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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?

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(a) 3$\hbar$(b) $L>L_{z}$(c) $30^{\circ}, 55^{\circ}, 73^{\circ}, 90^{\circ}, 107^{\circ}, 125^{\circ},$ and $150^{\circ}$

Physics 103

Chapter 29

Atoms, Molecules, and Solids

Atomic Physics

Nuclear Physics

University of Washington

Simon Fraser University

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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What shape is the feasible…

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Q1.6: Two arbitrary vector…

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Vector A points along the …

brothel number two. We are considering states that have an L value of three. 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 maximise this. And that is going to be Max, of course, had plus l. So the maximum value or illegal street, it's simply going to be a plus three h bar. Easy enough. But onto B, we want to find the value of L. And which is larger, the ah big L Value or the maximum possible LZ? Well, l is Give it here and we see that it is two thirds. Or Sergeant, there's two square root three Time's a tch bar. And this is of course, larger. Then this. And in fact, l will always be larger than LZ because L Z is just a component of El and finally for part. See, we are We'll give it a little splash of color. We are to assume a model. Where are el factor is described classically and we want to know for each possible value of Elsie, what angle does that vector make with the axis Well, given that we can say Cho, Santa is going to be equal to L Z over l. This is the geometric considerations then wait, Can sort of pieced together from stuff. So there are going to be some allowed values depending on what this Elsie is because it depends on em out. Thus, we have to sort of walk through them all, and, ah, find a way, uh, expression for the angle. So I'm going to go to go ahead and derive this So we'll say our theta is equal to universe co sign of LZ over l. We should just inverse co sign of ml each bar all on l l plus one h bar. And now we have a nice, simple expression that only depends on M l and L. And since l doesn't change for this case, we'll go to a new paint and write out our magnum opus for Percy. Our fate is just going to depend on ml all on to Route three, and that's it in ml goes from zero positive, minus one positive minus two. And since we are going up to L. A gal's three positive minus three, that means there are 1234567 angles that we're looking for And I will write it in terms of, say, the ML. So, like this start out big 30 degrees em illegals to we go down to or sorry, go up to 54.7 degrees. For one, we recover 73.2 degrees for zero. Uh, this one's nice, because his co sign a legal zero, which means that our angle is 90. Now we get into the negatives, which means our angles are going to be greater than greater than 90. So here we go 125 points three and then finally, for the maximum angle, we get 150 degrees. Even so, these are all the allowed values. It's a little absurd to box them all, but here we are

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