Harper Funeral Home Obituary, Ci Includes Only Offensive Activities, Most Fun Titan Quest Builds, The Last House On Needless Street Wiki, Willow Bark Skincare Pregnancy, Articles A

If measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the cos and sin below become switched. Lets see how we can normalize orbitals using triple integrals in spherical coordinates. (8.5) in Boas' Sec. If the radius is zero, both azimuth and inclination are arbitrary. $$ 16.4: Spherical Coordinates - Chemistry LibreTexts This statement is true regardless of whether the function is expressed in polar or cartesian coordinates. Latitude is either geocentric latitude, measured at the Earth's center and designated variously by , q, , c, g or geodetic latitude, measured by the observer's local vertical, and commonly designated . {\displaystyle (r,\theta ,\varphi )} Surface integrals of scalar fields. , Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (radius r, inclination , azimuth ), where r [0, ), [0, ], [0, 2), by, Cylindrical coordinates (axial radius , azimuth , elevation z) may be converted into spherical coordinates (central radius r, inclination , azimuth ), by the formulas, Conversely, the spherical coordinates may be converted into cylindrical coordinates by the formulae. We will see that \(p\) and \(d\) orbitals depend on the angles as well. Using the same arguments we used for polar coordinates in the plane, we will see that the differential of volume in spherical coordinates is not \(dV=dr\,d\theta\,d\phi\). The answer is no, because the volume element in spherical coordinates depends also on the actual position of the point. 4.4: Spherical Coordinates - Engineering LibreTexts Why do academics stay as adjuncts for years rather than move around? How to use Slater Type Orbitals as a basis functions in matrix method correctly? for any r, , and . Coming back to coordinates in two dimensions, it is intuitive to understand why the area element in cartesian coordinates is \(dA=dx\;dy\) independently of the values of \(x\) and \(y\). Two important partial differential equations that arise in many physical problems, Laplace's equation and the Helmholtz equation, allow a separation of variables in spherical coordinates. But what if we had to integrate a function that is expressed in spherical coordinates? $$. Because \(dr<<0\), we can neglect the term \((dr)^2\), and \(dA= r\; dr\;d\theta\) (see Figure \(10.2.3\)). Find \(A\). Both versions of the double integral are equivalent, and both can be solved to find the value of the normalization constant (\(A\)) that makes the double integral equal to 1. When you have a parametric representatuion of a surface $$y=r\sin(\phi)\sin(\theta)$$ Students who constructed volume elements from differential length components corrected their length element terms as a result of checking the volume element . $$ The volume element spanning from r to r + dr, to + d, and to + d is specified by the determinant of the Jacobian matrix of partial derivatives, Thus, for example, a function f(r, , ) can be integrated over every point in R3 by the triple integral. In cartesian coordinates the differential area element is simply \(dA=dx\;dy\) (Figure \(\PageIndex{1}\)), and the volume element is simply \(dV=dx\;dy\;dz\). ) Because of the probabilistic interpretation of wave functions, we determine this constant by normalization. Assume that f is a scalar, vector, or tensor field defined on a surface S.To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere.Let such a parameterization be r(s, t), where (s, t) varies in some region T in the plane. Why is this sentence from The Great Gatsby grammatical? ) , A bit of googling and I found this one for you! Chapter 1: Curvilinear Coordinates | Physics - University of Guelph Where $\color{blue}{\sin{\frac{\pi}{2}} = 1}$, i.e. The small volume is nearly box shaped, with 4 flat sides and two sides formed from bits of concentric spheres. Use the volume element and the given charge density to calculate the total charge of the sphere (triple integral). Calculating Infinitesimal Distance in Cylindrical and Spherical Coordinates Calculating \(d\rr\)in Curvilinear Coordinates Scalar Surface Elements Triple Integrals in Cylindrical and Spherical Coordinates Using \(d\rr\)on More General Paths Use What You Know 9Integration Scalar Line Integrals Vector Line Integrals Spherical coordinates, Finding the volume bounded by surface in spherical coordinates, Angular velocity in Fick Spherical coordinates, The surface temperature of the earth in spherical coordinates. As we saw in the case of the particle in the box (Section 5.4), the solution of the Schrdinger equation has an arbitrary multiplicative constant. gives the radial distance, polar angle, and azimuthal angle. Spherical coordinates are useful in analyzing systems that are symmetrical about a point. 