Some equations and constants

8. Some equations and constants#

8.1. Physical constants#

Table 8.1 Some physical constants with their values in standard units.#
Name	Symbol	Value
Speed of light	\(c\)	\(3.00 \cdot 10^{8}\;\mathrm{m}/\mathrm{s}\)
Elementary charge	\(e\)	\(1.60 \cdot 10^{-19}\;\mathrm{C}\)
Electron mass	\(m_e\)	\(9.11 \cdot 10^{-31}\;\mathrm{kg} = 0.511\;\mathrm{MeV}/c^2\)
Proton mass	\(m_p\)	\(1.67 \cdot 10^{-27}\;\mathrm{kg} = 938\;\mathrm{MeV}/c^2\)
Gravitational constant	\(G\)	\(6.67 \cdot 10^{-11}\;\mathrm{N}\cdot\mathrm{m}^2/\mathrm{kg}^2\)
Planck’s constant	\(h\)	\(6.63 \cdot 10^{-34}\;\mathrm{J}\cdot\mathrm{s}\)
	\(\hbar = h / 2 \pi\)	\(1.05 \cdot 10^{-34}\;\mathrm{J}\cdot\mathrm{s}\)
Permittivity of space	\(\varepsilon_0\)	\(8.85419 \cdot 10^{-12}\;\mathrm{C}^2/\mathrm{J}\cdot\mathrm{m}\)
Boltzmann’s constant	\(\kB\)	\(1.38 \cdot 10^{-23}\;\mathrm{J}/\mathrm{K}\)
Bohr radius	\(a\)	\(\frac{4\pi \varepsilon_0 \hbar^2}{e^2 m_\mathrm{e}} = 5.29177 \cdot 10^{-11}\;\mathrm{m}\)
Fine structure constant	\(\alpha\)	\(\frac{e^2}{4\pi\varepsilon_0 \hbar c} = \frac{1}{137.036}\)
Rydberg energy	\(R_\mathrm{E}\)	\(\frac{e^4 m_\mathrm{e}}{2(4\pi\varepsilon_0 \hbar)^2} = \frac12 \left(m_\mathrm{e} c^2\right) \alpha^2 = 13.6057\;\mathrm{eV}\).

8.2. Wave-particle duality#

The de Broglie energy and momentum are given by:

(8.1)#\[\begin{align*} E &= h f = \hbar \omega, \end{align*}\]

(8.2)#\[\begin{align*} p &= h/\lambda = \hbar k. \end{align*}\]

8.3. Schrödinger equation#

The general Schrödinger equation in three dimensions is given by:

(8.3)#\[ i \hbar \frac{\partial \Psi(\bm{x}, t)}{\partial t} = - \frac{\hbar^2}{2m} \nabla^2 \Psi(\bm{x}, t) + V(\bm{x}) \Psi(\bm{x}, t). \]

In one dimension, the time-independent version of the Schrödinger equation is:

(8.4)#\[ \hat{H} \psi(x) = -\frac{\hbar^2}{2m} \frac{\mathrm{d}^2 \psi(x)}{\mathrm{d}x^2} + V(x) \psi(x) = E \psi(x). \]

8.4. Operators#

The position and momentum operators are given by

(8.5)#\[\begin{align*} \hat{x} \Psi(x,t) &= x \Psi(x,t), \end{align*}\]

(8.6)#\[\begin{align*} \hat{p} \Psi(x,t) &= - i \hbar \frac{\partial \Psi(x,t)}{\partial x}. \end{align*}\]

The expectation value of any operator \(\hat{Q}\) can be calculated through

(8.7)#\[ \Braket{\hat{Q}} = \int_{-\infty}^\infty \Psi^*(x,t) \hat{Q} \Psi(x,t) \,\mathrm{d}x. \]

The commutator \(\left[ \hat{A}, \hat{B} \right]\) of two operators is defined as

(8.8)#\[ \left[ \hat{A}, \hat{B} \right] = \hat{A}\hat{B} - \hat{B}\hat{A}. \]

For any two Hermitian operators \(\hat{A}\) and \(\hat{B}\), we have the general uncertainty principle

(8.9)#\[ \sigma_A \sigma_B \geq \left| \frac{1}{2i} \Braket{\left[ \hat{A}, \hat{B} \right]} \right|. \]

The time evolution of the expectation value of an operator is given by

(8.10)#\[ \frac{\mathrm{d}}{\mathrm{d}t} \Braket{\hat{Q}} = \frac{i}{\hbar} \Braket{\left[ \hat{H}, \hat{Q} \right]} + \Braket{ \frac{\partial \hat{Q}}{\partial t}}, \]

where \(\hat{H}\) is the Hamiltonian of the system.

