(a) Construct the spatial wave function (ψ) for hydrogen in the state n = 3 , l = 2 , m = 1. Express your answer as a function of r, θ, ϕ, and a (the Bohr radius) only—no other variables (ρ, z, etc.) or functions (Y, v, etc.), or constants (A, c0, etc.), or derivatives, allowed (π is okay, and e,

Question

(a) Construct the spatial wave function (ψ) for hydrogen in the state n = 3 , l = 2 , m = 1. Express your answer as a function of r, θ, ϕ, and a (the Bohr radius) only—no other variables (ρ, z, etc.) or functions (ρ, v, etc.), or constants (A, [latex] c_0 [/latex], 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 r, θ, and ϕ.

(c) Find the expectation value of [latex] r^s [/latex] in this state. For what range of s (positive and negative) is the result finite?

Accepted Answer

(a) From Tables 4.3 and 4.7,

Table 4.3: The first few spherical harmonics [latex] Y_{\ell}^{m}( heta, \phi) [/latex]
[latex] Y_{2}^{+2}=\left(\frac{15}{32 \pi} ight)^{1 / 2} \sin ^{2} heta e^{\pm 2 i \phi} [/latex]	[latex] Y_{0}^{0}=\left(\frac{1}{4 \pi} ight)^{1 / 2} [/latex]
[latex] Y_{3}^{0}=\left(\frac{7}{16 \pi} ight)^{1 / 2}\left(5 \cos ^{3} heta-3 \cos heta ight) [/latex]	[latex] Y_{1}^{0}=\left(\frac{3}{4 \pi} ight)^{1 / 2} \cos heta [/latex]
[latex] Y_{3}^{\pm 1}=\mp\left(\frac{21}{64 \pi} ight)^{1 / 2} \sin heta\left(5 \cos ^{2} heta-1 ight) e^{\pm i \phi} [/latex]	[latex] Y_{1}^{\pm 1}=\mp\left(\frac{3}{8 \pi} ight)^{1 / 2} \sin heta e^{\pm i \phi} [/latex]
[latex] Y_{3}^{\pm 2}=\left(\frac{105}{32 \pi} ight)^{1 / 2} \sin ^{2} heta \cos heta e^{\pm 2 i \phi} [/latex]	[latex] Y_{2}^{0}=\left(\frac{5}{16 \pi} ight)^{1 / 2}\left(3 \cos ^{2} heta-1 ight) [/latex]
[latex] Y_{3}^{+3}=\mp\left(\frac{35}{64 \pi} ight)^{1 / 2} \sin ^{3} heta e^{\pm 3 i \phi} [/latex]	[latex] Y_{2}^{\pm 1}=\mp\left(\frac{15}{8 \pi} ight)^{1 / 2} \sin heta \cos heta e^{\pm i \phi} [/latex]

Table 4.7: The first few radial wave functions for hydrogen,[latex] R_{n \ell}(r) [/latex]

[latex] R_{10}=2 a^{-3 / 2} \exp (-r / a) [/latex]

[latex] R_{20}=\frac{1}{\sqrt{2}} a^{-3 / 2}\left(1-\frac{1}{2} \frac{r}{a} ight) \exp (-r / 2 a) [/latex].

[latex] R_{21}=\frac{1}{2 \sqrt{6}} a^{-3 / 2}\left(\frac{r}{a} ight) \exp (-r / 2 a) [/latex].

[latex] R_{30}=\frac{2}{3 \sqrt{3}} a^{-3 / 2}\left(1-\frac{2}{3} \frac{r}{a}+\frac{2}{27}\left(\frac{r}{a} ight)^{2} ight) \exp (-r / 3 a) [/latex].

[latex] R_{31}=\frac{8}{27 \sqrt{6}} a^{-3 / 2}\left(1-\frac{1}{6} \frac{r}{a} ight)\left(\frac{r}{a} ight) \exp (-r / 3 a) [/latex].

[latex] R_{32}=\frac{4}{81 \sqrt{30}} a^{-3 / 2}\left(\frac{r}{a} ight)^{2} \exp (-r / 3 a) [/latex].

[latex] R_{40}=\frac{1}{4} a^{-3 / 2}\left(1-\frac{3}{4} \frac{r}{a}+\frac{1}{8}\left(\frac{r}{a} ight)^{2}-\frac{1}{192}\left(\frac{r}{a} ight)^{3} ight) \exp (-r / 4 a) [/latex].

[latex] R_{41}=\frac{5}{16 \sqrt{15}} a^{-3 / 2}\left(1-\frac{1}{4} \frac{r}{a}+\frac{1}{80}\left(\frac{r}{a} ight)^{2} ight)\left(\frac{r}{a} ight) \exp (-r / 4 a) [/latex].