12.7: Cylindrical and Spherical Coordinates - Mathematics LibreTexts This statement is true regardless of whether the function is expressed in polar or cartesian coordinates. ) The spherical coordinate system is also commonly used in 3D game development to rotate the camera around the player's position[4]. The square-root factor comes from the property of the determinant that allows a constant to be pulled out from a column: The following equations (Iyanaga 1977) assume that the colatitude is the inclination from the z (polar) axis (ambiguous since x, y, and z are mutually normal), as in the physics convention discussed. Apply the Shell theorem (part a) to treat the sphere as a point particle located at the origin & find the electric field due to this point particle. ) It can also be extended to higher-dimensional spaces and is then referred to as a hyperspherical coordinate system. Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose. The symbol ( rho) is often used instead of r. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. The differential surface area elements can be derived by selecting a surface of constant coordinate {Fan in Cartesian coordinates for example} and then varying the other two coordinates to tIace out a small . $$ The vector product $\times$ is the appropriate surrogate of that in the present circumstances, but in the simple case of a sphere it is pretty obvious that ${\rm d}\omega=r^2\sin\theta\,{\rm d}(\theta,\phi)$. For positions on the Earth or other solid celestial body, the reference plane is usually taken to be the plane perpendicular to the axis of rotation. When using spherical coordinates, it is important that you see how these two angles are defined so you can identify which is which. E & F \\ Recall that this is the metric tensor, whose components are obtained by taking the inner product of two tangent vectors on your space, i.e. In polar coordinates: \[\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi} A^2 e^{-2ar^2}r\;d\theta dr=A^2\int\limits_{0}^{\infty}e^{-2ar^2}r\;dr\int\limits_{0}^{2\pi}\;d\theta =A^2\times\dfrac{1}{4a}\times2\pi=1 \nonumber\]. {\displaystyle (r,\theta ,\varphi )} 10.2: Area and Volume Elements - Chemistry LibreTexts r Vectors are often denoted in bold face (e.g. The distance on the surface of our sphere between North to South poles is $r \, \pi$ (half the circumference of a circle). If the inclination is zero or 180 degrees ( radians), the azimuth is arbitrary. [2] The polar angle is often replaced by the elevation angle measured from the reference plane towards the positive Z axis, so that the elevation angle of zero is at the horizon; the depression angle is the negative of the elevation angle. , r) without the arrow on top, so be careful not to confuse it with \(r\), which is a scalar. 1. 3. d dxdy dydz dzdx = = = az x y ddldl r dd2 sin ar r== Just as the two-dimensional Cartesian coordinate system is useful on the plane, a two-dimensional spherical coordinate system is useful on the surface of a sphere. The lowest energy state, which in chemistry we call the 1s orbital, turns out to be: This particular orbital depends on \(r\) only, which should not surprise a chemist given that the electron density in all \(s\)-orbitals is spherically symmetric. ( The relationship between the cartesian coordinates and the spherical coordinates can be summarized as: \[\label{eq:coordinates_5} x=r\sin\theta\cos\phi\], \[\label{eq:coordinates_6} y=r\sin\theta\sin\phi\], \[\label{eq:coordinates_7} z=r\cos\theta\]. so that our tangent vectors are simply Some combinations of these choices result in a left-handed coordinate system. In cartesian coordinates, all space means \(-\inftyPhysics Ch 67.1 Advanced E&M: Review Vectors (76 of 113) Area Element There are a number of celestial coordinate systems based on different fundamental planes and with different terms for the various coordinates. Regardless of the orbital, and the coordinate system, the normalization condition states that: \[\int\limits_{all\;space} |\psi|^2\;dV=1 \nonumber\]. But what if we had to integrate a function that is expressed in spherical coordinates? In this case, \(\psi^2(r,\theta,\phi)=A^2e^{-2r/a_0}\). 