An electron in an atom has an angular momentum quantum...

An electron in an atom has an angular momentum quantum number of $2 .$ (a) What is the magnitude of the angular momentum of this electron in terms of $\hbar$ ? (b) What are the possible values for the $z$ -components of this electron's angular momentum? (c) Draw a diagram showing the possible orientations of the angular momentum vector $\overrightarrow{\mathbf{L}}$ relative to the $z$ -axis. Indicate the angles with respect to the $z$ -axis.

Key Concepts

Visualization of Angular Momentum Orientation

The angular momentum vector model provides a way to visualize how the angular momentum is oriented relative to an axis, often shown as a cone of possible orientations. The angle between the angular momentum vector and the axis is related to the z-component, helping to illustrate the discrete nature of the directional quantization.

Space Quantization

Space quantization refers to the phenomenon where the orientation of angular momentum in space is limited to certain discrete directions, as signified by the quantized z-component. This concept underpins the idea that in quantum systems, not all orientations are allowed for angular momentum vectors.

Orbital Angular Momentum Magnitude

The magnitude of an orbital angular momentum is determined by the orbital quantum number (l) using the formula ?(l(l+1))?. This expression reveals that the angular momentum is not simply an integer multiple of ? but depends on the quantum number in a more complex, quantized way.

Angular Momentum Quantization

In quantum mechanics, angular momentum is quantized, meaning that its magnitude and directional projections can only take on discrete values. This quantization is a fundamental departure from classical physics and is described using quantum numbers.

Magnetic Quantum Number

The magnetic quantum number (m) specifies the component of angular momentum along a chosen axis, typically the z-axis. It can take on any integer value between -l and +l, which explains why the projection of the angular momentum is quantized into discrete levels.

Transcript

00:01 In this exercise, we have an electron that's bound to an atom, and this electron has an orbital quantum number l equals 2.

00:10 And in question a, we have to find what is the magnitude of the angular momentum of the electron.

00:16 And we know that the magnitude of the angular momentum, capital l squared, is equal to the orbital quantum number l times l plus 1 times hbar squared.

00:28 So in our case, l squared is equal to 2 times 3 times h bar squared.

00:35 So this is 6h bar squared.

00:39 And for the reason, the magnitude of the angular momentum is the square root of 6 times h ball.

00:49 In question b, we have to find what are the possible values of the z component of the angular momentum, lz.

00:59 And remember that the possible values of any of the components of the angular momentum range from minus lh bar to lh bar.

01:22 So in this case, the possible values of lz are minus 2h bar, minus h bar, minus h bar, 0, h bar, and 2h bar.

01:38 So the range is from minus lh bar to lh bar with steps of 1h bar.

01:45 So this here is the answer to question b.

01:53 And in question c, we have to sketch the possible orientations of the angular momentum vector.

02:04 So this here, i'm going to take this to be the x -axis.

02:10 Okay, and the angular momentum i'm going to draw in blue, but before i do that, in the x -axis we have, actually, this is the l -z -axis, and here we have the zero angular momentum.

02:31 Here we have the h -bar, and here we have the 2 -h -bar, down here, the minus h -bar, and down here minus 2 -h -bar.

02:46 So we know that the magnitude of the vector l is the square root of 6 times h bar.

02:56 That's a constant...

Please give Ace some feedback