8.5. Spin#

The operators for the \(x\), \(y\) and \(z\) component of the spin satisfy the commutation relations

(8.11)#\[\begin{split}\begin{align*} \left[ \hat{S}_x, \hat{S}_y \right] &= i \hbar \hat{S}_z, \\ \left[ \hat{S}_y, \hat{S}_z \right] &= i \hbar \hat{S}_x, \\ \left[ \hat{S}_z, \hat{S}_x \right] &= i \hbar \hat{S}_y. \end{align*}\end{split}\]

For a spin-1/2 particle in the basis of the eigenvectors \(\chi_{+} = \left(\begin{array}{c}1\\ 0 \end{array}\right)\) and \(\chi_{-} = \left(\begin{array}{c}0\\ 1 \end{array}\right)\) of the \(\hat{S}_z\) operator, the spin operators are given by the matrix expressions:

(8.12)#\[\begin{split}\begin{align*} \hat{S}_x &= \frac{\hbar}{2} \hat{\bm{\sigma}}_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \\ \hat{S}_y &= \frac{\hbar}{2} \hat{\bm{\sigma}}_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \\ \hat{S}_z &= \frac{\hbar}{2} \hat{\bm{\sigma}}_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \end{align*}\end{split}\]

8.6. Hydrogen wavefunctions#

8.6.1. Spherical harmonics#

The spherical harmonics, or Fourier modes on the sphere, are given by (equation (2.67)):

(8.13)#\[ Y_l^m(\theta, \phi) = \varepsilon \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^m(\cos\theta) e^{im\phi}, \]

where \(\varepsilon = 1\) if \(m \leq 0\) and \(\varepsilon = (-1)^m\) if \(m>0\). The \(P_l^m(x)\) are the associated Legendre functions, defined by (equation (2.66)):

(8.14)#\[ P_l^m(x) = (1-x^2)^{|m|/2} \left( \frac{\mathrm{d}}{\mathrm{d}x}\right)^{|m|} P_l(x), \]

for any integer \(m\) (but zero if \(|m| > l\)). The Legendre polynomials \(P_l(x)\) are most easily found using the Rodriguez formula, equation (2.65):

(8.15)#\[ P_l(x) = \frac{1}{2^l l!} \left( \frac{\mathrm{d}}{\mathrm{d}x}\right)^l (x^2-1)^l, \]

for any non-negative integer value of \(l\). The first six Legendre polynomials are:

(8.16)#\[\begin{split}\begin{align*} P_0(x) &= 1 \\ P_1(x) &= x \\ P_2(x) &= \frac12 \left(3 x^2 - 1\right) \\ P_3(x) &= \frac12 \left(5 x^3 - 3x\right) \\ P_4(x) &= \frac18 \left(35 x^4 - 30 x^2 + 3\right) \\ P_5(x) &= \frac18 \left(63 x^5 - 70 x^3 + 15x\right) \end{align*}\end{split}\]

../_images/ab8f14c428d67f83d7faddd3b89f8bbd96f47df55ded55476ecc76c061b1d1d4.svg — Fig. 8.1 The first six Legendre polynomials.#

The associated Legendre functions up to \(l=3\), expressed as a function of \(\cos(\theta)\), are:

