Show that unless $l=s$, L and S cannot be exactly opposite: that is, show that at its minimum possible value. Forwhich $j=l-s$. The magnitude of the total angular momentum is strictly greater than the difference $\left|L-S\right|$between the magnitudes of the orbital and intrinsic angular momentum vectors.

The given data can be listed below as:

• The total angular momentum’s magnitude is J .
• The orbital angular momentum is L .
• The spin angular momentum is S.
• The minimum value of the total angular momentum’s magnitude is j .
• The minimum value of the orbital angular momentum is l.
• The minimum value of the intrinsic angular momentum is s .

The angular momentum is described as the quantity of the rotation of a particular system. The quantity of the rotation is described as the product of the angular velocity and the moment of inertia of a system.

The equation of the total angular momentum is expressed as:

$J=\sqrt{j\left(j+1\right)\hslash }$

Here, J is the total angular momentum’s magnitude, j is the minimum value of the total angular momentum’s magnitude and $\hslash$is the Planck’s constant.

The equation of the orbital angular momentum is expressed as:

$L=\sqrt{l\left(l+1\right)\hslash }$

Here, L is the orbital angular momentum, l is the minimum value of orbital angular momentum and$\hslash$is the Planck’s constant.

The equation of the spin angular momentum is expressed as:

$S=\sqrt{s\left(s+1\right)\hslash }$

Here, is the spin angular momentum, is the minimum value of spin angular momentum and $\hslash$is the Planck’s constant.

The equation of the minimum value of the square of the total angular momentum is expressed as:

$\begin{array}{rcl}{j}_m^2& =& \left(l-s\right)\left(l-s+1\right){\hslash }^2\\ & =& \left(l\left(l+1\right)+s\left(s+1\right)-2s-2sl\right){\hslash }^2\\ & =& \left(l\left(l+1\right)+s\left(s+1\right)-2s\left(l+1\right)\right){\hslash }^2\\ & & \end{array}$…(i)

The equation of the difference amongst the spin and orbital angular momentum is expressed as:

$$\begin{gathered} \left|L-S\right|=\hslash \left|\sqrt{l\left(l+1\right)}-\sqrt{s\left(s+1\right)}\right| \\ {\left|L-S\right|}^2={\hslash }^2\left|l\left(l+1\right)+s\left(s+1\right)-2\sqrt{sl\left(l+1\right)\left(s+1\right)}\right| \end{gathered}$$…(ii)

Comparing the equation (i) and (ii), it can be identified that

$s\left(l+1\right)<\sqrt{ls\left(l+1\right)\left(s+1\right)}$

Squaring the above equation, the equation can be expressed as:

$$\begin{gathered} s\left(l+1\right)<l\left(s+1\right) \\ sl+s<sl+l \end{gathered}$$

Hence, $l>s$as then .$j_m>\left|L-S\right|$

Thus, magnitude J of the total angular momentum is strictly greater than the difference $\left|L-S\right|$between the magnitudes of the orbital and intrinsic angular momentum vectors.