Chapter 4: Quantum Mechanics in Three Dimensions

Work out the normalization factor for the spherical harmonics, as follows. From Section 4.1.2we know that

$Y_l^m=B_l^m{e}^{im\varphi }{P}_l^m\left(cos\theta \right)$

the problem is to determine the factor (which I quoted, but did not derive, in Equation 4.32). Use Equations 4.120, 4.121, and 4.130to obtain a recursion

relation giving $B_l^{m+1}$ in terms of $B_l^m$. Solve it by induction on to get $B_l^m$ up to an overall constant $C\left(l\right)$, .Finally, use the result of Problem 4.22 to fix the constant. You may find the following formula for the derivative of an associated Legendre function useful:

$\left(1-x^2\right)\frac{d{P}_l^m}{dx}=\sqrt{1-x^2}{P}_l^{m+1}-mx{P}_l^m$ [4.199]

The electron in a hydrogen atom occupies the combined spin and position state$R_{21}\left(\sqrt{1/3}{Y}_1^0{\chi }_{+}+\sqrt{2/3}{Y}_1^1{\chi }_{-}\right)$

(a) If you measured the orbital angular momentum squared $\left(L^2\right)$, what values might you get, and what is the probability of each?

(b) Same for the component of orbital angular momentum $\left(L_z\right)$.

(c) Same for the spin angular momentum squared$\left(S^2\right)$ .

(d) Same for the component of spin angular momentum $\left(S_z\right)$.

Let $\mathbf{J}\equiv \mathbf{L}+\mathbf{S}$be the total angular momentum.

(e) If you measureddata-custom-editor="chemistry" $J^2$ , what values might you get, and what is the probability of each?

(f) Same for$J_z$ .

(g) If you measured the position of the particle, what is the probability density for finding it at$r$ , $\theta$,$\varphi$ ?

(h) If you measured both the component of the spin and the distance from the origin (note that these are compatible observables), what is the probability density for finding the particle with spin up and at radius ?

(a) For a function$f\left(\varphi \right)$that can be expanded in a Taylor series, show that $f(\varphi +\phi )=e^{i{L}_z\phi /\hslash }f\left(\varphi \right)$ (where is an arbitrary angle). For this reason, $L_z/\hslash$ is called the generator of rotations about the Z-axis. Hint: Use Equation $4.129$ , and refer Problem $\mathbf{3}.\mathbf{39}$.More generally, $\mathbf{L}·\hat{n}/\hslash$ is the generator of rotations about the direction $\hat{n}$, in the sense that $exp(i\mathbf{L}·\hat{n}\phi /\hslash )$effects a rotation through angle$\phi$ (in the right-hand sense) about the axis $\hat{n}$ . In the case of spin, the generator of rotations is $\mathbf{S}·\hat{n}/\hslash$. In particular, for spin $1/2$ ${\chi }^{\text{'}}=e^{i(\mathbf{\sigma }·\hat{n})\phi /2}\chi$tells us how spinors rotate.

(b) Construct the $(2\times 2)$matrix representing rotation by ${180}^{\circ }$about the X-axis, and show that it converts "spin up" $\left({\chi }_{+}\right)$into "spin down"$\left({\chi }_{-}\right)$ , as you would expect.

(c) Construct the matrix representing rotation by ${90}^{\circ }$about the Y-axis, and check what it does to

${\chi }_{+}$

(d) Construct the matrix representing rotation by ${360}^{\circ }$about the -Zaxis, If the answer is not quite what you expected, discuss its implications.

