An electron is in the hydrogen atom with n=5. (a) Find the possible values of L and Lz for this

An electron is in the hydrogen atom with n=5. (a) Find the...

Key Concepts

Quantum Numbers

In quantum mechanics, the state of an electron in an atom is described using quantum numbers. The principal quantum number (n) specifies the energy level, the orbital (or azimuthal) quantum number (l) determines the magnitude of the angular momentum and can take on integers from 0 to n - 1, and the magnetic quantum number (m) defines the projection of the orbital angular momentum along a chosen axis (commonly the z-axis), taking on integer values from -l to l.

Orbital Angular Momentum Quantization

The magnitude of the orbital angular momentum is quantized and is given by L = sqrt[l(l+1)] multiplied by ?, where ? is the reduced Planck constant. This discrete quantization arises from the inherent quantum mechanical nature of angular momentum, which contrasts sharply with the continuous values allowed in classical physics.

Magnetic Quantum Number and Angular Momentum Projection

The magnetic quantum number (m) quantizes the z-component of the orbital angular momentum, such that Lz = m?. This quantization implies that even though the angular momentum vector has a fixed magnitude for a given l, its projection along the z-axis can only take specific, discrete values.

Angular Orientation of Angular Momentum

The angle between the total angular momentum vector and the z-axis is related to the ratio of its z-component to its magnitude. This relationship is expressed by cos(?) = m/?[l(l+1)], which shows that for each allowed m value corresponding to a given l, the angular momentum vector occupies a discrete set of possible orientations with respect to the z-axis.

Transcript

00:01 In this problem, we'll be talking about the quantization of angular momentum.

00:05 So we're given that we're talking about an electron in a hydrogen atom where the principal quantum number n is equal to five.

00:14 So as we know, this restricts the values of the orbital angular momentum quantum number l to a range so that l must be within a range from 0.

00:31 0 all the way up to n minus 1, which n minus 1 is going to be 4.

00:36 So l can only be in here.

00:38 And then we have to talk about the magnetic quantum number, m sub l, and m subel is restricted by l, in that it must be within the range plus or minus l all the way up to positive l.

00:54 So in this case, that's going to be 4, because 4 is a max value of l.

01:00 So m subel.

01:02 Must be in here.

01:03 So now we can go about finding which values of big l, the total angular momentum, are available, as well as the vertical values.

01:14 So if we write it as an array, i'm just gonna skip over the algebra, where the first element's gonna be l equals to zero, next one's gonna be l equals one and so on.

01:26 So for l, we're gonna have zero, root two, h, and then we're going to have six, root 6h, then we're going to have 2, root 3h, and finally, 2 root 5h.

01:45 And so these, oops, i grab my h.

01:49 Okay, so those are the available values of l.

01:54 Oh, i didn't want that.

01:57 Let's see if i can, no, it doesn't.

01:58 I'll just keep it there then.

01:59 Okay, so those are the available values of l.

02:01 What about l sub z? well l sub z is just going to look like integer multiples of h bar within this range.

02:10 So we're going to have l sub z.

02:14 I'm just going to write an array with plus or minus within it.

02:18 So i don't have to write all of them.

02:20 So basically we're going to have zero h bar, plus or minus h bar, that is, plus or minus 2h bar, plus or minus 3h bar, and plus or minus 4h bar.

02:35 Again, this is from this expression up here and the fact that m sub l is in this range.

02:41 But when we're talking about which values of l sub z are available to l, l sub z must be less than l, except when l is equal to zero, then this will also be equal to zero...

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