Chapter 4: Quantum Mechanics in Three Dimensions

The raising and lowering operators change the value of m by one unit:

$\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{L}}_{\mathbf{\pm }}{\mathit{f}}_{\mathbf{l}}^{\mathbf{m}}\mathbf{}\mathbf{=}\left(A_{\mathcal{l}}^m\right){\mathit{f}}_{\mathbf{l}}^{\mathbf{m}\mathbf{+}\mathbf{1}}\mathbf{}\mathbf{,}$ (4.120).

Where ${\mathbf{A}}_{\mathbf{l}}^{\mathbf{m}}\mathbf{}$are constant. Question: What is ${\mathbf{A}}_{\mathbf{l}}^{\mathbf{m}}\mathbf{}$, if the Eigen functions are to be normalized? Hint: First show that${\mathit{L}}_{\mathbf{\pm }}$is the Hermitian conjugate of ${\mathit{L}}_{\mathbf{\pm }}$(Since ${\mathit{L}}_x\mathbf{}\mathbf{and}\mathbf{}{\mathbf{L}}_{\mathbf{y}}$are observables, you may assume they are Hermitian…but prove it if you like); then use Equation 4.112.

(a) Starting with the canonical commutation relations for position and momentum (Equation 4.10), work out the following commutators:

$$\begin{gathered} \left[L_Z,X\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}\mathbf{y}\mathbf{,}\mathbf{}\left[L_Z,y\right]\mathbf{=}\mathbf{-}\mathbf{i}\sout{\mathbf{h}}\mathbf{x}\mathbf{,}\mathbf{}\left[L_Z,Z\right]\mathbf{=}\mathbf{0} \\ \left[L_Z,p_x\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}{\mathbf{p}}_{\mathbf{y}}\mathbf{}\mathbf{,}\left[L_Z,p_y\right]\mathbf{=}\mathbf{-}\mathbf{i}\sout{\mathbf{h}}{\mathbf{p}}_{\mathbf{x}}\mathbf{}\mathbf{,}\mathbf{}\left[L_Z,p_z\right]\mathbf{=}\mathbf{0} \end{gathered}$$

(b) Use these results to obtain $\left[L_Z,L_X\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}{\mathbf{L}}_{\mathbf{y}}$directly from Equation 4.96.

(c) Evaluate the commutators $\left[L_z,r^2\right]$and$\left[L_z,p^2\right]$(where, of course, ${\mathbf{r}}^{\mathbf{2}}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{p}}^{\mathbf{2}}\mathbf{=}{\mathbf{p}}_{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{p}}_{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{p}}_{\mathbf{z}}^{\mathbf{2}}\mathbf{}\mathbf{)}$

(d) Show that the Hamiltonian $\mathbf{H}\mathbf{=}\left(p^2/2m\right)\mathbf{+}\mathbf{V}$commutes with all three components of L, provided that V depends only on r . (Thus $\mathbf{H}\mathbf{,}{\mathbf{L}}^{\mathbf{2}}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{L}}_{\mathbf{Z}}$and are mutually compatible observables.)

(a) Work out all of the canonical commutation relations for components of the operator r and p : $\mathbf{[}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{]}\mathbf{,}\mathbf{[}\mathbf{x}\mathbf{,}{\mathbf{p}}_{\mathbf{y}}\mathbf{]}\mathbf{,}\mathbf{[}\mathbf{x}\mathbf{,}{\mathbf{p}}_{\mathbf{x}}\mathbf{]}\mathbf{,}\mathbf{[}{\mathbf{p}}_{\mathbf{y}}\mathbf{,}{\mathbf{p}}_{\mathbf{z}}\mathbf{]}\mathbf{,}$and so on.

(b) Confirm Ehrenfest’s theorem for 3 dimensions

_{$\frac{\mathbf{d}}{\mathbf{dt}}\mathbf{<}\mathbf{r}\mathbf{>}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{<}\mathbf{p}\mathbf{>}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\frac{\mathbf{d}}{\mathbf{dt}}\mathbf{<}\mathbf{p}\mathbf{>}\mathbf{=}\mathbf{<}\mathbf{-}\mathbf{\nabla }\mathbf{v}\mathbf{>}$}

(Each of these, of course, stand for three equations- one for each component.)

