(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 $(2) H_{9 / 2}, H_{11 / 2}$

(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 $(2) H_{9 / 2}, H_{11 / 2}$

Key Concepts

Angular Momentum Operators

In quantum mechanics, angular momentum is represented by operators such as L (orbital angular momentum) and S (spin angular momentum). These operators satisfy specific commutation relations and their squared operators have eigenvalues determined by quantum numbers. The overall behavior of angular momentum in quantum systems is captured through these operators and their algebra.

Addition of Angular Momentum

The total angular momentum J is defined as the vector sum of the orbital (L) and spin (S) angular momenta. This addition follows specific rules and leads to the relation J = L + S. Squaring both sides gives J² = L² + S² + 2L·S, which forms the basis for derivations involving the scalar product L·S in terms of the quantum numbers.

Quantum Number Eigenvalues

Quantum numbers such as l, s, and j correspond to the eigenvalues of the squared angular momentum operators L², S², and J², respectively. These eigenvalues are given by l(l+1), s(s+1), and j(j+1). These relationships are key when expressing the scalar product L·S in terms of j, l, and s.

Spin-Orbit Coupling and Scalar Product Derivation

By using the relation J² = L² + S² + 2L·S and substituting the eigenvalue expressions for the squared operators, one can derive an expression for the scalar product L·S. This derivation, which results in L·S = 1/2 [j(j+1) - l(l+1) - s(s+1)], is central to understanding spin-orbit interactions in quantum systems.

Determining Angles Between Angular Momentum Vectors

The dot product of two angular momentum vectors can also be expressed as |L||S| cos?, where ? is the angle between them. By equating this with the derived expression for L·S, one can solve for the cosine of the angle and hence determine the angle between the orbital and spin angular momenta. This concept is important for understanding the geometric interpretation of angular momentum coupling in quantum mechanics.

Transcript

00:01 In this question we need to find an expression for the dot product l .s.

00:05 We start by considering the equation for the angular, total angular momentum, which is j equals l plus s.

00:19 We square both sides of this equation to get j square equals l square plus s squared plus two, dot product l .s.

00:36 So now if we rearrange these, we get that the dot product of l and s is equal to 1 over 2 j square minus l square minus s square.

00:51 Now all we need to do is plug in the quantum numbers to finally get l .s.

00:58 S equals h bar square which we can take out over two, j, j plus 1, minus l, l plus 1 minus s times s plus 1.

01:21 And this is exactly what we wanted.

01:25 An expression for the dot product l .s in terms of quantum number l, j and s.

01:31 The second part of the question, we need to get the angle theta between l and s.

01:36 We know that the dot product, l .s, can be written in terms of the angle theta, as magnitude of l, magnitude of s, times cosine theta, where theta is exactly the angle that we need to calculate.

02:03 If we put the quantum numbers in and we rearrange this equation, we get that cosine of theta is equal to l .s.

02:23 Now we put the quantum numbers in, so we get h bar square times square root of l times l plus 1.

02:38 Square root of s times s plus 1.

02:45 So this way we can substitute the expression that we just got over here for l.

02:55 .

02:56 And we plug it in here.

03:01 So that way h bar and h bar will cancel and we will be left with j...