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In the book by Nouredine Zettili of quantum mechanics, he discusses about the geometrical representation of quantum mechanics. I have a conceptual doubt related to it?

How does he diagrammatically comes to a point by saying that average value of $L_x$ and $L_y$ are $0$. Like he gives the reason that all orientations are likely possible of the surface of a cone made by $L$, then projection of this $L$ on $x$ and $y$ plane is $0$. Can we see this diagrammatically, like I am unable to visualize it?

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  • $\begingroup$ What is wrong with Figure 5.1 Page 293 which has two diagrams for $\vec J$ and then Figure 5.3 Page 296 which has two diagrams for $\vec L$ to illustrate what happens? $\endgroup$
    – Farcher
    Jul 17, 2023 at 7:26

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This is a common way to visualize the spin up and spin down states

enter image description here

The up state is represented in the left image by the entire cone that is pointing upwards. One arrow is drawn to show a single instance of this cone, i.e. a possible measurement. To see how this works, we can express the up-states in terms of the $x$ and $y$ eigenstates. \begin{align} |x+\rangle&=\tfrac 1 {\sqrt 2}|\uparrow\rangle+\tfrac 1 {\sqrt 2}|\uparrow\rangle\\ |x-\rangle&=\tfrac 1 {\sqrt 2}|\uparrow\rangle-\tfrac 1 {\sqrt 2}|\uparrow\rangle\\ \\ |y+\rangle&=\tfrac 1 {\sqrt 2}|\uparrow\rangle+\tfrac i {\sqrt 2}|\uparrow\rangle\\ |y-\rangle&=\tfrac 1 {\sqrt 2}|\uparrow\rangle-\tfrac i {\sqrt 2}|\uparrow\rangle \end{align} How do we interpret these equations? When we are in the up state, which has with 100% certainty the probability of measuring $+\hbar/2$ in the z-direction, we are automatically in a superposition in the $x$ or $y$ basis. For example the probability amplitudes in $x$-direction are both 50%, so when we measure the up state in the $x$ direction we will get $+\hbar/2$ or $-\hbar/2$ with equal probability. The average over many measurements will give zero.

Another way to see this is the fact that the "norm" of this state is longer than the magnitude of $z$-component, which you can interpret as the state being at an angle with respect to the $z$-axis. $$S=\sqrt{\tfrac 1 2(\tfrac 1 2+1}\hbar=\tfrac{\sqrt{3}}4\hbar\\ S_z=\pm\hbar/2$$

You might also be interested in the Bloch sphere. On the Bloch sphere the eigenstates in the $x,y,z$ directions are nicely mapped to the Cartesian $x,y,z$ directions respectively, like you would expect probably. Note that the Bloch sphere lives in abstract state space, so while it is easier to interpret, it is less representative of physical space than the above picture.

Source of image: File:Spin half angular momentum.svg. (2020, September 28). Wikimedia Commons. Retrieved 07:30, July 17, 2023 from https://commons.wikimedia.org/w/index.php?title=File:Spin_half_angular_momentum.svg&oldid=473779714.

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