(e) Show that$e^{i(\mathbf{\sigma }·\hat{n})\phi /2}=cos(\phi /2)+i(\hat{n}·\mathbf{\sigma })sin(\phi /2)$

The fundamental commutation relations for angular momentum (Equation 4.99) allow for half-integer (as well as integer) eigenvalues. But for orbital angular momentum only the integer values occur. There must be some extra constraint in the specific form$\mathit{L}\mathbf{=}\mathit{r}\mathbf{\times }\mathit{p}$ that excludes half-integer values. Let be some convenient constant with the dimensions of length (the Bohr radius, say, if we're talking about hydrogen), and define the operators

$$\begin{gathered} {\mathit{q}}_{\mathbf{1}}\mathbf{\equiv }\frac{\mathbf{1}}{\sqrt{\mathbf{2}}}\left[x+\left(a^2/ħ\right)p_y\right]\mathbf{;} \\ {\mathit{p}}_{\mathbf{1}}\mathbf{\equiv }\frac{\mathbf{1}}{\sqrt{\mathbf{2}}}\mathbf{[}{\mathbf{p}}_{\mathbf{x}}\mathbf{-}\left(ħ/a^2\right)\mathbf{y}\mathbf{]}\mathbf{;} \\ {\mathit{q}}_{\mathbf{2}}\mathbf{\equiv }\frac{\mathbf{1}}{\sqrt{\mathbf{2}}}\mathbf{[}\mathbf{x}\mathbf{-}\left(a^2/ħ\right){\mathbf{p}}_{\mathbf{y}}\mathbf{]}\mathbf{;} \\ {\mathit{p}}_{\mathbf{2}}\mathbf{\equiv }\frac{\mathbf{1}}{\sqrt{\mathbf{2}}}\mathbf{[}{\mathbf{p}}_{\mathbf{x}}\mathbf{-}\mathbf{(}\mathbf{ħ}\mathbf{/}{\mathbf{a}}^{\mathbf{2}}\mathbf{)}\mathbf{y}\mathbf{]}\mathbf{;} \end{gathered}$$

(a) Verify that $\left[q_{1,}{q}_2\right]\mathbf{=}\left[p_1,p_2\right]\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{\left[}{\mathbf{q}}_{\mathbf{1}\mathbf{,}}{\mathbf{p}}_{\mathbf{1}}\mathbf{\right]}\mathbf{=}\mathbf{[}{\mathbf{p}}_{\mathbf{2}}\mathbf{,}{\mathbf{q}}_{\mathbf{2}}\mathbf{]}\mathbf{=}\mathit{i}\mathit{ħ}$. Thus the q's and the p's satisfy the canonical commutation relations for position and momentum, and those of index 1are compatible with those of index 2 .

(b) Show that$\mathbf{\left[}{\mathbf{q}}_{\mathbf{1}\mathbf{,}}{\mathbf{q}}_{\mathbf{2}}\mathbf{\right]}\mathbf{=}\mathbf{[}{\mathbf{p}}_{\mathbf{1}}\mathbf{,}{\mathbf{p}}_{\mathbf{2}}\mathbf{]}\mathbf{}\mathbf{}{\mathit{L}}_{\mathbf{z}}\mathbf{=}\frac{\mathbf{ħ}}{\mathbf{2}{\mathbf{a}}^{\mathbf{2}}}\left(q_1^2-q_2^2\right)\mathbf{+}\frac{{\mathbf{a}}^{\mathbf{2}}}{\mathbf{2}\mathbf{ħ}}\mathbf{(}{\mathbf{p}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{-}{\mathbf{p}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{)}$

(c) Check that , where each is the Hamiltonian for a harmonic oscillator with mass and frequency .

(d) We know that the eigenvalues of the harmonic oscillator Hamiltonian are , where (In the algebraic theory of Section this follows from the form of the Hamiltonian and the canonical commutation relations). Use this to conclude that the eigenvalues of must be integers.

Deduce the condition for minimum uncertainty in${\mathit{S}}_{\mathbf{x}}$ and${\mathit{S}}_{\mathbf{y}}$(that is, equality in the expression role="math" localid="1658378301742" ${\mathit{\sigma }}_{{\mathbf{S}}_{\mathbf{x}}}{\mathit{\sigma }}_{{\mathbf{S}}_{\mathbf{y}}}\mathbf{\ge }\mathbf{(}\mathbf{ħ}\mathbf{/}\mathbf{2}\mathbf{)}\mathbf{\left|}<S_z>\mathbf{\right|}$, for a particle of spin 1/2 in the generic state (Equation 4.139). Answer: With no loss of generality we can pick to be real; then the condition for minimum uncertainty is that bis either pure real or else pure imaginary.

