Chapter 4: Quantum Mechanics in Three Dimensions

Determine the commutator of ${\mathbf{S}}^{\mathbf{2}}$with${\mathbf{S}}_{\mathbf{Z}}^{\left(1\right)}$(where$\mathbf{S}\mathbf{\equiv }{\mathbf{S}}^{\left(1\right)}\mathbf{+}{\mathbf{S}}^{\left(2\right)}$) Generalize your result to show that

$\left[S^2,S^{\left(1\right)}\right]\mathbf{=}\mathbf{2}\mathbf{I}\sout{\mathbf{h}}\left(S^{\left(1\right)}\times S^{\left(2\right)}\right)$

Comment: Because ${\mathbf{S}}_{\mathbf{z}}^{\left(1\right)}$does not commute with ${\mathbf{S}}^{\mathbf{2}}$, we cannot hope to find states that are simultaneous eigenvectors of both. In order to form eigenstates of${\mathbf{S}}^{\mathbf{2}}$weneed linear combinations of eigenstates of${\mathbf{S}}_{\mathbf{z}}^{\left(1\right)}$. This is precisely what the Clebsch-Gordan coefficients (in Equation 4.185) do for us, On the other hand, it follows by obvious inference from Equation 4.187that the sumrole="math" localid="1655980965321" ${\mathit{S}}^{\left(1\right)}\mathbf{+}{\mathit{S}}^{\left(2\right)}$does commute withdata-custom-editor="chemistry" ${\mathit{S}}^{\mathbf{2}}$, which is a special case of something we already knew (see Equation 4.103).

Consider the three-dimensional harmonic oscillator, for which the potential is

$\mathbf{V}\left(r\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{m\omega }}^{\mathbf{2}}{\mathbf{r}}^{\mathbf{2}}$

(a) Show that separation of variables in cartesian coordinates turns this into three one-dimensional oscillators, and exploit your knowledge of the latter to determine the allowed energies. Answer:

${\mathbf{E}}_{\mathbf{n}}\mathbf{=}\left(n+3/2\right)\sout{\mathbf{h}}\mathbf{\omega }$

(b) Determine the degeneracyof$\mathbf{d}\left(n\right)\mathbf{}\mathbf{of}\mathbf{}{\mathbf{E}}_{\mathbf{n}}\mathbf{.}$

Because the three-dimensional harmonic oscillator potential (Equation 4.188)is spherically symmetric, the Schrödinger equation can be handled by separation of variables in spherical coordinates, as well as cartesian coordinates. Use the power series method to solve the radial equation. Find the recursion formula for the coefficients, and determine the allowed energies. Check your answer against Equation4.189.

Use equations 4.27 4.28 and 4.32 to construct ${\mathit{Y}}_{\mathbf{0}}^{\mathbf{0}}\mathbf{,}\mathbf{}{\mathit{Y}}_{\mathbf{2}}^{\mathbf{1}}$Check that they are normalized and orthogonal

Use equations 4.27 4.28 and 4.32 to construct${\mathit{y}}_{\mathbf{0}}^{\mathbf{0}}\mathbf{}\mathbf{,}\mathbf{}{\mathit{y}}_{\mathbf{2}}^{\mathbf{1}}$Check that they are normalized and orthogonal

(a) Prove the three-dimensional virial theorem

$\mathbf{2}\mathbf{⟨}\mathbf{T}\mathbf{⟩}\mathbf{=}\mathbf{⟨}\mathbf{r}\mathbf{\cdot }\mathbf{\nabla }\mathbf{V}\mathbf{⟩}$

(for stationary states). Hint: Refer to problem 3.31,

(b) Apply the virial theorem to the case of hydrogen, and show that

$\mathbf{⟨}\mathbf{T}\mathbf{⟩}\mathbf{=}\mathbf{-}{\mathbf{E}}_{\mathbf{n}}\mathbf{;}\mathbf{⟨}\mathbf{V}\mathbf{⟩}\mathbf{=}\mathbf{2}{\mathbf{E}}_{\mathbf{n}}$

