What is the angle between and in a (a) and (b) state of hydrogen?
(a) The angle between L and S when they're aligned is .
(b) The angle between L and S when they're anti-aligned is .
State of hydrogen and state of hydrogen atom.
When L and S are aligned, they look like in figure 1.
Here, is the angle between and .
The state is for when and are anti-aligned, look like in figure 2.
Use the law of cosines in order to find angle, by magnitude of the vectors.
Simplify further as shown below.
To find the magnitude of vectors L, S, and J, for which find quantum numbers l, s, and j.
For p shell, l=1.
The electron's spin s is 1/2 and 3/2 from the provide j.
Use these values in the equations.
Substitute the values in equation (1).
Therefore, the angle between L and S when they're aligned is role="math" localid="1658381059167" .
Find the value as follows:
Then that, along with the same L and S as before can be inserted into equation (2).
Therefore, the angle between L and S when they're anti-aligned is .
Two particles in a box occupy the and individual-particle states. Given that the normalization constant is the same as in Example (see Exercise 36), calculate for both the symmetric and antisymmetric states the probability that both particles would be found in the left side of the box (i.e., between 0 and )?
Suppose that the channel’s outgoing end is in the hydrogen Stem-Gerlach apparatus of the figure. You place a second such apparatus whose channel is aligned with the first but rotated about the -axis, so that its B –field lines point roughly in the -direction instead of the. What would you see emerging at the end of your added apparatus? Consider the behavior of the spin-up and spin-down beams separately. Assume that when these beams are separated in the first apparatus, we can choose to block one or the other for study, but also assume that neither deviates too far from the center of the channel.
The electron is known to have a radius no larger than . If actually produced by circulating mass, its intrinsic angular momentum of roughly would imply very high speed, even if all that mass were as far from the axis as possible.
(a) Using simply (from |r × p|) for the angular momentum of a mass at radius r, obtain a rough value of p and show that it would imply a highly relativistic speed.
(b) At such speeds, and combine to give (just as for the speedy photon). How does this energy compare with the known internal energy of the electron?
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