In classical electrodynamics the force on a particle of charge q

moving with velocity through electric and magnetic fields E and B is given

by the Lorentz force law:$\mathbf{F}\mathbf{=}\mathbf{q}\left(E+v\times B\right)$

This force cannot be expressed as the gradient of a scalar potential energy

function, and therefore the Schrödinger equation in its original form (Equation 1.1)

cannot accommodate it. But in the more sophisticated form $\mathit{i}\sout{\mathbf{h}}\frac{\mathbf{\partial }\mathbf{\psi }}{\mathbf{\partial }\mathbf{t}}\mathbf{=}\mathit{H}\mathit{\psi }$

there is no problem; the classical Hamiltonian is$\mathbf{H}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}\mathbf{m}}{\left(p-\mathrm{qA}\right)}^{\mathbf{2}}\mathbf{+}\mathbf{q\psi }$where A

is the vector potential$\left(B=\nabla \times A\right)$and $\mathbf{\psi }$is the scalar potential $\left(E=-\nabla \psi -\partial A/\partial t\right)$,

so the Schrödinger

equation (making the canonical substitution$\mathbf{p}\mathbf{\to }\left(\sout{h}/i\right)\mathbf{\nabla }\mathbf{)}$becomes$\mathbf{i}\sout{\mathbf{h}}\frac{\mathbf{\partial }\mathbf{\psi }}{\mathbf{\partial }\mathbf{t}}\mathbf{=}\left[\frac{1}{2m}{\left(\frac{\sout{h}}{i}\nabla -\mathrm{qA}\right)}^2+\mathrm{q\psi }\right]\mathbf{\psi }$

(a) Show that $\frac{\mathbf{d}\mathbf{<}\mathbf{r}\mathbf{>}}{\mathbf{dt}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{<}\left(p-\mathrm{qA}\right)\mathbf{>}$

(b) As always (see Equation ) we identify$\mathbf{d}\mathbf{<}\mathbf{r}\mathbf{>}\mathbf{/}\mathbf{dt}$with$\mathbf{<}\mathbf{v}\mathbf{>}$. Show that

$\mathbf{m}\frac{\mathbf{d}\mathbf{<}\mathbf{v}\mathbf{>}}{\mathbf{dt}}\mathbf{=}\mathbf{q}\mathbf{<}\mathbf{E}\mathbf{>}\mathbf{+}\frac{\mathbf{q}}{\mathbf{2}\mathbf{m}}\mathbf{<}\left(p\times B-B\times p\right)\mathbf{>}\mathbf{-}\frac{{\mathbf{q}}^{\mathbf{2}}}{\mathbf{m}}\mathbf{<}\left(A\times B\right)\mathbf{>}$

(c) In particular, if the fields and are uniform over the volume of the wave packet,

show that$\mathbf{m}\frac{\mathbf{d}\mathbf{<}\mathbf{v}\mathbf{>}}{\mathbf{dt}}\mathbf{=}\mathbf{q}\left(E+<V>\times B\right)$so the expectation value of $\left(v\right)$moves

according to the Lorentz force law, as we would expect from Ehrenfest's theorem.

Use Equation 4.32 to construct ${\mathit{Y}}_{\mathbf{l}}^{\mathbf{l}}\mathbf{(}\mathit{\theta }\mathbf{,}\mathit{\varphi }\mathbf{)}$and${\mathbf{y}}_{\mathbf{3}}^{\mathbf{2}}\mathbf{(}\mathbf{\theta }\mathbf{.}\mathbf{\varphi }\mathbf{)}$ . (You can take ${\mathit{P}}_{\mathbf{3}}^{\mathbf{2}}$from Table 4.2, but you'll have to work out${\mathit{P}}_{\mathbf{l}}^{\mathbf{l}}$ from Equations 4.27 and 4.28.) Check that they satisfy the angular equation (Equation 4.18), for the appropriate values of l and m .

