Math Reference

This annex provides formal definitions and historical context for the mathematical operators and functions used in the chapters.

A.1 Cartesian Coordinates ($x,y,z$)

In the familiar 3D grid, the operators are straightforward:

The Gradient ($\mathrm{\nabla }$)

Used for the momentum operator ($\mathbf{p}=-i\hslash \mathrm{\nabla }$):

$$\mathrm{\nabla }=\hat{\mathbf{x}}\frac{\partial }{\partial x}+\hat{\mathbf{y}}\frac{\partial }{\partial y}+\hat{\mathbf{z}}\frac{\partial }{\partial z}$$

The Laplacian (${\mathrm{\nabla }}^2$)

Used for kinetic energy in the Schrödinger equation:

$${\mathrm{\nabla }}^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}$$

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A.2 Spherical Coordinates ($r,\theta ,\varphi$)

In spherical systems, we define a point by its distance from the origin ($r$), the polar angle from the z-axis ($\theta$), and the azimuthal angle around the xy-plane ($\varphi$).

• $x=r\sin \theta \cos \varphi$
• $y=r\sin \theta \sin \varphi$
• $z=r\cos \theta$

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A.3 Derivation: From Cartesian to Spherical

To transform the Laplacian from $x,y,z$ to $r,\theta ,\varphi$, we use Metric Scale Factors ($h$). These factors act as "converters" between a small change in a coordinate (like $d\theta$) and the actual physical distance traveled in space.

1. Geometric Origin of the Scale Factors

Imagine moving a point by a tiny amount in each of the three directions. The total displacement $d\mathbf{s}$ is the sum of three arc lengths:

1. Radial Displacement ($dr$): Moving "outward" by $dr$ moves you exactly $dr$ units in space.

   $${\text{Arc length}}_r=1\cdot dr⟹{\mathbf{h}}_{\mathbf{r}}\mathbf{=}\mathbf{1}$$

2. Polar Displacement ($d\theta$): Rotating "down" by $d\theta$ at a distance $r$ traces an arc on a circle of radius $r$.

   $${\text{Arc length}}_{\theta }=r\cdot d\theta ⟹{\mathbf{h}}_{\theta }\mathbf{=}\mathbf{r}$$

3. Azimuthal Displacement ($d\varphi$): Rotating "around" the z-axis by $d\varphi$. However, your distance from the z-axis is not $r$; it is the "horizontal" projection $r\sin \theta$.

   $${\text{Arc length}}_{\varphi }=(r\sin \theta )\cdot d\varphi ⟹{\mathbf{h}}_{\varphi }\mathbf{=}\mathbf{r}\sin \theta$$

2. The Gradient

In any orthogonal coordinate system ($q_1,q_2,q_3$), the gradient is defined as:

$$\mathrm{\nabla }f=\frac{{\hat{\mathbf{q}}}_{\mathbf{1}}}{h_1}\frac{\partial f}{\partial q_1}+\frac{{\hat{\mathbf{q}}}_{\mathbf{2}}}{h_2}\frac{\partial f}{\partial q_2}+\frac{{\hat{\mathbf{q}}}_{\mathbf{3}}}{h_3}\frac{\partial f}{\partial q_3}$$

Substituting our $h$ factors:

$$\mathrm{\nabla }=\hat{\mathbf{r}}\frac{\partial }{\partial r}+\hat{\theta }\frac{1}{r}\frac{\partial }{\partial \theta }+\hat{\varphi }\frac{1}{r\sin \theta }\frac{\partial }{\partial \varphi }$$

3. The Laplacian

The Laplacian is the divergence of the gradient ($\mathrm{\nabla }\cdot \mathrm{\nabla }$). The general formula for a coordinate system with scale factors $h_1,h_2,h_3$ is:

$${\mathrm{\nabla }}^{2}f=\frac{1}{h_1{h}_2{h}_3}[\frac{\partial }{\partial q_1}\left(\frac{h_2{h}_3}{h_1}\frac{\partial f}{\partial q_1}\right)+\frac{\partial }{\partial q_2}\left(\frac{h_1{h}_3}{h_2}\frac{\partial f}{\partial q_2}\right)+\frac{\partial }{\partial q_3}\left(\frac{h_1{h}_2}{h_3}\frac{\partial f}{\partial q_3}\right)]$$

For spherical coordinates, the product $h_r{h}_{\theta }{h}_{\varphi }=r^2\sin \theta$.

