(a) Construct the wave function for hydrogen in the state n =4 , l = 3 , m = 3 . Express your answer as a function of the spherical coordinates r, θ, and ϕ.

(b) Find the expectation value of r in this state. (As always, look up any nontrivial integrals.)

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

Question

(a) Construct the wave function for hydrogen in the state n =4 , l = 3 , m = 3 . Express your answer as a function of the spherical coordinates r, θ, and ϕ.

(b) Find the expectation value of r in this state. (As always, look up any nontrivial integrals.)

(c) If you could somehow measure the observable [latex] L_{x}^{2}+L_{y}^{2} [/latex] on an atom in this state, what value (or values) could you get, and what is the probability of each?

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_{433}=R_{43} Y_{3}^{3}=\frac{1}{768 \sqrt{35}} \frac{1}{a^{3 / 2}}\left(\frac{r}{a} ight)^{3} e^{-r / 4 a}\left(-\sqrt{\frac{35}{64 \pi}} \sin ^{3} heta e^{3 i \phi} ight)= -\frac{1}{6144 \sqrt{\pi} a^{9 / 2}} r^{3} e^{-r / 4 a} \sin ^{3} heta e^{3 i \phi} [/latex].

(b)

[latex] \langle r angle=\int r|\psi|^{2} d^{3} r =\frac{1}{(6144)^{2} \pi a^{9}} \int r\left(r^{6} e^{-r / 2 a} \sin ^{6} heta ight) r^{2} \sin heta d r d heta d \phi [/latex].

[latex] =\frac{1}{(6144)^{2} \pi a^{9}} \int_{0}^{\infty} r^{9} e^{-r / 2 a} d r \int_{0}^{\pi} \sin ^{7} heta d heta \int_{0}^{2 \pi} d \phi [/latex].

[latex] =\frac{1}{(6144)^{2} \pi a^{9}}\left[9 !(2 a)^{10} ight]\left(2 \frac{2 \cdot 4 \cdot 6}{3 \cdot 5 \cdot 7} ight)(2 \pi)=18 a [/latex].

(c) Using Eq. 4.133: [latex] L_{x}^{2}+L_{y}^{2}=L^{2}-L_{z}^{2}=3(4) \hbar^{2}-(3 \hbar)^{2}= 3 \hbar^{2}[/latex]. with probability 1.

[latex] H \psi=E \psi, \quad L^{2} \psi=\hbar^{2} \ell(\ell+1) \psi, \quad L_{z} \psi=\hbar m \psi [/latex]. (4.133).

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 4.53: (a) Construct the wave function for hydrogen in the state n ...

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