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

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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

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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

Further Mechanics and Thermal Physics

Electric Charge Field and Potential

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Medical Physics

Radiation

Fluids

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 ${(L_{x}^{2} + L_{y}^{2})}^{1 / 2}$ ?