(8.17)#\[\begin{split}\begin{align*} P_0^0(\cos(\theta)) &= 1 \\ P_1^0(\cos(\theta)) &= \cos(\theta) \\ P_1^1(\cos(\theta)) &= \sin(\theta) \\ P_2^0(\cos(\theta)) &= \frac12 \left(3\cos^2(\theta) - 1 \right) \\ P_2^1(\cos(\theta)) &= -3 \sin(\theta) \cos(\theta) \\ P_2^2(\cos(\theta)) &= 3 \sin^2(\theta) \\ P_3^0(\cos(\theta)) &= \frac12 \left(5\cos^3(\theta) - 3 \cos(\theta) \right) \\ P_3^1(\cos(\theta)) &= -\frac32 \sin(\theta) \left( 5\cos^2(\theta) - 1\right) \\ P_3^2(\cos(\theta)) &= 15 \sin^2(\theta) \cos(\theta) \\ P_3^3(\cos(\theta)) &= -15 \sin(\theta) \left(1 - \cos^2(\theta)\right) \end{align*}\end{split}\]

../_images/d16177be5bf63c36d313ea01e3bc7f4341a01fb61bb8204813ff14b59411bccf.svg — Fig. 8.2 The first six associated Legendre functions as functions of the angle \(\theta\).#

Substituting the associated Legendre functions into the expression (8.13) for the spherical harmonics, we find up to \(l=3\):

(8.18)#\[\begin{split}\begin{align*} Y_0^0(\theta, \phi) &= \left(\frac{1}{4\pi}\right)^{1/2} \\ Y_1^0(\theta, \phi) &= \left(\frac{3}{4\pi}\right)^{1/2} \cos(\theta) e^{\pm i \phi}\\ Y_1^{\pm 1}(\theta, \phi) &= \mp \left(\frac{3}{8\pi}\right)^{1/2} \sin(\theta) \\ Y_2^0(\theta, \phi) &= \left(\frac{5}{16\pi}\right)^{1/2} \left(3\cos^2(\theta) - 1 \right) \\ Y_2^{\pm 1}(\theta, \phi) &= \mp \left(\frac{15}{8\pi}\right)^{1/2} \sin(\theta) \cos(\theta) e^{\pm i \phi}\\ Y_2^{\pm 2}(\theta, \phi) &= \left(\frac{15}{32\pi}\right)^{1/2} \sin^2(\theta) e^{\pm 2i \phi}\\ Y_3^0(\theta, \phi) &= \left(\frac{7}{16\pi}\right)^{1/2} \left(5\cos^3(\theta) - 3 \cos(\theta) \right) \\ Y_3^{\pm 1}(\theta, \phi) &= \mp \left(\frac{21}{64\pi}\right)^{1/2} \sin(\theta) \left( 5\cos^2(\theta) - 1\right) e^{\pm i \phi}\\ Y_3^{\pm 2}(\theta, \phi) &= \left(\frac{105}{32\pi}\right)^{1/2} \sin^2(\theta) \cos(\theta) e^{\pm 2i \phi}\\ Y_3^{\pm 3}(\theta, \phi) &= \mp \left(\frac{35}{64\pi}\right)^{1/2} \sin(\theta) \left(1 - \cos^2(\theta)\right) e^{\pm 3i \phi} \end{align*}\end{split}\]

8.6.2. Radial part of the hydrogen wavefunction#

Specifically for the hydrogen atom, the radial part of the wavefunction is given in terms of the associated Laguerre polynomials, given by (equation (2.84)):

(8.19)#\[\begin{split}\begin{align*} L_q^p(x) &= (-1)^p \left(\frac{\mathrm{d}}{\mathrm{d}x}\right)^p L_{p+q}(x) = \frac{x^{-p} e^x}{q!} \left(\frac{\mathrm{d}}{\mathrm{d}x}\right)^q \left( e^{-x} x^{p+q} \right), \\ L_q(x) &= \frac{e^x}{q!} \left(\frac{\mathrm{d}}{\mathrm{d}x}\right)^q \left( e^{-x} x^q \right). \end{align*}\end{split}\]

The first six Laguerre polynomials are:

(8.20)#\[\begin{split}\begin{align*} L_0(x) &= 1 \\ L_1(x) &= -x + 1 \\ L_2(x) &= \frac12 \left(x^2 - 4x + 2 \right)\\ L_3(x) &= \frac16 \left(-x^3 + 9 x^2 - 18x + 6 \right)\\ L_4(x) &= \frac{1}{24} \left(x^4 - 16 x^3 + 72 x^2 - 96x + 24 \right)\\ L_5(x) &= \frac{1}{120} \left(-x^5 + 25 x^4 - 200 x^3 + 600 x^2 - 600 x + 120 \right) \end{align*}\end{split}\]

