Chapter 4: Quantum Mechanics in Three Dimensions

For the most general normalized spinor (Equation 4.139),

compute$\left\{S_x\right\}\mathbf{,}\left\{S_y\right\}\mathbf{,}\left\{S_z\right\}\mathbf{,}\left\{S_x^2\right\}\mathbf{,}\left\{S_y^2\right\}\mathbf{,}\mathbf{}\mathbf{and}\left\{S_x^2\right\}\mathbf{.}\mathbf{}\mathbf{check}\mathbf{}\mathbf{that}\left\{S_x^2\right\}\mathbf{+}\left\{S_y^2\right\}\mathbf{+}\left\{S_z^2\right\}\mathbf{=}\left\{S^2\right\}\mathbf{.}$

$$\begin{gathered} \mathcal{X}\mathbf{=}\left(\begin{array}{c}a\\ b\end{array}\right)\mathbf{=}{{\mathbf{a}}_{\mathbf{X}}}_{\mathbf{+}}\mathbf{+}{\mathbf{b}}_{\mathbf{X}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left(4.139\right)\mathbf{.} \end{gathered}$$

Use separation of variables in Cartesian coordinates to solve infinite cubical well

_$V(x,y,z)=0$if x,y,z are all between 0 to a;

_{$V(x,y,z)=\infty$}Otherwise

a) Find the stationary states and the corresponding energies

b) Call the distinct energies $E_1,E_2,E_3,..$in the order of increasing energy. Find_{localid="1658127758806" $E_1,E_2,E_3,E_4,E_5,E_6$}determine their degeneracies (that is, the number of different states that share the same energy). Comment: In one dimension degenerate bound states do not occur but in three dimensions they are very common.

c) What is the degeneracy of E₁₄ and why is this case interesting?

Construct the matrix${\mathbf{S}}_{\mathbf{r}}$representing the component of spin angular momentum along an arbitrary direction$\stackrel{\mathbf{⏜}}{\mathbf{r}}$. Use spherical coordinates, for which

$\stackrel{\mathbf{⏜}}{\mathbf{r}}\mathbf{}\mathbf{sin\theta cos\Phi }\stackrel{\mathbf{⏜}}{\mathbf{ı}\mathbf{}}\mathbf{+}\mathbf{sin\theta sin\Phi }\stackrel{\mathbf{⏜}}{\mathbf{ø}}\mathbf{+}\mathbf{cos\theta }\stackrel{\mathbf{⏜}}{\mathbf{k}}\mathbf{}\mathbf{}\mathbf{}$ [4.154]

Find the eigenvalues and (normalized) eigen spinors of${\mathbf{S}}_{\mathbf{r}}$. Answer:

${\mathbf{x}}_{\mathbf{+}}^{\left(r\right)}\mathbf{=}\left(\begin{array}{c}\cos \left(\theta /2\right)\\ e^{\mathrm{i\varphi }}\sin \left(\theta /2\right)\end{array}\right)$; ${\mathbf{x}}_{\mathbf{+}}^{\left(r\right)}\mathbf{=}\mathbf{\left(}\begin{array}{c}{\mathbf{e}}^{\mathbf{i\varphi }}\mathbf{sin}\left(\theta /2\right)\\ \mathbf{-}\mathbf{cos}\left(\theta /2\right)\end{array}\mathbf{\right)}$ [4.155]

Note: You're always free to multiply by an arbitrary phase factor-say,${\mathit{e}}^{\mathbf{i}\mathbf{\varphi }}$-so your answer may not look exactly the same as mine.

Construct the spin matrices$(S_x,S_y\text{\hspace{0.17em}and}{S}_z)$ , for a particle of spin 1. Hint: How many eigenstates of$S_z$ are there? Determine the action of $S_z$, $S_{+}$, and $S_{-}$on each of these states. Follow the procedure used in the text for spin $1/2$.

(a) If you measured the component of spin angular momentum along the x direction, at time t, what is the probability that you would get $\mathbf{+}\mathbf{h}\mathbf{/}\mathbf{2}$?

(b) Same question, but for the ycomponent.

(c) Same, for the z component.