[Refer to. Problem 4.59for background.] Suppose $\mathit{A}\mathbf{=}\frac{{\mathbf{B}}_{\mathbf{0}}}{\mathbf{2}}\left(X\hat{\varnothing }-y\hat{I}\right)$ and$\mathit{\phi }\mathbf{=}\mathit{K}{\mathit{z}}^{\mathbf{2}}$, where ${\mathit{B}}_{\mathbf{0}}$ and Kare constants.

(a) Find the fields E and B.

(b) Find the allowed energies, for a particle of mass m and charge q , in these fields, Answer: $$\begin{gathered} \mathit{E}\left(n_1,n_2\right)\mathbf{=}\left(n_1+\frac{1}{2}\right)\mathit{ħ}{\mathit{\omega }}_{\mathbf{1}}\mathbf{+}\left(n_2+\frac{1}{2}\right)\mathit{ħ}{\mathit{\omega }}_{\mathbf{2}}\mathbf{}\mathbf{,}\left(n_1,n_2=0,1,2,...\right)\mathbf{}\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e} \\ {\mathit{\omega }}_{\mathbf{1}}\mathbf{\equiv }\mathit{q}{\mathit{B}}_{\mathbf{0}}\mathbf{/}\mathit{m}\mathbf{}\mathbf{}\mathbf{and}\mathbf{}\mathbf{}{\mathbf{\omega }}_{\mathbf{2}}\mathbf{\equiv }\sqrt{\mathbf{2}\mathbf{qK}\mathbf{/}\mathbf{m}}\mathbf{}\mathbf{}\mathbf{.} \end{gathered}$$ Comment: If K=0this is the quantum analog to cyclotron motion;${\mathit{\omega }}_{\mathbf{1}}$ is the classical cyclotron frequency, and it's a free particle in the z direction. The allowed energies,$\left(n_1+\frac{1}{2}\right)\mathit{ħ}{\mathit{\omega }}_{\mathbf{1}}$, are called Landau Levels.

[Refer to Problem 4.59 for background.] In classical electrodynamics the potentials Aand$\mathit{\phi }$are not uniquely determined; 47 the physical quantities are the fields, E and B.

(a) Show that the potentials

${\mathbf{\phi }}^{\mathbf{\text{'}}}\mathbf{\equiv }\mathbf{\phi }\mathbf{-}\frac{\mathbf{\partial }\mathbf{\Lambda }}{\mathbf{\partial }\mathbf{t}}\mathbf{,}\mathbf{}{\mathbf{A}}^{\mathbf{\text{'}}}\mathbf{\equiv }\mathbf{A}\mathbf{+}\mathbf{\nabla }\mathbf{\Lambda }$

(where$\mathbf{\wedge }$is an arbitrary real function of position and time). yield the same fields as$\mathit{\phi }$and A. Equation 4.210 is called a gauge transformation, and the theory is said to be gauge invariant.

(b) In quantum mechanics the potentials play a more direct role, and it is of interest to know whether the theory remains gauge invariant. Show that

${\mathit{\Psi }}^{\mathbf{\text{'}}}\mathbf{\equiv }{\mathit{e}}^{\mathbf{i}\mathbf{q}\mathbf{\Lambda }\mathbf{/}\mathbf{\hslash }}\mathit{\Psi }$

satisfies the Schrödinger equation (4.205) with the gauge-transformed potentials${\mathit{\phi }}^{\mathbf{\text{'}}}$and${\mathit{A}}^{\mathbf{\text{'}}}$, Since${\mathit{\Psi }}^{\mathbf{\text{'}}}$differs from$\mathit{\psi }$only by a phase factor, it represents the same physical state, 48and the theory is gauge invariant (see Section 10.2.3for further discussion).

Starting from the Rodrigues formula, derive the orthonormality condition for Legendre polynomials:

$\underset{\mathbf{-}\mathbf{1}}{\overset{\mathbf{1}}{\mathbf{\int }}}{\mathbf{P}}_{\mathbf{l}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}{\mathbf{P}}_{\mathbf{I}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dx}\mathbf{=}\left(\frac{2}{2l+1}\right){\mathbf{\delta }}_{\mathbf{II}}\mathbf{.}$

Hint: Use integration by parts.