Question

Find the probability distributions of the orbital angular momentum
variables L2 and Lz for the following orbital state functions:
(a) Ψ(x) = f(r) sin θ cos φ,
(b) Ψ(x) = f(r)(cos θ)2,
(c) Ψ(x) = f(r) sin θ cos θ sin φ.
Here r, θ, φ are the usual spherical coordinates, and f(r) is an arbitrary
radial function (not necessarily the same in each case) into which the
normalization constant has been absorbed

Answer 1

Answer on Question #84387 – Physics – Quantum Mechanics

Find the probability distributions of the orbital angular momentum variables L2 and Lz for the following orbital state functions:

(a) $\Psi (x) = f(r)\sin \theta \cos \phi ,$

(b) $\Psi (x) = f(r)(\cos \theta)2,$

(c) $\Psi (x) = f(r)\sin \theta \cos \theta \sin \phi .$

Here $r, \theta, \phi$ are the usual spherical coordinates, and $f(r)$ is an arbitrary radial function (not necessarily the same in each case) into which the normalization constant has been absorbed

Solution. Distributions are as follows:

$$L _ {2} \Psi (x) = - \left[ \frac {h}{2 \pi} \frac {\partial}{\partial \varphi} \psi ; L ^ {2} \Psi (x) = - \frac {h ^ {2}}{4 \pi^ {2}} \left[ \frac {1}{\sin \theta} \frac {\partial}{\partial \theta} \left(\sin \theta \frac {\partial}{\partial \theta}\right) + \frac {1}{\sin^ {2} \theta} \frac {\partial}{\partial \varphi^ {2}} \right] \psi . \right.$$

Then:

a) $L_{2} \Psi(x) = -\frac{h}{2\pi} \frac{\partial}{\partial \varphi} \psi = -\frac{h}{2\pi} \frac{\partial f(r)}{\partial \varphi} \sin \theta \cos \phi + \frac{h}{2\pi} f(r) \sin \theta \sin \phi = -\frac{h}{2\pi} \sin \theta \left( \frac{\partial f(r)}{\partial \varphi} \cos \phi - f(r) \sin \phi \right)$ ;

$L^2\Psi (x) = -\frac{h^2}{4\pi^2}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\Bigl (sin\theta \frac{\partial}{\partial\theta}\Bigr) + \frac{1}{\sin^2\theta}\frac{\partial}{\partial\varphi^2}\right]\psi = -\frac{h^2}{4\pi^2}\left[\frac{1}{\sin\theta}\left(\frac{\partial^2f(r)}{\partial\theta^2}\sin^2\theta cos\varphi +3\sin \theta cos\theta cos\phi \frac{\partial f(r)}{\partial\theta} + \cos 2\theta cos\phi f(r)\right) + \frac{1}{\sin^2\theta}\left(\frac{\partial^2f(r)}{\partial\varphi^2}\sin \theta cos\varphi -2\frac{\partial f(r)}{\partial\varphi}\sin \theta sin\varphi +f(r)\sin \theta cos\varphi\right)\right]$

$= -\frac{h^2}{4\pi^2}\left[\frac{\partial^2f(r)}{\partial\theta^2}\sin \theta cos\varphi +3\cos \theta cos\phi \frac{\partial f(r)}{\partial\theta} +\frac{\cos 2\theta}{\sin\theta}\cos \phi f(r) + \frac{\partial^2f(r)}{\partial\varphi^2}\frac{\cos\varphi}{\sin\theta} -2\frac{\partial f(r)}{\partial\varphi}\frac{\sin\varphi}{\sin\theta} +f(r)\frac{\cos\varphi}{\sin\theta}\right];$

b) $L_{2} \Psi(x) = -\frac{h}{2\pi} \frac{\partial}{\partial \varphi} \psi = -\frac{h}{2\pi} \frac{\partial f(r)}{\partial \varphi} (\cos \theta)^{2}$ ;

$$\begin{array}{l} L ^ {2} \Psi (x) = - \frac {h ^ {2}}{4 \pi^ {2}} \left[ \frac {1}{\sin \theta} \frac {\partial}{\partial \theta} \left(\sin \theta \frac {\partial}{\partial \theta}\right) + \frac {1}{\sin^ {2} \theta} \frac {\partial}{\partial \varphi^ {2}} \right] \psi = - \frac {h ^ {2}}{4 \pi^ {2}} \quad \left[ \frac {\partial^ {2} f (r)}{\partial \theta^ {2}} \cos^ {2} \theta - 8 \cos \theta \sin \theta \frac {\partial f (r)}{\partial \theta} + \right. \\ + \frac {\partial f (r)}{\partial \theta} \frac {\cos^ {3} \theta}{\sin \theta} - 4 \sin \theta \cos^ {2} \theta f (r) + 2 \sin^ {2} \theta f (r) + \frac {\partial^ {2} f (r)}{\partial \varphi^ {2}} \cos^ {2} \theta \rbrack ; \\ \end{array}$$

