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

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

Step by step solution

01

Step 1. Given information

The equation of total angular momentum,

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

Here,

$L = \sqrt{l (l + 1)} 鈩�$

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

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

{(L_{z})}_{\max} = l 鈩�

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

02

Step 2. Finding the minimum value of Lx2+Ly212

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

$= l (l + 1) {鈩�}^{2} - l^{2} {鈩�}^{2}$

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

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

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

03

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

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

$= l (l + 1) {鈩�}^{2} - 0$

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

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

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

$= \sqrt{6} 鈩�$

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

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Most popular questions from this chapter

Find the distance, in terms of $a_{B}$ , between the two peaks in the radial probability density of the $2 s$ state of hydrogen.

Hint: This problem requires a numerical solution.

In a multielectron atom, the lowest $- l$ state for each $n (2 s, 3 s, 4 s, e t c .)$ is significantly lower in energy than the hydrogen state having the same $n$ . But the highest $- l$ state for each $n (2 p, 3 d, 4 f e t c)$ is very nearly equal in energy to the hydrogen state with the same $n$ . Explain.

Identify the element for each of these electron configurations. Then determine whether this configuration is the ground state or an excited state.

a. $1 s^{2} 2 s^{2} 2 p^{5} 3 s$

b. $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{2}$

An electrical discharge in a neon-filled tube maintains a steady population of $1.0 脳 10^{9}$ atoms in an excited state with $蟿 = 20 n s$ . How many photons are emitted per second from atoms in this state?

An excited state of an atom has a $25 n s$ lifetime. What is the probability that an excited atom will emit a photon during a $0.50 n s$ interval?

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