180 The spherical coordinate system generalizes the two-dimensional polar coordinate system. Understand how to normalize orbitals expressed in spherical coordinates, and perform calculations involving triple integrals. ), geometric operations to represent elements in different Explain math questions One plus one is two. (b) Note that every point on the sphere is uniquely determined by its z-coordinate and its counterclockwise angle phi, $0 \leq\phi\leq 2\pi$, from the half-plane y = 0, r X_{\phi} = (-r\sin(\phi)\sin(\theta),r\cos(\phi)\sin(\theta),0), \\ Spherical coordinates are the natural coordinates for physical situations where there is spherical symmetry (e.g. Coming back to coordinates in two dimensions, it is intuitive to understand why the area element in cartesian coordinates is \(dA=dx\;dy\) independently of the values of \(x\) and \(y\). ( $$. $$\int_{0}^{ \pi }\int_{0}^{2 \pi } r \, d\theta * r \, d \phi = 2 \pi^2 r^2$$. To a first approximation, the geographic coordinate system uses elevation angle (latitude) in degrees north of the equator plane, in the range 90 90, instead of inclination. $$ This simplification can also be very useful when dealing with objects such as rotational matrices. [Solved] . a} Cylindrical coordinates: i. Surface of constant Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid. One can add or subtract any number of full turns to either angular measure without changing the angles themselves, and therefore without changing the point. 4: Thus, we have In spherical coordinates, all space means \(0\leq r\leq \infty\), \(0\leq \phi\leq 2\pi\) and \(0\leq \theta\leq \pi\). (g_{i j}) = \left(\begin{array}{cc} Learn more about Stack Overflow the company, and our products. 6. PDF V9. Surface Integrals - Massachusetts Institute of Technology Spherical coordinates (r, , ) as commonly used in physics ( ISO 80000-2:2019 convention): radial distance r (distance to origin), polar angle ( theta) (angle with respect to polar axis), and azimuthal angle ( phi) (angle of rotation from the initial meridian plane). We see that the latitude component has the $\color{blue}{\sin{\theta}}$ adjustment to it. $$h_1=r\sin(\theta),h_2=r$$ 167-168). Notice that the area highlighted in gray increases as we move away from the origin. {\displaystyle (-r,\theta {+}180^{\circ },-\varphi )} If you are given a "surface density ${\bf x}\mapsto \rho({\bf x})$ $\ ({\bf x}\in S)$ then the integral $I(S)$ of this density over $S$ is then given by Legal. Use your result to find for spherical coordinates, the scale factors, the vector ds, the volume element, the basis vectors a r, a , a and the corresponding unit basis vectors e r, e , e . where we do not need to adjust the latitude component. ( The differential of area is \(dA=r\;drd\theta\). The lowest energy state, which in chemistry we call the 1s orbital, turns out to be: This particular orbital depends on \(r\) only, which should not surprise a chemist given that the electron density in all \(s\)-orbitals is spherically symmetric. You can try having a look here, perhaps you'll find something useful: Yea I saw that too, I'm just wondering if there's some other way similar to using Jacobian (if someday I'm asked to find it in a self-invented set of coordinates where I can't picture it). ) Spherical coordinates are somewhat more difficult to understand. ) , For a wave function expressed in cartesian coordinates, \[\int\limits_{all\;space} |\psi|^2\;dV=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\psi^*(x,y,z)\psi(x,y,z)\,dxdydz \nonumber\]. r The geometrical derivation of the volume is a little bit more complicated, but from Figure \(\PageIndex{4}\) you should be able to see that \(dV\) depends on \(r\) and \(\theta\), but not on \(\phi\). This can be very confusing, so you will have to be careful. The value of should be greater than or equal to 0, i.e., 0. is used to describe the location of P. Let Q be the projection of point P on the xy plane. We know that the quantity \(|\psi|^2\) represents a probability density, and as such, needs to be normalized: \[\int\limits_{all\;space} |\psi|^2\;dA=1 \nonumber\]. We also mentioned that spherical coordinates are the obvious choice when writing this and other equations for systems such as atoms, which are symmetric around a point. Spherical coordinates (continued) In Cartesian coordinates, an infinitesimal area element on a plane containing point P is In spherical coordinates, the infinitesimal area element on a sphere through point P is x y z r da , or , or . In the cylindrical coordinate system, the location of a point in space is described using two distances (r and z) and an angle measure (). Phys. Rev. Phys. Educ. Res. 15, 010112 (2019) - Physics students or The blue vertical line is longitude 0. For example a sphere that has the cartesian equation x 2 + y 2 + z 2 = R 2 has the very simple equation r = R in spherical coordinates. Can I tell police to wait and call a lawyer when served with a search warrant? The spherical coordinates of a point in the ISO convention (i.e. This convention is used, in particular, for geographical coordinates, where the "zenith" direction is north and positive azimuth (longitude) angles are measured eastwards from some prime meridian. PDF Week 7: Integration: Special Coordinates - Warwick Jacobian determinant when I'm varying all 3 variables). 