../_images/3fb3e6c76cd3be3e7dc262806196863e074e7582bbfae070bf2390c4d5f53454.svg — Fig. 8.3 The first six Laguerre polynomials.#

For the associated Laguerre polynomials up to \(q = 3\) we find:

(8.21)#\[\begin{split}\begin{align*} L_0^p(x) &= 1 \\ L_1^p(x) &= -x + p + 1 \\ L_2^p(x) &= \frac{1}{2} \left(p^2-2 p x+3 p+x^2-4 x+2\right)\\ L_3^p(x) &= \frac16 \left(p^3-3 p^2 x+6 p^2+3 p x^2-15 p x+11 p-x^3+9 x^2-18 x+6\right) \end{align*}\end{split}\]

../_images/d8c58ad047a7b7a43b9f12ec33a15e4444f614a52f1a73033860635fc4e76e6c.svg — Fig. 8.4 The first ten associated Laguerre polynomials.#

The (normalized) first ten radial wave functions for hydrogen are given by

(8.22)#\[\begin{split}\begin{align*} R_{10}(r) &= \frac{2}{\sqrt{a_0^3}} e^{-r/a_0}, \\ R_{20}(r) &= \frac{1}{\sqrt{2 a_0^3}} \left(1-\frac{r}{2 a_0} \right) e^{-r/2a_0}, \\ R_{21}(r) &= \frac{1}{\sqrt{6 a_0^3}} \left(\frac{r}{2 a_0} \right) e^{-r/2a_0}, \\ R_{30}(r) &= \frac{2}{3\sqrt{3 a_0^3}} \left(1 - \frac{2r}{3 a_0} + \frac23 \left(\frac{r}{3 a_0}\right)^2 \right) e^{-r/3a_0}, \\ R_{31}(r) &= \frac{8}{9\sqrt{6 a_0^3}} \left(1 - \frac12 \frac{r}{3 a_0} \right) \left( \frac{r}{3 a_0} \right) e^{-r/3a_0}, \\ R_{32}(r) &= \frac{4}{9\sqrt{30 a_0^3}} \left( \frac{r}{3 a_0} \right)^2 e^{-r/3a_0}, \\ R_{40}(r) &= \frac{1}{4\sqrt{3 a_0^3}} \left(1 - \frac{3r}{4a_0} + 2 \left(\frac{r}{4a_0}\right)^2 - \frac13 \left(\frac{r}{4a_0}\right)^3 \right) e^{-r/4a_0}, \\ R_{41}(r) &= \frac{5}{4\sqrt{15 a_0^3}} \left(1 - \frac{r}{4a_0} + \frac15 \left(\frac{r}{4a_0}\right)^2 \right) \left(\frac{r}{4a_0}\right) e^{-r/4a_0}, \\ R_{42}(r) &= \frac{1}{4\sqrt{5 a_0^3}} \left(1 - \frac13 \frac{r}{4a_0} \right) \left( \frac{r}{4a_0} \right)^2 e^{-r/4a_0}, \\ R_{43}(r) &= \frac{1}{12\sqrt{35 a_0^3}} \left( \frac{r}{4a_0} \right)^3 e^{-r/4a_0}. \end{align*}\end{split}\]

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%config InlineBackend.figure_formats = ['svg']
import numpy as np
import matplotlib.pyplot as plt
from myst_nb import glue

def R10(r):
    # Returns R_{10}(r), with r in units of the Bohr radius.
    return 2 * np.exp(-r)

def R20(r):
    # Returns R_{20}(r), with r in units of the Bohr radius.
    return (1 / np.sqrt(2)) * (1 - r/2) * np.exp(-r/2)

def R21(r):
    # Returns R_{21}(r), with r in units of the Bohr radius.
    return (1/np.sqrt(6)) * (r/2) * np.exp(-r/2)

