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In a particular state of the hydrogen atom, the angle between the angular momentum vector $\overline { L }$ and the $z$ axis is $\theta = 26.6 ^ { \circ } .$ (See Figure $29.2 . )$ If this is the smallest angle for this particular value of the angular momentum quantum number $l ,$ what is $l$ ?

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Physics 103

Chapter 29

Atoms, Molecules, and Solids

Atomic Physics

Nuclear Physics

Simon Fraser University

Hope College

McMaster University

Lectures

02:42

Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and the processes by which these arrangements change. The theory of quantum mechanics, a set of mathematical rules that describe the behaviour of matter and its interactions, provides a good model for the description of atomic structure and properties.

02:26

In physics, nuclear physics is the field of physics that studies the constituents and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those in nuclear medicine and magnetic resonance imaging, ion implantation in materials engineering, and radiocarbon dating in geology and archaeology.

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question Number eight says we have a hydrogen atom located are rather it's in a particular quantum state and the angle of our angular momentum vector l and the zero axis. So let's draw like this is some angle fada, which in this case is 26.6 degrees. So if this is the smallest angle for this value, then what is El on? The way that we can figure this out is from this simple geometric ratio here of co sign favor being ableto LZ over l and the, uh, What we know is that the smallest non zero angle forgiven l happens when so that's the smallest Saito when ml is equal to its max value plus l. So we have that are co sign of data which is given here. So we haven't is equal to l over. No, no, plus one. There's a lot of Ellis here, and we can rearrange things a little bit because we want to solve for l. We get that l s one co sign Squared Data's He was l squared. Cancel out hell, we can solve the equation to get El is equal to co sign squared Seita divided by one minus co sign squared data. Well, all right, So if you plugged this stuff in, then you get No. For one thing, this is signs that you basically have attention, But you get that l is equal to four and ah, net. Yeah, because it's what we're looking for. We're looking for the quantum number. So there you go. You can probably get this in a closed form solution as well, but you'd have to check that.

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