Consider an electron in the N shell. (a) What is the...

Consider an electron in the $N$ shell. (a) What is the smallest orbital angular momentum it could have? (b) What is the largest orbital angular momentum it could have? Express your answers in terms of $\hslash$ and in SI units. (c) What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of $\hslash$ and in SI units. (d) 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. (e) 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?

Key Concepts

Orbital Angular Momentum Quantum Number (l)

In atomic physics, electrons are characterized by a set of quantum numbers, one of which is the orbital angular momentum quantum number, denoted by l. This quantum number can take on integer values ranging from 0 up to n?1, where n is the principal quantum number or energy level. The value of l determines the magnitude of the orbital angular momentum and influences the shape of the electron’s orbital. The magnitude of the orbital angular momentum is given by ?[l(l+1)]?, and its component along a chosen axis is quantized in units of ?, with possible values given by m (where m ranges from ?l to +l).

Quantization of Angular Momentum

Angular momentum in quantum mechanics is intrinsically quantized, which means it can only take certain discrete values. Both orbital and spin angular momentum follow this principle. The total magnitude of an angular momentum vector is determined by an expression that involves the corresponding quantum number (l for orbital momentum or s for spin), while the measured component along any axis, such as the z-axis, can only be one of a set of allowed values. This quantization is a fundamental property of quantum systems and is central to understanding the behavior of microscopic particles.

Measurement of Angular Momentum Projection

Although the total angular momentum of a particle is fixed by its quantum number, the measurement of its component along a specific direction (like the z-direction) yields different possible values. For orbital angular momentum, the maximum value of the z-component is achieved when the magnetic quantum number m equals l, giving a value of l?. Similarly, for spin angular momentum (with s=1/2 for an electron), the projection along a chosen axis can be either +½? or ?½?. This concept explains why even though the magnitude of the angular momentum is greater, a measurement along any axis is always limited to specific quantized components.

Spin Angular Momentum Quantum Number (s)

Spin angular momentum is an intrinsic property of particles, independent of their orbital motion. For electrons, this quantum number is s = 1/2, which means that the magnitude of the spin angular momentum is given by ?[s(s+1)]?, and its possible projections along any axis are restricted to +½? or ?½?. This intrinsic spin contributes to many of the unique quantum behaviors observed in electrons, such as fine structure in atomic spectra and the overall magnetic properties of materials.

Transcript

00:01 In this problem, we have an electron in an shell of hydrogen atom and we are asked to calculate some quantum numbers and related angular momentum.

00:11 So before starting the problem, i'm just giving some quick review.

00:16 This is a principal quantum number which from which we can easily calculate the angular, orbital angular momentum quantum number which is the small l, small l, which can, which can, which can.

00:31 Can be written that goes from 0, 1, 2, 3 to n minus 1.

00:36 And for the orbital angular momentum, which is uppercase l, is it now as under root l and 2l plus 1h bar.

00:44 And if you want to calculate an orbital angular momentum in some specific direction, which is totally arbitrary to be consistent with the both and take the z direction, which is given as lz equal to n -o -h, where ml is the magnetic and loomentum quantum number which goes from negative l to plus l.

01:08 And the same definition holds for the spin concept which is for spinning momentum and spinning momentum in dz direction.

01:17 Where ms for electron is like positive half or negative half.

01:24 So let's start with the problem.

01:28 So in the first part there are asking us to calculate the minimum value of orbital angular momentum in n -shell.

01:40 To make it more easy to understand, i'll take n equal to 4.

01:48 So if n equal to 4, the possible value of orbital angular momentum are 0, 1, 2, 3.

02:01 So what is the maximum value among these? it is 3 and the minimum is 0.

02:07 So to calculate the minimum angular the minimum, which is it should be minimum...

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