• The $r$ term: $\frac{\partial }{\partial r}(r\cdot r\sin \theta \cdot \frac{1}{1}\frac{\partial f}{\partial r})=\sin \theta \frac{\partial }{\partial r}(r^2\frac{\partial f}{\partial r})$
• The $\theta$ term: $\frac{\partial }{\partial \theta }(1\cdot r\sin \theta \cdot \frac{1}{r}\frac{\partial f}{\partial \theta })=\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })$
• The $\varphi$ term: $\frac{\partial }{\partial \varphi }(1\cdot r\cdot \frac{1}{r\sin \theta }\frac{\partial f}{\partial \varphi })=\frac{1}{\sin \theta }\frac{{\partial }^{2}f}{\partial {\varphi }^2}$

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A.4 Associated Legendre Polynomials, $P_{\ell }^{m_{\ell }}(x)$

In Chapter 4, the solution to the $\theta$ equation involves Associated Legendre Polynomials, $P_{\ell }^{m_{\ell }}(\cos \theta )$.

Formal Definition

These polynomials can be expressed using Rodrigues' Formula:

$$P_{\ell }^{m_{\ell }}(x)=\frac{(-1{)}^{m_{\ell }}}{2^{\ell }\ell !}(1-x^2{)}^{m_{\ell }/2}\frac{d^{\ell +m_{\ell }}}{d{x}^{\ell +m_{\ell }}}(x^2-1{)}^{\ell }$$

Historical Context

These polynomials were not invented for quantum mechanics. They were developed by the French mathematician Adrien-Marie Legendre (1752–1833) in the late 18th century. Legendre was studying Celestial Mechanics—specifically, the gravitational potential of planets. Schrödinger adopted this math because atoms, like planets, are spherically symmetric systems.

Why use a 3D plot for 2D angles?

The angular part of the wavefunction, $Y(\theta ,\varphi )$, depends only on two angles. To visualize it in 3D:

1. Mapping: We set the radius $\rho$ in our plots equal to the absolute value: $\rho =|Y_{\ell }^{m_{\ell }}(\theta ,\varphi )|$.
2. Interpretation: The surface "bulge" in a specific direction indicates the probability of finding the electron there.
3. Independence: These shapes represent only the angular likelihood and are independent of the radial "shells" $R(r)$ discussed in Chapter 5.

Visualizing the Solutions in Desmos 3D

Visualize the solutions? 🤔

3D Atomic Orbital Visualizer — Interactive graph plotting angular probability density $|Y_{\ell }^{m_{\ell }}{|}^2$.

⚠️ Notation alert: In standard textbooks (and Chapter 4), $\theta$ is polar and $\varphi$ is azimuthal. In Desmos 3D these names are swapped: phi is polar, theta is azimuthal. The formulas below use the Desmos convention.

Subshell	$m_{\ell }$	Name	Formula (Desmos convention)
$s$ ($\ell =0$)	$0$	—	rho = 1
$p$ ($\ell =1$)	$0$	$p_z$	rho = |cos(phi)|
	$+1$	$p_x$	rho = |sin(phi)*cos(theta)|
	$-1$	$p_y$	rho = |sin(phi)*sin(theta)|
$d$ ($\ell =2$)	$0$	$d_{z^2}$	rho = 0.5 * |3*cos^2(phi) - 1|
	$+1$	$d_{xz}$	rho = |2*sin(phi)*cos(phi)*cos(theta)|
	$-1$	$d_{yz}$	rho = |2*sin(phi)*cos(phi)*sin(theta)|
	$+2$	$d_{x^2-y^2}$	rho = |sin^2(phi)*cos(2*theta)|
	$-2$	$d_{xy}$	rho = |sin^2(phi)*sin(2*theta)|
$f$ ($\ell =3$)	$0$	$f_{z^3}$	rho = 0.5 * |5*cos^3(phi) - 3*cos(phi)|
	$+1$	$f_{x{z}^2}$	rho = 0.73 * |sin(phi)*(5*cos^2(phi) - 1)*cos(theta)|
	$-1$	$f_{y{z}^2}$	rho = 0.73 * |sin(phi)*(5*cos^2(phi) - 1)*sin(theta)|
	$+2$	$f_{z(x^2-y^2)}$	rho = 2.6 * |sin^2(phi)*cos(phi)*cos(2*theta)|
	$-2$	$f_{zxy}$	rho = 2.6 * |sin^2(phi)*cos(phi)*sin(2*theta)|
	$+3$	$f_{x(x^2-3y^2)}$	rho = |sin^3(phi)*cos(3*theta)|
	$-3$	$f_{y(3x^2-y^2)}$	rho = |sin^3(phi)*sin(3*theta)|

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A.5 Associated Laguerre Polynomials, $L_q^p(x)$

In Chapter 5, the solution to the radial equation involves Associated Laguerre Polynomials, $L_{n-\ell -1}^{2\ell +1}(x)$.

