Chapter 8: Spin and Atomic Physics

Question: Show that the frequency at which an electron’s intrinsic magnetic dipole moment would process in a magnetic field is given by $\mathit{\omega }\mathbf{\cong }\raisebox{1ex}{${\displaystyle \mathbf{e}\mathbf{B}}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle {\mathbf{m}}_{\mathbf{e}}}$}\right.$. Calculate the frequency for a field of 1.0 T.

Question: In the Stern-Gerlach experiment how much would a hydrogen atom emanating from a 500 K oven$\left(KE=\frac{3}{2}{k}_{B}T\right)$be deflected in traveling 1 m through a magnetic field whose rate of change is 10 T/m?

What angles might the intrinsic angular momentum vector make with the z-axis for a deuteron? (See Table 8.1)

Does circulating charge require both angular momentum and magnetic? Consider positive and negative charges simultaneously circulating and counter circulating.

A hydrogen atom in its ground state is subjected to an external magnetic field of 1.0 T. What is the energy difference between the spin-up and spin-down states?

The subatomic omega particle has spin $\mathbf{s}\mathbf{=}\frac{\mathbf{3}}{\mathbf{2}}$. What angles might its intrinsic angular in momentum vector make with the z-axis?

Is intrinsic angular momentum "real" angular momentum? The famous Einstein-de Haas effect demonstrates it. Although it actually requires rather involved techniques and high precision, consider a simplified case. Suppose you have a cylinder $\mathbf{2}\mathbf{}\mathbf{\text{cm}}$in diameter hanging motionless from a thread connected at the very center of its circular top. A representative atom in the cylinder has atomic mass 60 and one electron free to respond to an external field. Initially, spin orientations are as likely to be up as down, but a strong magnetic field in the upward direction is suddenly applied, causing the magnetic moments of all free electrons to align with the field.

(a) Viewed from above, which way would the cylinder rotate?

(b) What would be the initial rotation rate?

Figureshows the Stern-Gerlach apparatus. It reveals that spin-$\frac{\mathbf{1}}{\mathbf{2}}$particles have just two possible spin states. Assume that when these two beams are separated inside the channel (though still near its centreline). we can choose to block one or the other for study. Now a second such apparatus is added after the first. Their channels are aligned. But the second one is rotated about the-axis by an angle \(\phi\) from the first. Suppose we block the spin-down beam in the first apparatus, allowing only the spin-up beam into the second. There is no wave function for spin. but we can still talk of a probability amplitude, which we square to give a probability. After the first apparatus' spin-up beam passes through the second apparatus, the probability amplitude is$\mathbf{cos}\mathbf{(}\mathbf{\varphi }\mathbf{/}\mathbf{2}\mathbf{)}{\mathbf{\uparrow }}_{\mathbf{2}\mathbf{\text{nd}}}\mathbf{+}\mathbf{sin}\mathbf{(}\mathbf{\varphi }\mathbf{/}\mathbf{2}\mathbf{)}{\mathbf{\downarrow }}_{\mathbf{2}\mathbf{nd}}$where the arrows indicate the two possible findings for spin in the second apparatus.

(a) What is the probability of finding the particle spin up in the second apparatus? Of finding it spin down? Argue that these probabilities make sense individually for representative values of$\mathit{\varphi }$and their sum is also sensible.

(b) By contrasting this spin probability amplitude with a spatial probability amplitude. Such as$\mathbf{\psi }\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{Ae}}^{\mathbf{-}{\mathbf{te}}^{\mathbf{2}}}$. Argue that although the arbitrariness of$\mathbf{\varphi }$gives the spin cases an infinite number of solves. it is still justified to refer to it as a "two-state system," while the spatial case is an infinite-state system.

Show that the symmetric and anti symmetric combinations of $\mathbf{(}\mathbf{8}\mathbf{-}\mathbf{19}\mathbf{)}$and$\mathbf{(}\mathbf{8}\mathbf{-}\mathbf{20}\mathbf{)}$are solutions of the two. Particle Schrödinger equation$\mathbf{(}\mathbf{8}\mathbf{-}\mathbf{13}\mathbf{)}$of the same energy as${\mathbf{\psi }}_{\mathbf{n}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}{\mathbf{\psi }}_{\mathbf{m}}\mathbf{\left(}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}$, the unsymmetrized product$\mathbf{(}\mathbf{8}\mathbf{-}\mathbf{17}\mathbf{)}$.

Two particles in a box occupy the $\mathbf{n}\mathbf{=}\mathbf{1}$and${\mathbf{n}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{2}$individual-particle states. Given that the normalization constant is the same as in Example$\mathbf{8}\mathbf{.}\mathbf{2}$(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$\frac{\mathbf{1}}{\mathbf{3}}\mathbf{L}$)?