[latex] R_{42}=\frac{1}{64 \sqrt{5}} a^{-3 / 2}\left(1-\frac{1}{12} \frac{r}{a} ight)\left(\frac{r}{a} ight)^{2} \exp (-r / 4 a) [/latex].

[latex] R_{43}=\frac{1}{768 \sqrt{35}} a^{-3 / 2}\left(\frac{r}{a} ight)^{3} \exp (-r / 4 a) [/latex].

(a)

[latex] \psi_{321}=R_{32} Y_{2}^{1}=\frac{4}{81 \sqrt{30}} \frac{1}{a^{3 / 2}}\left(\frac{r}{a} ight)^{2} e^{-r / 3 a}\left[-\sqrt{\frac{15}{8 \pi}} \sin heta \cos heta e^{i \phi} ight]= -\frac{1}{\sqrt{\pi}} \frac{1}{81 a^{7 / 2}} r^{2} e^{-r / 3 a} \sin heta \cos heta e^{i \phi} [/latex].

(b)

[latex] \int|\psi|^{2} d^{3} r =\frac{1}{\pi} \frac{1}{(81)^{2} a^{7}} \int\left(r^{4} e^{-2 r / 3 a} \sin ^{2} heta \cos ^{2} heta ight) r^{2} \sin heta d r d heta d \phi [/latex].

[latex] =\frac{1}{\pi(81)^{2} a^{7}} 2 \pi \int_{0}^{\infty} r^{6} e^{-2 r / 3 a} d r \int_{0}^{\pi}\left(1-\cos ^{2} heta ight) \cos ^{2} heta \sin heta d heta [/latex].

[latex] =\left.\frac{2}{(81)^{2} a^{7}}\left[6 !\left(\frac{3 a}{2} ight)^{7} ight]\left[-\frac{\cos ^{3} heta}{3}+\frac{\cos ^{5} heta}{5} ight] ight|_{0} ^{\pi} [/latex].

[latex] =\frac{2}{3^{8} a^{7}} 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \frac{3^{7} a^{7}}{2^{7}}\left[\frac{2}{3}-\frac{2}{5} ight]=\frac{3 \cdot 5}{4} \cdot \frac{4}{15}=1 [/latex].

(c)

[latex] \left\langle r^{s} ight angle=\int_{0}^{\infty} r^{s}\left|R_{32} ight|^{2} r^{2} d r=\left(\frac{4}{81} ight)^{2} \frac{1}{30} \frac{1}{a^{7}} \int_{0}^{\infty} r^{s+6} e^{-2 r / 3 a} d r [/latex].

[latex] =\frac{8}{15(81)^{2} a^{7}}(s+6) !\left(\frac{3 a}{2} ight)^{s+7}= (s+6) !\left(\frac{3 a}{2} ight)^{s} \frac{1}{720} =\frac{(s+6) !}{6 !}\left(\frac{3 a}{2} ight)^{s} [/latex].

Finite for s > -7.

Table 4.3: The first few spherical harmonics $Y_{\ell}^{m}(\theta, \phi)$
$Y_{2}^{+2}=\left(\frac{15}{32 \pi}\right)^{1 / 2} \sin ^{2} \theta e^{\pm 2 i \phi}$	$Y_{0}^{0}=\left(\frac{1}{4 \pi}\right)^{1 / 2}$
$Y_{3}^{0}=\left(\frac{7}{16 \pi}\right)^{1 / 2}\left(5 \cos ^{3} \theta-3 \cos \theta\right)$	$Y_{1}^{0}=\left(\frac{3}{4 \pi}\right)^{1 / 2} \cos \theta$
$Y_{3}^{\pm 1}=\mp\left(\frac{21}{64 \pi}\right)^{1 / 2} \sin \theta\left(5 \cos ^{2} \theta-1\right) e^{\pm i \phi}$	$Y_{1}^{\pm 1}=\mp\left(\frac{3}{8 \pi}\right)^{1 / 2} \sin \theta e^{\pm i \phi}$
$Y_{3}^{\pm 2}=\left(\frac{105}{32 \pi}\right)^{1 / 2} \sin ^{2} \theta \cos \theta e^{\pm 2 i \phi}$	$Y_{2}^{0}=\left(\frac{5}{16 \pi}\right)^{1 / 2}\left(3 \cos ^{2} \theta-1\right)$
$Y_{3}^{+3}=\mp\left(\frac{35}{64 \pi}\right)^{1 / 2} \sin ^{3} \theta e^{\pm 3 i \phi}$	$Y_{2}^{\pm 1}=\mp\left(\frac{15}{8 \pi}\right)^{1 / 2} \sin \theta \cos \theta e^{\pm i \phi}$

Question 4.52: (a) Construct the spatial wave function (ψ) for hydrogen in...

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