Book edition 2nd Edition

Author(s) Randy Harris

Pages 633 pages

ISBN 9780805303087

Answers without the blur.

Just sign up for free and you're in.

Short Answer

What is the angle between $L$ and $S$ in a (a) $2 p_{3 / 2}$ and (b) $2 p_{1 / 2}$ state of hydrogen?

(a) The angle between L and S when they're aligned is $φ = 66^{o}$ .

(b) The angle between L and S when they're anti-aligned is $φ = 145^{o}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given data

$2 p_{3 / 2}$ State of hydrogen and $2 p_{1 / 2}$ state of hydrogen atom.

Step 2: Concept used

When L and S are aligned, they look like in figure 1.

Figure 1.

Here, $θ$ is the angle between and .

The state is for when and are anti-aligned, look like in figure 2.

Figure 2.

Step 3: Use the law of cosines in order to find angle

Use the law of cosines in order to find angle, by magnitude of the vectors.

$c^{2} = a^{2} + b^{2} - 2 ab cosθ J^{2} = L^{2} + S^{2} - 2 LS \cos J^{2} = L^{2} + S^{2} - 2 LS \cos (180^{o} - ϕ)$

Simplify further as shown below.

$\begin{array}{rcl} J^{2} & = & L^{2} + S^{2} - 2 LS \cos ϕ \\ J^{2} - L^{2} - S^{2} & = & 2 LS cosϕ \\ cosϕ & = & \frac{J^{2} - L^{2} - S^{2}}{2 LS} \\ ϕ & = & \cos^{- 1} (\frac{J^{2} - L^{2} - S^{2}}{2 LS}) \end{array}$ ……. (1)

Step 4: Find the magnitude of vectors L, S, and J

To find the magnitude of vectors L, S, and J, for which find quantum numbers l, s, and j.

For p shell, l=1.

The electron's spin s is 1/2 and 3/2 from the $2 p_{3 / 2}$ provide j.

Use these values in the equations.

$L = \sqrt{l (l + 1)} h, S = \sqrt{s (s + 1)} h, J = \sqrt{j (j + 1)} h L = \sqrt{(1) [(1) + 1]} h, S = \sqrt{(\frac{1}{2}) [(\frac{1}{2}) + 1]} h, J = \sqrt{(\frac{3}{2}) [(\frac{3}{2}) + 1]} h L = \sqrt{2} h, S = \frac{\sqrt{3}}{2} h, J = \frac{\sqrt{15}}{2} h$

Substitute the values in equation (1).

$\begin{array}{rcl} φ & = & \cos^{- 1} (\frac{J^{2} - L^{2} - S^{2}}{2 LS}) \\ = & \cos^{- 1} [\frac{{(\frac{\sqrt{15}}{2} h)}^{2} - {(\sqrt{2} h)}^{2} - {(\frac{\sqrt{3}}{2} h)}^{2}}{2 (\sqrt{2} h) (\frac{\sqrt{3}}{2} h)}] \\ = & \cos^{- 1} (\frac{1}{\sqrt{6}}) \\ = & 65 . 9^{0} \end{array}$

Therefore, the angle between L and S when they're aligned is role="math" localid="1658381059167" $66^{o}$ .

Step 5: Find the value of J

(b)

Find the value as follows:

$\begin{array}{rcl} J & = & \sqrt{j (j + 1) h} \\ = & \sqrt{\frac{1}{2} (\frac{1}{2} + 1) h} \\ = & \frac{\sqrt{3}}{2} h \end{array}$

Step 6: Find the angle between L and S when they're anti-aligned

Then that, along with the same L and S as before can be inserted into equation (2).

$\begin{array}{rcl} φ & = & \cos^{- 1} (\frac{J^{2} - L^{2} - S^{2}}{2 LS}) \\ = & \cos^{- 1} [\frac{{(\frac{\sqrt{3}}{2} h)}^{2} - {(\sqrt{2} h)}^{2} - {(\frac{\sqrt{3}}{2} h)}^{2}}{2 (\sqrt{2} h) (\frac{\sqrt{3}}{2} h)}] \\ = & \cos^{- 1} (- \frac{2}{\sqrt{6}}) \\ = & 144 . 7^{o} \end{array}$

Therefore, the angle between L and S when they're anti-aligned is $φ = 145^{o}$ .

Find Study Materials for

Create Study Materials

Select your language

Modern Physics

Answers without the blur.

Short Answer

What is the angle between $L$ and $S$ in a (a) $2 p_{3 / 2}$ and (b) $2 p_{1 / 2}$ state of hydrogen?

Step by Step Solution

Step 1: Given data

Step 2: Concept used

Step 3: Use the law of cosines in order to find angle

Step 4: Find the magnitude of vectors L, S, and J

Step 5: Find the value of J

Step 6: Find the angle between L and S when they're anti-aligned

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Select your language

Modern Physics

Answers without the blur.

Short Answer

What is the angle between L and S in a (a) 2p3/2 and (b) 2p1/2 state of hydrogen?

Step by Step Solution

Step 1: Given data

Step 2: Concept used

Step 3: Use the law of cosines in order to find angle

Step 4: Find the magnitude of vectors L, S, and J

Step 5: Find the value of J

Step 6: Find the angle between L and S when they're anti-aligned

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Work Energy and Power

Radiation

Astrophysics

Physics of Motion

Scientific Method Physics

Fluids

Torque and Rotational Motion

Nuclear Physics

Fields in Physics

Measurements

Space Physics

Electricity

Physical Quantities and Units

Medical Physics

Electrostatics

Further Mechanics and Thermal Physics

Mechanics and Materials

Engineering Physics

Energy Physics

Force

Particle Model of Matter

Waves Physics

Magnetism

Modern Physics

Atoms and Radioactivity

Dynamics

Electricity and Magnetism

Conservation of Energy and Momentum

Fundamentals of Physics

Kinematics Physics

Circular Motion and Gravitation

Linear Momentum

Rotational Dynamics

Oscillations

Translational Dynamics

Geometrical and Physical Optics

Thermodynamics

Magnetism and Electromagnetic Induction

Electromagnetism

Electric Charge Field and Potential

Turning Points in Physics

What is the angle between $L$ and $S$ in a (a) $2 p_{3 / 2}$ and (b) $2 p_{1 / 2}$ state of hydrogen?