(c) Apply the virial theorem to the three-dimensional harmonic oscillator and show that in this case

$\mathbf{⟨}\mathbf{T}\mathbf{⟩}\mathbf{=}\mathbf{⟨}\mathbf{V}\mathbf{⟩}\mathbf{=}{\mathbf{E}}_{\mathbf{n}}\mathbf{/}\mathbf{2}$

[Attempt this problem only if you are familiar with vector calculus.] Define the (three-dimensional) probability current by generalization of Problem 1.14:

$\mathbf{J}\mathbf{=}\frac{\mathbf{i}\sout{\mathbf{h}}}{\mathbf{2}\mathbf{m}}\left(\psi \nabla {\psi }^{*}-{\psi }^{*}\nabla \psi \right)$

(a) Show that satisfies the continuity equation $\mathbf{\nabla }\mathbf{.}\mathbf{J}\mathbf{=}\mathbf{-}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{t}}{\left|\psi \right|}^{\mathbf{2}}$which expresses local conservation of probability. It follows (from the divergence theorem) that ${\mathbf{\int }}_{\mathbf{s}}\mathbf{J}\mathbf{.}\mathbf{da}\mathbf{=}\mathbf{-}\frac{\mathbf{d}}{\mathbf{dt}}{\mathbf{\int }}_{\mathbf{v}}{\left|\psi \right|}^{\mathbf{2}}{\mathbf{d}}^{\mathbf{3}}\mathbf{r}$where $\mathbf{V}$is a (fixed) volume and is its boundary surface. In words: The flow of probability out through the surface is equal to the decrease in probability of finding the particle in the volume.

(b) Find$\mathbf{J}$for hydrogen in the state$\mathbf{n}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{l}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{m}\mathbf{=}\mathbf{1}$ . Answer:

$\frac{\sout{\mathbf{h}}}{\mathbf{64}{\mathbf{ma}}^{\mathbf{5}}}{\mathbf{re}}^{\mathbf{-}\mathbf{r}\mathbf{/}\mathbf{a}}\mathbf{sin\theta }\hat{\mathbf{\varphi }}$

(c) If we interpret$\mathbf{mJ}$as the flow of mass, the angular momentum is

$\mathbf{L}\mathbf{=}\mathbf{m}\mathbf{\int }\left(r\times J\right){\mathbf{d}}^{\mathbf{3}}\mathbf{r}$

Use this to calculate ${\mathbf{L}}_{\mathbf{z}}$for the state${\mathbf{\psi }}_{\mathbf{211}}$, and comment on the result.

The (time-independent) momentum space wave function in three dimensions is defined by the natural generalization of Equation 3.54:

$$\begin{gathered} \mathbf{\Phi }\mathbf{(}\mathbf{p}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{2}\mathbf{\pi }\sout{\mathbf{h}}}}{\mathbf{\int }}_{\mathbf{\infty }}^{\mathbf{\infty }}{\mathbf{e}}^{\mathbf{-}\mathbf{ipx}\mathbf{/}\sout{\mathbf{h}}}\mathbf{\psi }\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{dx} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{54}\mathbf{)}\mathbf{.} \\ \mathbf{\varphi }\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{\equiv }\frac{\mathbf{1}}{{\mathbf{\left(}\mathbf{2}\mathbf{\pi }\sout{\mathbf{h}}\mathbf{\right)}}^{\mathbf{3}\mathbf{/}\mathbf{2}}}\mathbf{\int }{\mathbf{e}}^{\mathbf{-}\mathbf{i}\mathbf{(}\mathbf{p}\mathbf{.}\mathbf{r}\mathbf{)}\mathbf{I}\sout{\mathbf{h}}}\mathbf{\psi }\mathbf{\left(}\mathbf{r}\mathbf{\right)}{\mathbf{d}}^{\mathbf{3}}\mathbf{r}\mathbf{.}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{223}\mathbf{)}\mathbf{.} \end{gathered}$$

