Consider an electron in the NN shell. What is the smallest orbital angular momentum it could have?

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 7a

Chapter 41, Problem 7a

Consider an electron in the $N$ shell. What is the smallest orbital angular momentum it could have?

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Understand the quantum number system: The N shell corresponds to the principal quantum number $ n = 4 $. The orbital angular momentum of an electron is determined by the azimuthal quantum number $ l $, which can take integer values from $ 0 $ to $ n-1 $.

Identify the smallest possible value of $ l $: Since $ l $ starts at $ 0 $, the smallest orbital angular momentum corresponds to $ l = 0 $.

Recall the formula for orbital angular momentum: The magnitude of the orbital angular momentum is given by $ \sqrt{l(l+1)} \hbar $, where $ \hbar $ is the reduced Planck's constant.

Substitute $ l = 0 $ into the formula: When $ l = 0 $, the orbital angular momentum becomes $ \sqrt{0(0+1)} \hbar = 0 $.

Conclude: The smallest orbital angular momentum an electron in the N shell can have is $ 0 $, which occurs when $ l = 0 $.

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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. It is quantized and can be expressed using the formula L = mvr, where L is the angular momentum, m is the mass of the electron, v is its velocity, and r is the radius of its orbit. In quantum mechanics, the angular momentum is also described by quantum numbers.

Quantum Numbers

Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. The principal quantum number (n) indicates the energy level and size of the orbital, while the azimuthal quantum number (l) determines the shape of the orbital. For an electron in the N shell, n = 4, and the smallest value of l is 0, corresponding to an s orbital.

Quantization of Angular Momentum

In quantum mechanics, angular momentum is quantized, meaning it can only take on specific discrete values. The quantization condition for orbital angular momentum is given by L = √(l(l+1))ħ, where l is the azimuthal quantum number and ħ is the reduced Planck's constant. For the smallest orbital angular momentum, we use l = 0, leading to L = 0, indicating no angular momentum for that state.

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A photon is emitted when an electron in a three-dimensional cubical box of side length $8.00 \times 10^{- 11}$ m makes a transition from the $n sub X equals 2$ , $n sub Y equals 2$ , $n sub Z equals 1$ state to the $n sub X equals 1$ , $n sub Y equals 1$ , $n sub Z equals 1$ state. What is the wavelength of this photon?

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Textbook Question

Consider an electron in the $N$ shell. What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of $ℏ$ and in SI units.

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Textbook Question

Consider an electron in the $N$ shell. What is the largest orbital angular momentum it could have? Express your answers in terms of $\hslash$ and in SI units.

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Consider an electron in the $N$ shell. What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of $\hslash$ and in SI units.

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Textbook Question

Model a hydrogen atom as an electron in a cubical box with side length $L$ . Set the value of $L$ so that the volume of the box equals the volume of a sphere of radius $a = 5.29 \times 10^{- 11}$ m, the Bohr radius. Calculate the energy separation between the ground and first excited levels, and compare the result to this energy separation calculated from the Bohr model.

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