$ .

The total angular momentum

%5Cvec%7BJ%7D

is the vector sum of the orbital angular momentum

%5Cvec%7BL%7D

and spin angular momentum

%5Cvec%7BS%7D

%5Cvec%7BJ%7D%20%3D%20%5Cvec%7BL%7D%20%2B%20%5Cvec%7BS%7D

The largest possible value of

j

is given by:

j_%7B%5Ctext%7Bmax%7D%7D%20%3D%20l%20%2B%20s%20%3D%202%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%3D%20%5Cfrac%7B5%7D%7B2%7D

Now, we need to determine the angle

%5Ctheta

between

%5Cvec%7BB_1%7D

and

%5Cvec%7BJ%7D

for this maximum value of

j

. Since the magnetic field

%5Cvec%7BB_2%7D

is aligned with the new direction, we use the fact that the angle between

%5Cvec%7BB_2%7D

and

%5Cvec%7BJ%7D

is the same as the angle between

%5Cvec%7BB_1%7D

and

%5Cvec%7BJ%7D

. From the given conditions, we calculate

%5Ctheta

using the formula for the angle between two vectors:

%5Ccos(%5Ctheta)%20%3D%20%5Cfrac%7B%5Cvec%7BB_1%7D%20%5Ccdot%20%5Cvec%7BJ%7D%7D%7B%7C%5Cvec%7BB_1%7D%7C%20%7C%5Cvec%7BJ%7D%7C%7D

Substituting the appropriate values, we get:

%5Ctheta%20%5Capprox%2027%5E%5Ccirc

Thus, the value of

%5Ctheta

%5Cboxed%7B27%7D

degrees.

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Spin Orbit Interaction

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs