Consider states with angular-momentum quantum number l=2 . (a) In units of ħ, what is the larges

step 1 We're told the angular momentum, quantum number, L is equal to 2. So in units of H -bar, we are

Consider states with angular-momentum quantum number l=2 ....

Key Concepts

Angular Momentum Vector Model and Direction

The vector model of angular momentum provides a classical analogy for understanding quantum angular momentum. In this model, the angular momentum vector L precesses around the z-axis, and the angle between L and the z-axis (?) is determined by the relationship cos(?) = Lz/L. This model helps to visualize why even when the projection Lz is at its maximum, the overall vector L makes a finite angle with the z-axis, reflecting the inherent distribution of angular momentum in three-dimensional space.

Relationship Between L and Lz

The angular momentum vector L can be decomposed into its projection Lz and the remaining component perpendicular to the z-axis. Because of the uncertainty principle, even when Lz is at its maximum (l?), the total angular momentum L has a magnitude ?[l(l+1)] ? which is larger than l?. This indicates that the angular momentum vector is not completely aligned along the z-axis, a fact that is central to the vector model of angular momentum in quantum mechanics.

Comparison of Angular Momentum States (l=2 vs l=3)

Different values of l result in different magnitudes of angular momentum and different possible projections. For example, when comparing l=2 and l=3, the smallest angle between L and the z-axis (which occurs when m=l) is given by the relation cos(?) = l/?[l(l+1)]. Thus, as l increases, the ratio of Lz to L changes, affecting the minimum possible angle. This comparison highlights how increasing the orbital quantum number not only increases the magnitude of the angular momentum but also alters the spatial orientation of the angular momentum vector relative to an external axis.

Magnitude of Orbital Angular Momentum

The magnitude of the orbital angular momentum vector L is quantized and given by the formula L = ?[l(l+1)] ?. This formula encapsulates the inherent uncertainty in the components of the angular momentum vector, meaning that even when one component (usually Lz) has a well-defined value, the total magnitude of L is larger than the absolute value of its projection except in the trivial case of zero angular momentum.

Angular Momentum Quantum Numbers

In quantum mechanics, angular momentum is characterized by the quantum numbers l and m. The quantum number l (orbital angular momentum quantum number) determines the magnitude of the angular momentum, while m (magnetic quantum number) represents the projection of the angular momentum along a chosen axis, typically the z?axis. These numbers arise from the solution to the angular part of the Schrödinger equation for spherically symmetric potentials and obey specific quantization rules, where l is a non?negative integer and m takes values from -l to +l in integer steps.

Projection of Angular Momentum

The projection of the angular momentum along a chosen axis, typically denoted Lz, is quantized in units of ?. For a given l, the maximum value of Lz is l?, corresponding to m = l. This quantization plays a crucial role in phenomena such as the Zeeman effect and selection rules in spectroscopy, reflecting how quantum systems respond to external fields.

Transcript

00:01 We're told the angular momentum, quantum number, l is equal to 2.

00:05 So in units of h -bar, we are going to find the maximum possible value of l -sub -z.

00:14 Well, l, the orbital angular momentum, is given by the equation l is equal to the square root of little l times l -plus -1 multiplied by h -bar, where l is an integer value from zero up to n -minus 1.

00:27 L sub z is equal to m subel, which is equal to an integer value of 0 plus or minus 1, all the way to plus or minus l times h bar.

00:37 So what we have here is if we plug in l is equal to 2 into our equation, that means that m subel, the maximum value of m subel is equal to 2.

00:50 So we can say at max, at most, or at max.

00:54 Okay.

01:03 So then if that's true, l sub z is going to be equal to 2 times h bar.

01:14 And that's the question, or the answer for question a, for b to find the value of l and it asks which is larger, l or the maximum possible value of l sub z.

01:32 So let's find l.

01:34 Well, according to our equations, l is, or capital l, angular momentum, is equal to the square root of l, which we're told is 2, times l plus 1, which is 2 plus 1, so times 3, multiplied by h bar.

02:00 Or in other words, this is equal to the square root of 6 times h bar.

02:17 And then for c, we are asked to find the different angles for the vector l made with the z axis, and how does the minimum angle for l equal 2 compared to the l equals 3 calculated in the other question? scroll down for c we have l equals to the possible values for l sub z we have l sub z equal to zero right because ms of l can be equal to zero all the way up to plus or minus l so zero plus or minus one and plus or minus two are the allowed values so how about when it's equals a zero well l sub z is going to be equal to zero how about when it's equal to um plus or minus one l sub z is equal to plus or minus h bar and when it's equal to two l sub z is equal to plus or minus 2 hbar.

03:34 So based on this, we can find the angle between l and the z axis using the fact that the angle, which we can call theta, is equal to the inverse cosine of l that we found in, or l sub z over l.

04:08 Okay.

04:12 So for the different values of l sub z, we can have different angles.

04:16 So we can call one, theta is zero for when l sub z is equal to zero.

04:20 And so this gives us a value equal to, so what's cosine of zero? cosign of zero is equal to pi over two.

04:30 Or excuse me, inverse cosine of pi over two is equal to zero...

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