(a) Starting with the expression 𝐉=𝐋+𝐒 for the total angular momentum of an electron, derive an

(a) Starting with the expression 𝐉=𝐋+𝐒 for the total angular...

(a) Starting with the expression $\mathbf{J}=\mathbf{L}+\mathbf{S}$ for the total angular momentum of an electron, derive an expression for the scalar product $\mathbf{L} \cdot \mathbf{S}$ in terms of the quantum numbers $j, \ell,$ and $s .$ (b) Using $\mathbf{L} \cdot \mathbf{S}=|\mathbf{L}||\mathbf{S}| \cos \theta$ where $\theta$ is the angle between $\mathbf{L}$ and $\mathbf{S},$ find the angle between the electron's orbital angular momentum and spin angular momentum for the following states: (1) $P_{1 / 2}, P_{3 / 2}$ and $(2) H_{9 / 2}, H_{11 / 2}$

Transcript

00:01 Hello students, so in this question take j equals to l plus s and squaring both sides, both sides we will get j square equals l plus s whole square so from here we will get j square equals to l square plus s square plus 2 mod of l and mod of s so so which is equal to j multiplied by j plus 1 multiplied by h square which is equal to l square plus s square plus 2 l .s.

00:41 So from here which is equal to j, j plus 1, h square equals to l, l plus 1, h square plus s, s plus 1, h square plus 2 ls.

00:57 So this implies ls equals to j j plus 1 minus l l plus 1 plus s plus s plus h squared divided by 2.

01:15 So from here we will get mod of l mod of s cos theta equals to j j j plus 1 minus l l l plus 1 plus s s plus s plus so from here we will get cos theta, cost theta equals to j, j plus 1 minus l, l plus 1 plus s plus 1, divided by 2 root root l plus 1.

01:54 So this will be our cause theta now.

01:58 So this will be our cost theta now.

02:00 Now for p half, half state, for p half state, l equals to 1, l equals to 1 and s equals to plus half and j equals to half.

02:15 So substituting the value, substituting, substituting the value in the above equation.

02:21 So cos theta equals to 3 divided by 4 minus 2 plus 3 divided by 4 and divided by 2 and divided by 2.

02:32 Root 2 multiplied by root 3 by 2.

02:36 So from here we will get cos theta equals to cos theta equals to minus 8 divided by 4 root 6 so theta equals to theta equals to 145 degree.

02:48 So from here we will get theta equals to 145 degree.

02:52 Now for p 3 by 2 state we we have l equals to 1 and s equals to 1 divided by 2 and j equals to 3 divided by 2...

Question

Please give Ace some feedback