Historical Context

These were developed by the French mathematician Edmond Laguerre (1834–1886). His polynomials are the only solutions that satisfy the radial Schrödinger equation while remaining finite at the origin ($r=0$) and decaying to zero at infinity ($r\to \mathrm{\infty }$).

Physical Interpretation: Onion Layers

The number of radial nodes is $n-\ell -1$. These are spherical shells where probability is zero. A $2s$ orbital ($1$ node) looks like a sphere inside another sphere.

Visualizing the Radial Solutions in Desmos 3D

Visualize the solutions? 🤔

3D Radial Node Visualizer — Plotting isosurfaces of $(R_{n\ell }{)}^2$ to reveal the "onion-layer" and radial node structure.

First define $r=\sqrt{x^2+y^2+z^2}$. These formulas use atomic units ($a_0=1$). Plot $(R_{n\ell }{)}^2=c$ (try $c=0.01$).

Shell	Orbital	Radial function
K	$1s$	R_10 = 2 * exp(-r)
L	$2s$	R_20 = (sqrt(2)/4) * (2 - r) * exp(-r/2)
	$2p$	R_21 = (sqrt(6)/12) * r * exp(-r/2)
M	$3s$	R_30 = (2*sqrt(3)/27) * (3 - 2*r) * exp(-r/3)
	$3p$	R_31 = (2*sqrt(6)/243) * r * (6 - r) * exp(-r/3)
	$3d$	R_32 = (2*sqrt(30)/1215) * r^2 * exp(-r/3)
N	$4s$	R_40 = (1/32) * (r^2 - 6*r + 8) * exp(-r/4)
	$4p$	R_41 = (sqrt(15)/192) * r * (4 - r) * exp(-r/4)
	$4d$	R_42 = (sqrt(5)/3840) * r^2 * (12 - r) * exp(-r/4)
	$4f$	R_43 = (sqrt(35)/26880) * r^3 * exp(-r/4)
O	$5s$	R_50 = (2*sqrt(5)/625) * (4*r^2 - 20*r + 25) * exp(-r/5)
	$5p$	R_51 = (2*sqrt(30)/46875) * r * (6*r^2 - 75*r + 250) * exp(-r/5)
	$5d$	R_52 = (2*sqrt(70)/46875) * r^2 * (15 - 2*r) * exp(-r/5)
	$5f$	R_53 = (2*sqrt(70)/1640625) * r^3 * (20 - r) * exp(-r/5)
	$5g$	R_54 = (1/(15000*sqrt(70))) * (2r/5)^4 * exp(-r/5) (never occupied in ground state)
P	$6s$	R_60 = (sqrt(6)/2916) * (-2*r^3 + 30*r^2 - 135*r + 162) * exp(-r/6)
	$6p$	R_61 = (sqrt(210)/306180) * r * (-r^3 + 30*r^2 - 315*r + 945) * exp(-r/6)
	$6d$	R_62 = (sqrt(105)/306180) * r^2 * (r^2 - 21*r + 126) * exp(-r/6)
Q	$7s$	R_70 = (2*sqrt(7)/117649) * (-40*r^3 + 476*r^2 - 2058*r + 2401) * exp(-r/7)
	$7p$	R_71 = (4*sqrt(21)/2470629) * r * (-8*r^3 + 196*r^2 - 1715*r + 4802) * exp(-r/7)

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A.6 Summary: The Four Quantum Numbers

Number	Symbol	Allowed Values	Physical Meaning
Principal	$n$	$1,2,3,\dots$	Shell number, size and energy of the orbital.
Azimuthal	$\ell$	$0,1,\dots ,n-1$	Subshell shape ($s,p,d,f$). Orbital angular momentum.
Magnetic	$m_{\ell }$	$-\ell ,\dots ,+\ell$	Orbital orientation in space.
Spin	$m_s$	$+\frac{1}{2},-\frac{1}{2}$	Intrinsic angular momentum (Spin-Up or Spin-Down).