Ch 41: Quantum Mechanics II: Atomic Structure

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 41: Quantum Mechanics II: Atomic Structure

Problem 13

Chapter 41, Problem 13

Calculate, in units of $U$ , the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of $2$ , $20$ , and $200$ . Compare each with the value of $n h$ postulated in the Bohr model. What trend do you see?

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Understand the problem: The orbital angular momentum of an electron in a hydrogen atom is quantized and given by the formula $ L = \sqrt{l(l+1)} \hbar $, where $ l $ is the orbital quantum number. For a given principal quantum number $ n $, $ l $ can take integer values from 0 to $ n-1 $. The maximum orbital angular momentum corresponds to $ l = n-1 $.

Step 1: Write the formula for the maximum orbital angular momentum. Substituting $ l = n-1 $ into the formula, we get $ L_{\text{max}} = \sqrt{(n-1)n} \hbar $.

Step 2: Express $ \hbar $ in terms of the unit $ U $. In the Bohr model, $ U = \hbar $, so $ L_{\text{max}} $ in units of $ U $ becomes $ L_{\text{max}} = \sqrt{(n-1)n} U $.

Step 3: Calculate $ L_{\text{max}} $ for $ n = 2, 20, $ and $ 200 $. For each case, substitute the value of $ n $ into the formula $ L_{\text{max}} = \sqrt{(n-1)n} U $. For example, for $ n = 2 $, $ L_{\text{max}} = \sqrt{(2-1) \cdot 2} U = \sqrt{2} U $. Repeat this for $ n = 20 $ and $ n = 200 $.

Step 4: Compare $ L_{\text{max}} $ with the Bohr model prediction. In the Bohr model, the angular momentum is postulated to be $ L = nU $. For each $ n $, compare the calculated $ L_{\text{max}} $ with $ nU $. Observe that as $ n $ increases, $ L_{\text{max}} $ approaches $ nU $, indicating that the quantum mechanical result converges to the classical Bohr model prediction for large $ n $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Orbital Angular Momentum

Orbital angular momentum is a measure of the rotational motion of an electron around the nucleus of an atom. In quantum mechanics, it is quantized and given by the formula L = mvr, where m is the mass, v is the velocity, and r is the radius of the orbit. For hydrogen, the angular momentum is also expressed in terms of the principal quantum number n, with L = nħ, where ħ is the reduced Planck's constant.

Principal Quantum Number (n)

The principal quantum number (n) indicates the energy level of an electron in an atom and determines the size of the electron's orbit. It can take positive integer values (1, 2, 3, ...), with higher values corresponding to higher energy levels and larger orbits. In the context of the hydrogen atom, the maximum orbital angular momentum is directly proportional to n, leading to the relationship L = nħ.

Bohr Model

The Bohr model of the hydrogen atom describes the electron as moving in fixed orbits around the nucleus, with quantized energy levels. According to this model, the angular momentum of the electron is quantized and given by L = nħ, where n is the principal quantum number. This model provides a foundational understanding of atomic structure and helps predict the behavior of electrons in hydrogen-like atoms, allowing for comparisons of angular momentum across different energy levels.

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Textbook Question

A hydrogen atom undergoes a transition from a $2 p$ state to the $1 s$ ground state. In the absence of a magnetic field, the energy of the photon emitted is $122$ nm. The atom is then placed in a strong magnetic field in the $z$ -direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. How many different photon wavelengths are observed for the $2 p right arrow 1 s$ transition? What are the $m_{l}$ values for the initial and final states for the transition that leads to each photon wavelength?

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Textbook Question

In a particular state of the hydrogen atom, the angle between the angular momentum vector $\vec{L}$ and the $z$ -axis is $u equals 26.6$ °. If this is the smallest angle for this particular value of the orbital quantum number $l$ , what is $l$ ?

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Textbook Question

A hydrogen atom in a $3 p$ state is placed in a uniform external magnetic field $\vec{B}$ . Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. What field magnitude $B$ is required to split the $3 p$ state into multiple levels with an energy difference of $2.71 \times 10^{- 5}$ eV between adjacent levels?

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Textbook Question

Consider an electron in the $N$ shell. For the electron in part (c), what is the ratio of its spin angular momentum in the $z$ -direction to its orbital angular momentum in the $z$ -direction? Note: Part (c) asked for the largest orbital angular momentum this electron could have in any chosen direction.

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Textbook Question

A hydrogen atom is in a $d$ state. In the absence of an external magnetic field, the states with different $m_{l}$ values have (approximately) the same energy. Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. Calculate the splitting (in electron volts) of the ml levels when the atom is put in a $0.800$ T magnetic field that is in the $+ z$ -direction

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Textbook Question

The orbital angular momentum of an electron has a magnitude of $4.716 \times 10^{- 34}$ kg-m²/s. What is the angular momentum quantum number $l$ for this electron?

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Calculate, in units of UU, the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of 22, 2020, and 200200. Compare each with the value of nhnh postulated in the Bohr model. What trend do you see?

Verified video answer for a similar problem:

Key Concepts

Orbital Angular Momentum

Principal Quantum Number (n)

Bohr Model

Calculate, in units of $U$ , the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of $2$ , $20$ , and $200$ . Compare each with the value of $n h$ postulated in the Bohr model. What trend do you see?