An electron is at rest in an oscillating magnetic field

$\mathbf{B}\mathbf{=}{\mathbf{B}}_{\mathbf{0}}\mathbf{cos}\mathbf{\left(}\mathbf{\omega t}\mathbf{\right)}\hat{\mathbf{k}}$

where${\mathit{B}}_{\mathbf{0}}$ and$\mathbf{\omega }$ are constants.

(a) Construct the Hamiltonian matrix for this system.

(b) The electron starts out (at t=0 ) in the spin-up state with respect to the x-axis (that is:$\mathbf{\chi }\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathbf{\chi }}_{\mathbf{+}}^{\left(x\right)}\mathbf{}\mathbf{)}$. Determine $\mathbf{X}\left(t\right)$at any subsequent time. Beware: This is a time-dependent Hamiltonian, so you cannot get in the usual way from stationary states. Fortunately, in this case you can solve the timedependent Schrödinger equation (Equation 4.162) directly.

(c) Find the probability of getting$\mathbf{-}\mathbf{h}\mathbf{/}\mathbf{2}$ , if you measure ${\mathbf{S}}_{\mathbf{x}}$. Answer:

${\mathbf{sin}}^{\mathbf{2}}\left(\frac{{\mathrm{\gamma B}}_0}{2\omega }\sin \left(\mathrm{\omega t}\right)\right)$

(d) What is the minimum field$\left(B_0\right)$ required to force a complete flip in${\mathit{S}}_{\mathbf{x}}$ ?

(a) Apply ${\mathit{S}}_{\mathbf{-}\mathbf{\text{to}}}\mathbf{|}\mathbf{10}\mathbf{⟩}$ (Equation$\mathbf{}\mathbf{4}\mathbf{.}\mathbf{177}$ ), and confirm that you get $\sqrt{\mathbf{2}}\mathbf{|}\mathbf{1}\mathbf{-}\mathbf{1}\mathbf{⟩}$

(b) Apply ${\mathit{S}}_{\mathbf{\pm }}$to$\mathbf{[}\mathbf{00}\mathbf{⟩}$ (Equation 4.178), and confirm that you get zero.

(c) Show that $\mathbf{|}\mathbf{11}\mathbf{⟩}$ and $\mathbf{|}\mathbf{1}\mathbf{-}\mathbf{1}\mathbf{⟩}$ (Equation 4.177) are eigenstates of ${\mathbf{S}}^2$, with the appropriate eigenvalue

(a) Apply $\mathit{S}\mathit{\_}$tolocalid="1656131461017" $\overline{)10>}$(Equation$\mathbf{4}\mathbf{.}\mathbf{177}$), and confirm that you getlocalid="1656131442455" $\sqrt{\mathbf{2}}\sout{\mathbf{h}}\overline{)\mathbf{1}\mathbf{-}\mathbf{1}}\mathbf{>}$.

(b) Apply${\mathbf{S}}_{\mathbf{+}}$to$\mathbf{[}\mathbf{00}\mathbf{>}$(Equation$\mathbf{4}\mathbf{.}\mathbf{178}$), and confirm that you get zero.

(c) Show thatlocalid="1656131424007" $\overline{)\mathbf{11}}\mathbf{>}$andlocalid="1656131406083" $\overline{)\mathbf{1}\mathbf{-}\mathbf{1}\mathbf{>}}$(Equation$\mathbf{4}\mathbf{.}\mathbf{177}$) are eigenstates of${\mathit{S}}^{\mathbf{2}}$, with the appropriate eigenvalue

Quarks carry spin $\mathbf{1}\mathbf{/}\mathbf{2}$. Three quarks bind together to make a baryon (such as the proton or neutron); two quarks (or more precisely a quark and an antiquark) bind together to make a meson (such as the pion or the kaon). Assume the quarks are in the ground state (so the orbital angular momentum is zero).

(a) What spins are possible for baryons?

(b) What spins are possible for mesons?

(a) A particle of spin1and a particle of spin 2 are at rest in a configuration such that the total spin is 3, and its z component is $\mathbf{\hslash }$ . If you measured the z component of the angular momentum of the spin-2particle, what values might you get, and what is the probability of each one?

(b) An electron with spin down is in the state${\mathit{\psi }}_{\mathbf{510}}$of the hydrogen atom. If you could measure the total angular momentum squared of the electron alone (not including the proton spin), what values might you get, and what is the probability of each?