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

Consider states with angular-momentum quantum number $l=2$ (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) For each allowed value of $L_{v}$ what angle does the vector $\vec{L}$ make with the $+z$ -axis? How does the minimum angle for $l=2$ compare to the minimum angle for $l=3$ calculated in Example 41.2$?$

Key Concepts

Angle between Angular Momentum Vector and z-axis

The angle between the angular momentum vector and the z-axis is determined by the ratio of the z-component to the total magnitude of the angular momentum. Specifically, the cosine of this angle is given by m/?[l(l+1)]. This relationship illustrates that even though the maximum value of Lz occurs when m = l, the overall vector generally lies at an angle to the z-axis, unless l is zero.

Comparison of Angular Momentum and its z-Component

There is a distinction between the magnitude of the angular momentum vector and its maximum possible z-component. The magnitude ?[l(l+1)] is always larger than the maximum z-component l (in units of ?), except in the trivial case when l equals zero. This difference reflects the fact that quantum angular momentum is inherently a three-dimensional vector, so even its maximum projection represents only one component of the total angular momentum.

Quantization of Angular Momentum Components

The component of the angular momentum along any axis, particularly Lz, is quantized and takes on discrete values proportional to ? times the magnetic quantum number m. This ensures that even though the overall magnitude of the angular momentum vector is well defined, its projections are restricted to specific discrete values.

Magnetic Quantum Number (m)

The magnetic quantum number, m, defines the projection of the angular momentum along a chosen axis (commonly the z-axis). Its allowed values range from -l to +l in integer steps. The largest possible value of the angular momentum's z-component, Lz, corresponds to m = l, reflecting the quantized nature of angular momentum in quantum mechanics.

Angular Momentum Quantum Number (l)

The angular momentum quantum number, denoted by l, characterizes the magnitude of the orbital angular momentum of a quantum system. It determines the number of possible orientations the angular momentum vector can have and defines the ranges of associated quantities such as the magnetic quantum number. A higher l implies a larger overall magnitude of the angular momentum vector.

Magnitude of Angular Momentum

In quantum mechanics, the magnitude of the angular momentum vector is given by the expression ?[l(l+1)] (in units of ?). This result emerges from the quantization of angular momentum and accounts for the fact that the components of angular momentum do not commute, leading to an inherent uncertainty in their simultaneous values.

Transcript

00:01 Here, let's say that we're looking at states of the hydrogen atom where the angular momentum quantum number l is equal to two.

00:09 And so some things we're going to find out are what is the maximum angular momentum in the vertical direction that we can have given this restriction.

00:20 And what are the available angles between the l vector and its corresponding z axis, which looks sort of like this? so if you have z, this is the direction of z.

00:36 If you have some vector l, this is going to be the height of the vertical component here.

00:43 So l sub z.

00:45 So we're going to figure out l sub z max, the angles available.

00:50 And finally, we're going to compare that to l equals 3.

00:55 So this is what we have.

00:58 This is what we're going to find.

00:59 And okay, so starting with the vertical component, the max that we can have, we have to think about what l sub z is actually equal to l sub z is actually going to be equal to m subel, the magnetic quantum number, times h bar.

01:19 So what values for m subal do we have available, given that l is equal to two? we know that m subel has to be in the range from negative l to positive l.

01:31 So we're going to have negative 2 positive 2.

01:35 And those are integers.

01:37 So this means that the maximum of this range is going to be 2, which means that the maximum of the vertical component of angle of momentum is going to be 2 times h bar.

01:52 So that's our first result.

01:54 Next, we're going to be looking at the angles that are available to the system given what we know about the angular momentum quantum number l being two.

02:09 So given that this is true, there are five different values of m -sabell that we can have...

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