(c) Formulate Heisenberg’s uncertainty principle in three dimensions Answer:

_{${\mathit{\sigma }}_{\mathbf{x}}{\mathit{\sigma }}_{\mathbf{p}}\mathbf{\ge }\frac{\mathbf{h}}{\mathbf{2}}\mathbf{;}{\mathit{\sigma }}_{\mathbf{y}}{\mathit{\sigma }}_{\mathbf{p}}\mathbf{\ge }\frac{\mathbf{h}}{\mathbf{2}}\mathbf{;}{\mathit{\sigma }}_{\mathbf{z}}{\mathit{\sigma }}_{\mathbf{p}}\mathbf{\ge }\frac{\mathbf{h}}{\mathbf{2}}$}

But there is no restriction on, say, ${\mathit{\sigma }}_{\mathbf{x}}{\mathit{\sigma }}_{\mathbf{py}}\mathbf{.}$

(a)Derive Equation 4.131 from Equation 4.130. Hint: Use a test function; otherwise you're likely to drop some terms.

(b)Derive Equation 4.132 from Equations 4.129 and 4.131 .Hint : Use Equation 4.112.

(a) What is${\mathbf{L}}_{\mathbf{+}}{\mathbf{Y}}_{\mathbf{1}}^{\mathbf{I}}$? (No calculation allowed!)

(b) Use the result of (a), together with Equation 4.130 and the fact that${\mathbf{L}}_z{\mathbf{Y}}_{\mathbf{1}}^{\mathbf{I}}\mathbf{=}\sout{\mathit{h}}\mathit{I}{\mathit{Y}}_{\mathit{I}}^{\mathit{I}}$ to determine${\mathit{Y}}_{\mathit{I}}^{\mathit{I}}\left(\theta ,\varphi \right)$ , up to a normalization constant.

(c) Determine the normalization constant by direct integration. Compare your final answer to what you got in Problem 4.5.

In Problem4.3 you showed that ${\mathit{Y}}_{\mathbf{2}}^{\mathbf{1}}\left(\theta ,\varphi \right)\mathbf{=}\mathbf{-}\sqrt{\mathbf{15}\mathbf{/}\mathbf{8}\mathbf{\pi }}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }{\mathit{e}}^{\mathbf{i}\mathbf{\varphi }}$. Apply the raising operator to find localid="1656065252558" ${\mathit{Y}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{(}\mathbf{\theta }\mathbf{,}\mathbf{\varphi }\mathbf{)}$. Use Equation 4.121to get the normalization.

Two particles of mass mare attached to the ends of a massless rigid rod of length a. The system is free to rotate in three dimensions about the center (but the center point itself is fixed).

(a) Show that the allowed energies of this rigid rotor are

${\mathbf{E}}_{\mathbf{n}}\mathbf{=}\frac{{\sout{\mathbf{h}}}^{\mathbf{2}}\mathbf{n}\left(n+1\right)}{{\mathbf{ma}}^{\mathbf{2}}}$, for n=0,1,2,...

Hint: First express the (classical) energy in terms of the total angular momentum.

(b) What are the normalized Eigen functions for this system? What is the degeneracy of the${\mathit{n}}^{\mathbf{t}\mathbf{h}}$energy level?

If the electron were a classical solid sphere, with radius

${\mathbf{r}}_{\mathbf{c}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{2}}}{\mathbf{4}\mathbf{\pi }\dot{{\mathbf{O}}_{\mathbf{0}}}{\mathbf{mc}}^{\mathbf{2}}}$

(the so-called classical electron radius, obtained by assuming the electron's mass is attributable to energy stored in its electric field, via the Einstein formula $\mathbf{E}\mathbf{=}{\mathbf{mc}}^{\mathbf{2}}$), and its angular momentum is $\left(1/2\right)\mathbf{h}$ then how fast (in $\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s}$) would a point on the "equator" be moving? Does this model make sense? (Actually, the radius of the electron is known experimentally to be much less than$\mathbf{5}\mathbf{.}\mathbf{156}\mathbf{\times }{\mathbf{10}}^{\mathbf{10}}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{}{\mathbf{r}}_{\mathbf{c}}$, but this only makes matters worse).

a) Check that the spin matrices (Equations 4.145 and 4.147) obey the fundamental commutation relations for angular momentum, Equation 4.134.