(a)Find the momentum space wave function for the ground state of hydrogen (Equation 4.80). Hint: Use spherical coordinates, setting the polar axis along the direction of p. Do the θ integral first. Answer:

$$\begin{gathered} {\mathbf{\psi }}_{\mathbf{100}}\left(r,\theta ,\varphi \right)\mathbf{=}\frac{\mathbf{1}}{\sqrt{{\mathbf{\pi a}}^{\mathbf{3}}}}{\mathbf{e}}^{\mathbf{-}\mathbf{r}\mathbf{/}\mathbf{a}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left(4.80\right)\mathbf{.} \\ \mathbf{\varphi }\left(p\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{\pi }}{\left(\frac{2a}{\sout{h}}\right)}^{\mathbf{3}\mathbf{/}\mathbf{2}}\mathbf{}\mathbf{}\mathbf{}\frac{\mathbf{1}}{{\left[1+{\left(\mathrm{ap}/\sout{h}\right)}^2\right]}^{\mathbf{2}\mathbf{.}}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left(4.224\right)\mathbf{.} \end{gathered}$$

(b) Check that $\mathbf{\Phi }\left(p\right)$is normalized.

(c) Use $\mathbf{\Phi }\left(p\right)$to calculate $<p^2>$, in the ground state of hydrogen.

(d) What is the expectation value of the kinetic energy in this state? Express your answer as a multiple of ${\mathbf{E}}_{\mathbf{1}}$, and check that it is consistent with the virial theorem (Equation 4.218).

$<T>\mathbf{=}\mathbf{-}{\mathbf{E}}_{\mathbf{n}}\mathbf{;}\mathbf{}<V>\mathbf{=}\mathbf{2}{\mathbf{E}}_{\mathbf{n}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left(4.218\right)\mathbf{.}$

(a) Construct the spatial wave function $\mathbf{\left(}\mathbf{\psi }\mathbf{\right)}$for hydrogen in the state $\mathbf{n}\mathbf{=}\mathbf{3}\mathbf{,}\mathbf{}\mathbf{I}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{m}\mathbf{=}\mathbf{1}\mathbf{.}$Express your answer as a function of $\mathbf{r}\mathbf{,}\mathbf{}\mathbf{\theta }\mathbf{,}\mathbf{\varphi }\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}\mathbf{a}$(the Bohr radius) only—no other variables ($\mathbf{p}\mathbf{,}\mathbf{z}\mathbf{}\mathbf{,}$etc.) or functions ($\mathit{p}\mathbf{,}\mathit{v}\mathbf{,}$etc.), or constants ($\mathbf{A}\mathbf{,}{\mathbf{c}}_{\mathbf{0}}\mathbf{,}$etc.), or derivatives, allowed (π is okay, and e, and 2, etc.).

(b) Check that this wave function is properly normalized, by carrying out the appropriate integrals over, $\mathbf{\theta }\mathbf{,}\mathbf{}\mathbf{and}\mathbf{}\mathbf{\varphi }\mathbf{.}$

(c) Find the expectation value of ${\mathbf{r}}^{\mathbf{s}}$in this state. For what range of s (positive and negative) is the result finite?

(a) Construct the wave function for hydrogen in the state $\mathbf{n}\mathbf{=}\mathbf{4}\mathbf{,}\mathbf{I}\mathbf{=}\mathbf{3}\mathbf{,}\mathbf{m}\mathbf{=}\mathbf{3}$. Express your answer as a function of the spherical coordinates $\mathit{r}\mathbf{,}\mathit{\theta }\mathbf{}\mathbf{and}\mathbf{}\mathit{\varphi }$.

(b) Find the expectation value of role="math" localid="1658391074946" $\mathit{r}$in this state. (As always, look up any nontrivial integrals.)

(c) If you could somehow measure the observable ${\mathit{L}}_{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathit{L}}_{\mathbf{y}}^{\mathbf{2}}$on an atom in this state, what value (or values) could you get, and what is the probability of each?