Chapter 41: Q. 28 (page 1207)

A hydrogen atom has $l = 2$ . What are the (a) minimum (as a multiple of $h$ ) and (b) maximum values of the quantity ${(L_{x}^{2} + L_{y}^{2})}^{1 / 2}$ ?

Short Answer

Expert verified

Part (a) The minimum value of ${(L_{x}^{2} + L_{y}^{2})}^{1 / 2} = \sqrt{2} ℏ$

Part (b) The maximum value of ${(L_{x}^{2} + L_{y}^{2})}^{1 / 2} = \sqrt{6} ℏ$

Step by step solution

Step 1. Given information

The equation of total angular momentum,

$L^{2} = L_{x}^{2} + L_{y}^{2} + L_{z}^{2}$

Here,

$L = \sqrt{l (l + 1)} ℏ$

$(L_{x}^{2} + L_{y}^{2}) = L^{2} - L_{z}^{2}$

Here $L_{x}, L_{y}, L_{z}$ are the components of angular momentum

{(L_{z})}_{\max} = l ℏ

${(L_{z})}_{\min} = 0$

Step 2. Finding the minimum value of Lx2+Ly212

$(L_{x}^{2} + L_{y}^{2}) = L^{2} - {(L_{z})}_{\max}^{2}$

$= l (l + 1) ℏ^{2} - l^{2} ℏ^{2}$

${(L_{x}^{2} + L_{y}^{2})}^{1 / 2} = \sqrt{l} ℏ$

${(L_{x}^{2} + L_{y}^{2})}^{1 / 2} = \sqrt{2} ℏ$

The minimum value of ${(L_{x}^{2} + L_{y}^{2})}^{1 / 2} = \sqrt{2} ℏ$

Step 3. Finding the maximum value of Lx2+Ly21/2=6ℏ

$(L_{x}^{2} + L_{y}^{2}) = L^{2} - {(L_{z})}_{\min}^{2}$

$= l (l + 1) ℏ^{2} - 0$

$(L_{x}^{2} + L_{y}^{2}) = l (l + 1) ℏ^{2}$

${(L_{x}^{2} + L_{y}^{2})}^{1 / 2} = \sqrt{l (l + 1)} ℏ$

${(L_{x}^{2} + L_{y}^{2})}^{1 / 2} = \sqrt{2 (2 + 1)} ℏ$

$= \sqrt{6} ℏ$

The maximum value of ${(L_{x}^{2} + L_{y}^{2})}^{1 / 2} = \sqrt{6} ℏ$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

A hydrogen atom has $l = 2$ . What are the (a) minimum (as a multiple of $h$ ) and (b) maximum values of the quantity ${(L_{x}^{2} + L_{y}^{2})}^{1 / 2}$ ?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the minimum value of Lx2+Ly212

Step 3. Finding the maximum value of Lx2+Ly21/2=6ℏ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Circular Motion and Gravitation

Electricity and Magnetism

Particle Model of Matter

Fluids

Famous Physicists

Study anywhere. Anytime. Across all devices.

A hydrogen atom has l=2. What are the (a) minimum (as a multiple of h ) and (b) maximum values of the quantity Lx2+Ly21/2 ?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the minimum value of Lx2+Ly212

Step 3. Finding the maximum value of Lx2+Ly21/2=6ℏ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Circular Motion and Gravitation

Electricity and Magnetism

Particle Model of Matter

Fluids

Famous Physicists

Study anywhere. Anytime. Across all devices.

A hydrogen atom has $l = 2$ . What are the (a) minimum (as a multiple of $h$ ) and (b) maximum values of the quantity ${(L_{x}^{2} + L_{y}^{2})}^{1 / 2}$ ?