def R30(r):
    # Returns R_{30}(r), with r in units of the Bohr radius.
    return (2 / np.sqrt(27)) * (1 - (2*r/3) + (2/3) * pow(r/3, 2)) * np.exp(-r/3)

def R31(r):
    # Returns R_{31}(r), with r in units of the Bohr radius.
    return (8/np.sqrt(486)) * (1- r/6) * (r/3) * np.exp(-r/3)

def R32(r):
    # Returns R_{32}(r), with r in units of the Bohr radius.
    return (4/np.sqrt(2430)) * pow(r/3,2) * np.exp(-r/3)

def R40(r):
    # Returns R_{40}(r), with r in units of the Bohr radius.
    return (0.25 / np.sqrt(3)) * (1 - (3*r/4) + 2 * pow(r/4, 2) - (1/3) * pow(r/4, 3)) * np.exp(-r/4)

def R41(r):
    # Returns R_{41}(r), with r in units of the Bohr radius.
    return (1.25/np.sqrt(15)) * (1- r/4 + 0.2 * pow(r/4, 2)) * (r/4) * np.exp(-r/4)

def R42(r):
    # Returns R_{42}(r), with r in units of the Bohr radius.
    return (0.25/np.sqrt(5)) * (1 - r/12) * pow(r/4,2) * np.exp(-r/4)

def R43(r):
    # Returns R_{43}(r), with r in units of the Bohr radius.
    return (1/(12*np.sqrt(35))) * pow(r/4, 3) * np.exp(-r/4)


r = np.linspace(0, 10, 1000)

fig, ax = plt.subplots(figsize=(6,4))

line10 = ax.plot(r, R10(r), label='$R_{10}(r)$')
line20 = ax.plot(r, R20(r), label='$R_{20}(r)$')
line21 = ax.plot(r, R21(r), label='$R_{21}(r)$')

line30 = ax.plot(r, R30(r), label='$R_{30}(r)$')
line31 = ax.plot(r, R31(r), label='$R_{31}(r)$')
line32 = ax.plot(r, R32(r), label='$R_{32}(r)$')

line40 = ax.plot(r, R40(r), label='$R_{40}(r)$')
line41 = ax.plot(r, R41(r), label='$R_{41}(r)$')
line42 = ax.plot(r, R42(r), label='$R_{42}(r)$')
line43 = ax.plot(r, R43(r), label='$R_{43}(r)$')

ax.axhline(y = 0, color = 'k', linestyle = ':')

ax.set_xlim(0,10)
ax.set_ylim(-0.15,0.4)
ax.set_xlabel('$r/a_0$')
ax.set_ylabel('$R_{nl}$')
ax.legend()
# Save graph to load in figure later (special Jupyter Book feature)
glue("hydrogenradialfunctions-appendix", fig, display=False)

../_images/3ac32501c3aae3e828e15795a1ab1059012d53538a2419d31f3da863fbb25437.svg — Fig. 8.5 Radial part of the first ten eigenfunctions of the hydrogen Hamiltonian.#

8.7. Mathematical identities#

8.7.1. Goniometric functions#

\[\begin{split}\begin{align*} e^{i\phi} &= \cos(\phi) + i \sin(\phi) \\ \sin(\phi) &= \frac{1}{2i} \left( e^{i \phi} - e^{-i \phi} \right) \\ \cos(\phi) &= \frac12 \left( e^{i \phi} + e^{-i \phi} \right) \\ 1 &= \sin^2(\phi) + \cos^2(\phi) \\ \sin(2 \phi) &= 2 \sin(\phi) \cos(\phi)\\ \cos(2\phi) &= \cos^2(\phi)-\sin^2(\phi) \\ \sin(a \pm b) &= \sin(a) \cos(b) \pm \sin(b) \cos(a) \\ \cos(a \pm b) &= \cos(a) \cos(b) \pm \sin(a) \sin(b) \end{align*}\end{split}\]

8.7.2. Some goniometric integrals#

The following hold for any integers \(m\) and \(n\).