$$\begin{gathered} {\mathbf{S}}_{\mathbf{z}}\mathbf{=}\frac{\sout{\mathbf{h}}}{\mathbf{2}}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\mathbf{}\mathbf{}\left(4.145\right)\mathbf{.} \\ {\mathbf{S}}_{\mathbf{x}}\mathbf{=}\frac{\sout{\mathbf{h}}}{\mathbf{2}}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{s}}_{\mathbf{y}}\mathbf{=}\frac{\sout{\mathbf{h}}}{\mathbf{2}}\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\mathbf{}\mathbf{}\mathbf{}\left(4.147\right)\mathbf{.} \\ \left[S_x,S_y\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}{\mathbf{S}}_{\mathbf{z}}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left[S_y,S_z\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}{\mathbf{S}}_{\mathbf{x}}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left[S_z,S_x\right]\mathbf{=}\mathbf{i}\sout{\mathbf{h}}{\mathbf{S}}_{\mathbf{y}}\mathbf{}\mathbf{}\mathbf{}\left(4.134\right)\mathbf{.} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{Show}\mathbf{}\mathbf{that}\mathbf{}\mathbf{the}\mathbf{}\mathbf{Pauli}\mathbf{}\mathbf{spin}\mathbf{}\mathbf{matrices}\mathbf{}\mathbf{(}\mathbf{Equation}\mathbf{}\mathbf{4}\mathbf{.}\mathbf{148}\mathbf{)}\mathbf{}\mathbf{satisfy}\mathbf{}\mathbf{the}\mathbf{}\mathbf{product}\mathbf{}\mathbf{rule} \\ {\mathbf{\sigma }}_{\mathbf{x}}\mathbf{\equiv }\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\mathbf{,}\mathbf{}{\mathbf{\sigma }}_{\mathbf{y}}\mathbf{\equiv }\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\mathbf{,}\mathbf{}{\mathbf{\sigma }}_{\mathbf{z}}\mathbf{\equiv }\mathbf{\left(}\begin{array}{cc}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{-}\mathbf{1}\end{array}\mathbf{\right)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left(4.148\right)\mathbf{.} \\ {\mathbf{\sigma }}_{\mathbf{j}}{\mathbf{\sigma }}_{\mathbf{k}}\mathbf{=}{\mathbf{\delta }}_{\mathbf{jk}}\mathbf{}\mathbf{+}\mathbf{i}{\underset{\mathbf{I}}{\mathbf{\sum }\stackrel{\mathbf{\text{'}}}{\mathbf{o}}}}_{\mathbf{jkl}}{\mathbf{\sigma }}_{\mathbf{I}}\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{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left(4.153\right)\mathbf{}\mathbf{}\mathbf{.}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{} \end{gathered}$$

$$\begin{gathered} \mathbf{Where}\mathbf{}\mathbf{the}\mathbf{}\mathbf{indices}\mathbf{}\mathbf{stand}\mathbf{}\mathbf{for}\mathbf{}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{}\mathbf{or}\mathbf{}\mathbf{z}\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}{\stackrel{\mathbf{\text{'}}}{\mathbf{o}}}_{\mathbf{jkl}}\mathbf{}\mathbf{is}\mathbf{}\mathbf{the}\mathbf{}\mathbf{Levi}\mathbf{-}\mathbf{Civita}\mathbf{}\mathbf{symbol}\mathbf{:} \\ \mathbf{+}\mathbf{1}\mathbf{ifjkl}\mathbf{=}\mathbf{123}\mathbf{,}\mathbf{231}\mathbf{,}\mathbf{}\mathbf{or}\mathbf{}\mathbf{2}\mathbf{=}\mathbf{312}\mathbf{;}\mathbf{-}\mathbf{1}\mathbf{}\mathbf{if}\mathbf{}\mathbf{jkl}\mathbf{=}\mathbf{132}\mathbf{,}\mathbf{213}\mathbf{,}\mathbf{}\mathbf{or}\mathbf{}\mathbf{321}\mathbf{;}\mathbf{}\mathbf{otherwise}\mathbf{.} \end{gathered}$$

An electron is in the spin state

$\chi =A\left(\begin{array}{c}3i\\ 4\end{array}\right)$

(a) Determine the normalization constant .

(b) Find the expectation values of $S_x$,$S_y$ , and $S_z$.

(c) Find the "uncertainties" ,${\sigma }_{S_x}$ , ${\sigma }_{S_y}$and${\sigma }_{S_z}$ . (Note: These sigmas are standard deviations, not Pauli matrices!)

(d) Confirm that your results are consistent with all three uncertainty principles (Equation 4.100 and its cyclic permutations - only with in place ofL, of course).