Consider states with l = 3 . (a) In units of ħ, what is the largest possible value of L z ? (

Consider states with l = 3 . (a) In units of ħ, what is the...

Consider states with $l = 3 .$ (a) In units of $\hbar ,$ what is the largest possible value of $L _ { z } ?$ (b) In units of $\hbar ,$ what is the value of $L ?$ Which is larger, $L$ or the maximum possible $L _ { z } ?$ (c) Assume a model in which $\vec { \boldsymbol { L } }$ is described as a classical vector. For each allowed value of $L _ { z } ,$ what angle does the vector $\vec { L }$ make with the $+ z$ axis?

Key Concepts

Semiclassical Vector Model of Angular Momentum

This concept involves visualizing the angular momentum as a classical vector that precesses about the z-axis. In the semiclassical picture, for each allowed value of Lz, one can associate an angle between the angular momentum vector and the z-axis using the relation cos(?) = Lz/|L|. This model helps bridge the gap between the fully quantized description and classical intuition.

Quantization of Angular Momentum Components

In quantum mechanics, the total angular momentum and its z-component are quantized. While the total magnitude of the angular momentum vector is fixed at ?(l(l+1)) ?, its projection along a single axis (like z) can only take specific discrete values (m ?) that are always less than or equal to the total magnitude except in the trivial case where l = 0.

Orbital Angular Momentum Quantum Number (l)

This is a quantum number that determines the magnitude of the orbital angular momentum. For any state with a given l, the magnitude of the angular momentum is given by the formula ?(l(l+1)) times ?. It also sets the range of possible magnetic quantum numbers, which are the possible z-components of the angular momentum.

Magnetic Quantum Number (m)

This quantum number specifies the projection of the angular momentum along the z-axis. Its allowed values range in integer steps from –l to +l. The maximum value of the z-component of angular momentum is achieved by setting m equal to l, giving Lz = l ?.

Transcript

00:01 Problem number two, we are considering states that have an l value of 3, and for part a in units of h -bar, what is the largest possible value of lz? well, if lz is defined as such, then we want to maximize this, and that is going to be max, of course, at plus l.

00:29 So the maximum value for l equals 3 is simply going to be plus 3h bar.

00:40 Easy enough, but on to b, we want to find the value of l, and which is larger, the big l value or the maximum possible l.

00:53 Z.

00:56 Well, l is given here, and we see that it is two.

01:08 Thirds or sorry two thirds two square root three times h bar and this is of course larger than this and in fact l will always be larger than l z because l z is just a component of l and finally for part c we are we'll give it a little splash of color uh we are to assume a model where our l vector is described classically, and we want to know for each possible value of lz, what angle does that vector make with the axis? well, given that, we can say cosine theta is going to be equal to lz over l, that's just a geometric considerations, then we can sort of piece together some stuff.

02:21 So there are going to be some allowed values depending on what this lz is, because it depends on ml.

02:40 Thus, we have to sort of walk through them all and find a expression for the angle.

02:51 So i'm going to go ahead and derive this.

02:56 So we'll say our theta is equal to inverse cosine of lz over l, which is just inverse cosine of mlh bar, all on ll plus 1h bar.

03:24 And now we have a nice simple expression that only depends on ml and l.

03:31 And since l doesn't change for this case, we'll go to a new page and write out our magnum opus for part c.

03:41 Our theta is just going to depend on ml, all on 2, root 3, and that's it.

04:01 In ml goes from 0, positive minus 1, positive minus 2, and since we are going up to l equals 3, positive minus 3, positive minus three...

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