c) $L_{2} \Psi(x) = -\frac{h}{2\pi} \frac{\partial}{\partial \varphi} \psi = -\frac{h}{2\pi} \left( \frac{\partial f(r)}{\partial \varphi} \sin \theta \cos \theta \sin \phi + f(r) \sin \theta \cos \theta \cos \phi \right)$ ;

$L^2\Psi (x) = -\frac{h^2}{4\pi^2}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\Bigl (sin\theta \frac{\partial}{\partial\theta}\Bigr) + \frac{1}{\sin^2\theta}\frac{\partial}{\partial\varphi^2}\right]\psi = -\frac{h^2}{4\pi^2}\bigl [\frac{\partial^2f(r)}{\partial\theta^2}\sin \theta cos\theta sin\varphi +3\frac{\partial f(r)}{\partial\theta}\cos^2\theta sin\varphi -$

$-2\frac{\partial f(r)}{\partial\theta}\sin \varphi \sin^2\theta +f(r)\sin \varphi \frac{\cos^3\theta}{\sin\theta} -5\cos \theta \sin \theta \sin \varphi f(r) + \frac{\partial^2f(r)}{\partial\varphi^2}\sin \theta \cos \theta \sin \varphi +$

$+2\frac{\partial f(r)}{\partial\varphi}\sin\theta\cos\theta\cos\varphi -f(r)\sin\theta\cos\theta\sin\varphi].$

Answer: a) $L_{2} \Psi(x) = -\frac{h}{2\pi} \sin \theta \left( \frac{\partial f(r)}{\partial \varphi} \cos \phi - f(r) \sin \phi \right)$ ; $L^{2} \Psi(x) = -\frac{h^{2}}{4\pi^{2}} \left[ \frac{\partial^{2} f(r)}{\partial \theta^{2}} \sin \theta \cos \varphi + 3 \cos \theta \cos \phi \frac{\partial f(r)}{\partial \theta} + \frac{\cos 2 \theta}{\sin \theta} \cos \phi f(r) + \frac{\partial^{2} f(r)}{\partial \varphi^{2}} \frac{\cos \varphi}{\sin \theta} - 2 \frac{\partial f(r)}{\partial \varphi} \frac{\sin \varphi}{\sin \theta} + f(r) \frac{\cos \varphi}{\sin \theta} \right]$ ;

b) $L_{2} \Psi(x) = -\frac{h^{2}}{2\pi} \frac{\partial f(r)}{\partial \varphi} (\cos \theta)^{2}$ ; $L^{2} \Psi(x) = -\frac{h^{2}}{4\pi^{2}} \left[ \frac{\partial^{2} f(r)}{\partial \theta^{2}} \cos^{2} \theta - 8 \cos \theta \sin \theta \frac{\partial f(r)}{\partial \theta} + \frac{\partial f(r)}{\partial \theta} \frac{\cos^{3} \theta}{\sin \theta} - 4 \sin \theta \cos^{2} \theta f(r) + 2 \sin^{2} \theta f(r) + \frac{\partial^{2} f(r)}{\partial \varphi^{2}} \cos^{2} \theta \right]$ ;

c) $L_{2} \Psi(x) = -\frac{h}{2\pi} \left( \frac{\partial f(r)}{\partial \varphi} \sin \theta \cos \theta \sin \phi + f(r) \sin \theta \cos \theta \cos \phi \right)$ ; $L^{2} \Psi(x) = -\frac{h^{2}}{4\pi^{2}} \left( \frac{\partial^{2} f(r)}{\partial \theta^{2}} \sin \theta \cos \theta \sin \varphi + 3 \frac{\partial f(r)}{\partial \theta} \cos^{2} \theta \sin \varphi - -2 \frac{\partial f(r)}{\partial \theta} \sin \varphi \sin^{2} \theta + f(r) \sin \varphi \frac{\cos^{3} \theta}{\sin \theta} - 5 \cos \theta \sin \theta \sin \varphi f(r) + \frac{\partial^{2} f(r)}{\partial \varphi^{2}} \sin \theta \cos \theta \sin \varphi + +2 \frac{\partial f(r)}{\partial \varphi} \sin \theta \cos \theta \cos \varphi - f(r) \sin \theta \cos \theta \sin \varphi \right]$ .

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