\[\begin{split}\begin{align*} \int_0^L \sin\left(\frac{m \pi x}{L} \right) \sin\left(\frac{n \pi x}{L} \right) \,\mathrm{d}x &= \frac{L}{2} \delta_{mn}.\\ \int_0^L \cos\left(\frac{m \pi x}{L} \right) \cos\left(\frac{n \pi x}{L} \right) \,\mathrm{d}x &= \frac{L}{2} \delta_{mn}. \\ \int_0^L \sin\left(\frac{n \pi x}{L} \right) \cos\left(\frac{n \pi x}{L} \right) \,\mathrm{d}x &= 0. \\ \int_0^L \sin\left(\frac{m \pi x}{L} \right) \cos\left(\frac{n \pi x}{L} \right) \,\mathrm{d}x &= \frac{1-(-1)^{m+n}}{m^2-n^2} \frac{L m}{\pi}. \\ \int_0^L x \sin^2\left(\frac{n \pi x}{L} \right) \,\mathrm{d}x = \int_0^L x \cos^2\left(\frac{n \pi x}{L} \right) \,\mathrm{d}x &= \frac{L^2}{4}. \\ \int_0^L x \sin\left(\frac{\pi x}{L} \right) \sin\left(\frac{2 \pi x}{L} \right) \,\mathrm{d}x &= - \frac{8 L^2}{9 \pi^2}. \\ \int_0^L x \cos\left(\frac{\pi x}{L} \right) \cos\left(\frac{2 \pi x}{L} \right) \,\mathrm{d}x &= - \frac{10 L^2}{9 \pi^2}. \\ \int_0^L x \sin\left(\frac{n \pi x}{L} \right) \cos\left(\frac{n \pi x}{L} \right) \,\mathrm{d}x &= -\frac{L^2}{4 n \pi}. \end{align*}\end{split}\]

For any \(a \neq 0\), we have the indefinite integrals

\[\begin{split}\begin{align*} \int x \sin(ax) \,\mathrm{d}x &= \frac{1}{a^2} \sin(ax) - \frac{x}{a} \cos(ax), \\ \int x \cos(ax) \,\mathrm{d}x &= \frac{1}{a^2} \cos(ax) + \frac{x}{a} \sin(ax). \end{align*}\end{split}\]

8.7.3. Exponential integrals#

(8.23)#\[ \int_0^\infty x^n e^{-x/a} \,\mathrm{d}x = n!\, a^{n+1} \qquad \text{for any integer } n. \]

8.7.4. Some Gaussian integrals#

Gaussian integrals are integrals over the Gaussian distribution \(\exp(-x^2)\) (aka the normal distribution). In general, these cannot be evaluated in closed form, and we define the error function as the integral:

(8.24)#\[ \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-y^2} \,\mathrm{d}y. \]

There are however a number of Gaussian integrals that can be evaluated explicitly: those over all of \(\mathbb{R}\). Of course, as the Gaussian distribution is symmetric, we can also find the integrals from plus or minus infinity to the point of symmetry.

(8.25)#\[\begin{split}\begin{align*} \int_{-\infty}^\infty e^{-a x^2} \,\mathrm{d}x &= \sqrt{\frac{\pi}{a}}. \\ \int_{-\infty}^\infty e^{-a x^2 + b x + c} \,\mathrm{d}x &= \sqrt{\frac{\pi}{a}} \exp\left(\frac{b^2}{4a}+c\right). \end{align*}\end{split}\]

(8.26)#\[\begin{split}\begin{align*} \int_{-\infty}^\infty x^n e^{-a x^2} \,\mathrm{d}x &= 0 \qquad \text{for any odd value of } n. \\ \int_{-\infty}^\infty x^n e^{-a x^2} \,\mathrm{d}x &= \frac{1 \cdot 3 \cdot 5 \cdots (n-1) \sqrt{\pi}}{2^{n/2} a^{(n+1)/2}} \qquad \text{for any even value of } n. \end{align*}\end{split}\]

Note that we can get (8.26) by repeated differentiation of (8.25) with respect to \(a\).

8.7.5. Vector derivatives#

The expressions below are in Cartesian (\(x, y, z\)), cylindrical (\(\rho, \phi, z\)) and spherical (\(r, \theta, \phi\)) coordinates.

