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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 4: Q16P (page 158)

A hydrogenic atom consists of a single electron orbiting a nucleus with Z protons. ( $Z = 1$ would be hydrogen itself, $Z = 2$ is ionized helium , $Z = 3$ is doubly ionized lithium, and so on.) Determine the Bohr energies $E n (Z)$ , the binding energy $E_{1} (Z)$ , the Bohr radius $a (Z)$ , and the Rydberg constant R $(Z)$ for a hydrogenic atom. (Express your answers as appropriate multiples of the hydrogen values.) Where in the electromagnetic spectrum would the Lyman series fall, for $Z = 2$ and $Z = 3$ ? Hint: There鈥檚 nothing much to calculate here鈥� in the potential (Equation 4.52) $Z e^{2}$ , so all you have to do is make the same substitution in all the final results.
$V (r) = - \frac{e^{2}}{4 蟺 \overset{藱}{o_{0}}} \frac{1}{r}$ (4.52).

Short Answer

Expert verified

The electronic spectrum would be the Lyman seires fall, for $z = 2$ and $z = 3$ is at, $z = 2$ is $3.04 脳 10^{- 8} m .$

At $z = 3$ is $1.35 脳 10^{- 8} m .$

Step by step solution

01

Given

The potential is given by:

$V (r) = - \frac{e^{2}}{4 蟺 \overset{藱}{o_{0}}} \frac{1}{r}$

02

Electronic spectrum

Consider a hydrogen, if irradiated with light it will be excited to another energy level. The energies of these photons are easily calculated from the Bohr energy formula:

$E_{n} = - \frac{1}{n^{2}} \frac{m e^{4}}{2 h^{2} {(4 韦韦 {鈭�}_{0})}^{2}}$

Lyman lines range from $n_{i} = 2 t o n_{i} = 鈭� (w i t h n_{f} = 1)$ ;

the wavelengths range from

03

Finding the electronic spectrum

$\frac{1}{位_{2}} = R (1 - \frac{1}{4}) = \frac{3}{4} R 位_{2} = \frac{4}{3 R}$

Down to

role="math" localid="1658210586427" $\frac{1}{位_{1}} = R (1 - \frac{1}{鈭�}) = R 鈬� R_{1} = \frac{1}{R}$

For $Z = 2$

$位_{1} = \frac{1}{4 R} = \frac{1}{4 (1.097 脳 10^{7})} = 2.28 脳 10^{- 8} m$

to

$位_{2} = \frac{1}{3 R} = 3.04 脳 10^{- 8} m,$

Ultraviolet.

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Most popular questions from this chapter

What is the most probable value of r, in the ground state of hydrogen? (The answer is not zero!) Hint: First you must figure out the probability that the electron would be found between r and r + dr.

a) Check that the spin matrices (Equations 4.145 and 4.147) obey the fundamental commutation relations for angular momentum, Equation 4.134.

$S_{z} = \frac{h}{2} (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) (4.145) . S_{x} = \frac{h}{2} (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), s_{y} = \frac{h}{2} (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) (4.147) . [S_{x}, S_{y}] = i h S_{z}, [S_{y}, S_{z}] = i h S_{x}, [S_{z}, S_{x}] = i h S_{y} (4.134) . (b) Show that the Pauli spin matrices (Equation 4.148) satisfy the product rule 蟽_{x} 鈮� (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), 蟽_{y} 鈮� (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}), 蟽_{z} 鈮� (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) (4.148) . 蟽_{j} 蟽_{k} = 未_{jk} + i {\underset{I}{鈭� \overset{'}{o}}}_{jkl} 蟽_{I}, (4.153) .$

$Where the indices stand for x, y, or z, and {\overset{'}{o}}_{jkl} is the Levi - Civita symbol : + 1 ifjkl = 123, 231, or 2 = 312; - 1 if jkl = 132, 213, or 321; otherwise .$

Consider the observables $A = x^{2}$ and $B = L_{z}$ .

(a) Construct the uncertainty principle for $蟽_{A} 蟽_{B}$

(b) Evaluate $蟽_{B}$ in the hydrogen state $蠄_{n / m}$ .

(c) What can you conclude about $< xy >$ in this state?

(a) Work out the Clebsch-Gordan coefficients for the case $s_{1} = 1 / 2, s_{2}$ =anything. Hint: You're looking for the coefficients A and Bin

$| sm 鉄� = A | \frac{1}{2} \frac{1}{2} 鉄� | s_{2} (m - \frac{1}{2}) 鉄� + B | \frac{1}{2} (- \frac{1}{2}) 鉄� | s_{2} (m + \frac{1}{2}) 鉄�$

such that $| sm 鉄�$ is an eigenstate of . Use the method of Equations 4.179 through 4.182. If you can't figure out what $S_{x}^{(2)}$ (for instance) does to $| s_{2} m_{2} 鉄�$ , refer back to Equation 4.136 and the line before Equation 4.147. Answer:

;role="math" localid="1658209512756" $A = \sqrt{\frac{s_{2} + \frac{1}{2} 卤 m}{2 s_{2} + 1}}; B = 卤 \sqrt{\frac{s_{2} + \frac{1}{2} 卤 m}{2 s_{2} + 1}}$

where, the signs are determined by $s = s_{2} 卤 1 / 2$ .

(b) Check this general result against three or four entries in Table 4.8.

(a) Use the recursion formula (Equation 4.76) to confirm that when $I = n - 1$ the radial wave function takes the form

$R_{n (n - 1)} = N_{n} r^{n - 1} e^{- r / n a}$ and determine the normalization constant by direct integration.

(b) Calculate 200a and $< r^{2} >$ for states of the form $蠄_{n (n - 1) m} 路$

(c) Show that the "uncertainty" in $r (未_{r})$ is $< r > / \sqrt{2 n + 1}$ for such states. Note that the fractional spread in decreases, with increasing (in this sense the system "begins to look classical," with identifiable circular "orbits," for large ). Sketch the radial wave functions for several values of, to illustrate this point.

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