Gradient:

(8.27)#\[\begin{split}\begin{align*} \bm{\nabla} f(\bm{r}) &= \bm{\nabla} f(x, y, z) = \begin{pmatrix} \partial_x f \\ \partial_y f \\ \partial_z f \end{pmatrix} = \frac{\partial f}{\partial x} \bm{\hat{x}} + \frac{\partial f}{\partial y} \bm{\hat{y}} + \frac{\partial f}{\partial z} \bm{\hat{z}}, \\ &= \bm{\nabla} f(\rho, \phi, z) = \frac{\partial f}{\partial \rho} \bm{\hat{\rho}} + \frac{1}{\rho} \frac{\partial f}{\partial \phi} \bm{\hat{\phi}} + \frac{\partial f}{\partial z} \bm{\hat{z}}, \\ &= \bm{\nabla} f(r, \theta, \phi) = \frac{\partial f}{\partial r} \bm{\hat{r}} + \frac{1}{r} \frac{\partial f}{\partial \theta} \bm{\hat{\theta}} + \frac{1}{r \sin(\theta)} \frac{\partial f}{\partial \phi} \bm{\hat{\phi}}. \end{align*}\end{split}\]

Divergence:

(8.28)#\[\begin{split}\begin{align*} \bm{\nabla} \cdot \bm{v} &= \left( \partial_x, \partial_y, \partial_z \right) \cdot \begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}, \\ &= \frac{1}{\rho} \frac{\partial }{\partial \rho} \left( \rho v_\rho \right) + \frac{1}{\rho} \frac{\partial v_\phi}{\partial \phi} + \frac{\partial v_z}{\partial z}, \\ &= \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 v_r \right) + \frac{1}{r \sin(\theta)} \frac{\partial }{\partial \theta} \left(\sin(\theta) v_\theta\right) + \frac{1}{r \sin(\theta)} \frac{\partial v_\phi}{\partial \phi}. \end{align*}\end{split}\]

Curl:

(8.29)#\[\begin{split}\begin{align*} \bm{\nabla} \times \bm{A} &= \left( \partial_x, \partial_y, \partial_z \right) \times \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix} = \begin{pmatrix} \partial_y A_z - \partial_z A_y \\ \partial_z A_x - \partial_x A_z \\ \partial_x A_y - \partial_y A_x \end{pmatrix}, \\ &= \left( \frac{1}{\rho} \frac{\partial v_z}{\partial \phi} - \frac{\partial v_\phi}{\partial z} \right) \bm{\hat{\rho}} + \left( \frac{\partial v_\rho}{\partial z} - \frac{\partial v_z}{\partial \rho} \right) \bm{\hat{\phi}} + \frac{1}{\rho} \left( \frac{\partial }{\partial \rho} \left( \rho v_\phi \right) - \frac{\partial v_\rho}{\partial \phi} \right) \bm{\hat{z}}, \\ &= \frac{1}{r \sin(\theta)} \left[ \frac{\partial }{\partial \theta} \left(v_\phi \sin(\theta) \right) - \frac{\partial \phi}{\partial v_\theta} \right] \bm{\hat{r}} + \frac{1}{r} \left[ \frac{1}{\sin(\theta)} \frac{\partial v_r}{\partial \phi} - \frac{\partial }{\partial r}\left(r v_\phi\right) \right] \bm{\hat{\theta}} + \frac{1}{r} \left[ \frac{\partial }{\partial r}\left(r v_\phi\right) - \frac{\partial v_r}{\partial \theta} \right] \bm{\hat{\phi}} \end{align*}\end{split}\]

Laplacian:

(8.30)#\[\begin{split}\begin{align*} \nabla^2 f &= \bm{\nabla} \cdot \left(\bm{\nabla} f \right) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}, \\ &= \frac{1}{\rho} \frac{\partial }{\partial \rho} \left( \rho \frac{\partial f}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2}, \\ &= \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin(\theta)} \frac{\partial }{\partial \theta} \left(\sin(\theta) \frac{\partial f}{\partial \theta}\right) + \frac{1}{r^2 \sin^2(\theta)} \frac{\partial^2 f}{\partial \phi^2}. \end{align*}\end{split}\]