6. Find \( d s^{2} \) in spherical coordinates by the | Chegg.com Find d s 2 in spherical coordinates by the method used to obtain Eq. Would we just replace \(dx\;dy\;dz\) by \(dr\; d\theta\; d\phi\)? Equivalently, it is 90 degrees (.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}/2 radians) minus the inclination angle. Total area will be $$r \, \pi \times r \, 2\pi = 2 \pi^2 \, r^2$$, Like this I am trying to find out the area element of a sphere given by the equation: r 2 = x 2 + y 2 + z 2 The sphere is centered around the origin of the Cartesian basis vectors ( e x, e y, e z). To make the coordinates unique, one can use the convention that in these cases the arbitrary coordinates are zero. ( ( The del operator in this system leads to the following expressions for the gradient, divergence, curl and (scalar) Laplacian, Further, the inverse Jacobian in Cartesian coordinates is, In spherical coordinates, given two points with being the azimuthal coordinate, The distance between the two points can be expressed as, In spherical coordinates, the position of a point or particle (although better written as a triple ( Lets see how this affects a double integral with an example from quantum mechanics. conflicts with the usual notation for two-dimensional polar coordinates and three-dimensional cylindrical coordinates, where is often used for the azimuth.[3]. A common choice is. I'm able to derive through scale factors, ie $\delta(s)^2=h_1^2\delta(\theta)^2+h_2^2\delta(\phi)^2$ (note $\delta(r)=0$), that: In geography, the latitude is the elevation. Notice that the area highlighted in gray increases as we move away from the origin. :URn{\displaystyle \varphi :U\to \mathbb {R} ^{n}} Perhaps this is what you were looking for ? . {\displaystyle (r,\theta ,\varphi )} The volume of the shaded region is, \[\label{eq:dv} dV=r^2\sin\theta\,d\theta\,d\phi\,dr\]. In this case, \(n=2\) and \(a=2/a_0\), so: \[\int\limits_{0}^{\infty}e^{-2r/a_0}\,r^2\;dr=\dfrac{2! (25.4.6) y = r sin sin . The same situation arises in three dimensions when we solve the Schrdinger equation to obtain the expressions that describe the possible states of the electron in the hydrogen atom (i.e. Notice the difference between \(\vec{r}\), a vector, and \(r\), the distance to the origin (and therefore the modulus of the vector). The best answers are voted up and rise to the top, Not the answer you're looking for? See the article on atan2. , The spherical coordinates of a point P are then defined as follows: The sign of the azimuth is determined by choosing what is a positive sense of turning about the zenith. {\displaystyle (r,\theta ,\varphi )} In two dimensions, the polar coordinate system defines a point in the plane by two numbers: the distance \(r\) to the origin, and the angle \(\theta\) that the position vector forms with the \(x\)-axis. It can be seen as the three-dimensional version of the polar coordinate system. The result is a product of three integrals in one variable: \[\int\limits_{0}^{2\pi}d\phi=2\pi \nonumber\], \[\int\limits_{0}^{\pi}\sin\theta \;d\theta=-\cos\theta|_{0}^{\pi}=2 \nonumber\], \[\int\limits_{0}^{\infty}e^{-2r/a_0}\,r^2\;dr=? We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. In any coordinate system it is useful to define a differential area and a differential volume element. x >= 0. These formulae assume that the two systems have the same origin and same reference plane, measure the azimuth angle in the same senses from the same axis, and that the spherical angle is inclination from the cylindrical z axis. The precise standard meanings of latitude, longitude and altitude are currently defined by the World Geodetic System (WGS), and take into account the flattening of the Earth at the poles (about 21km or 13 miles) and many other details. We will see that \(p\) and \(d\) orbitals depend on the angles as well. In polar coordinates: \[\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi} A^2 e^{-2ar^2}r\;d\theta dr=A^2\int\limits_{0}^{\infty}e^{-2ar^2}r\;dr\int\limits_{0}^{2\pi}\;d\theta =A^2\times\dfrac{1}{4a}\times2\pi=1 \nonumber\]. , Would we just replace \(dx\;dy\;dz\) by \(dr\; d\theta\; d\phi\)? Coordinate systems - Wikiversity The differential of area is \(dA=dxdy\): \[\int\limits_{all\;space} |\psi|^2\;dA=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty} A^2e^{-2a(x^2+y^2)}\;dxdy=1 \nonumber\], In polar coordinates, all space means \(0Calculating Infinitesimal Distance in Cylindrical and Spherical Coordinates The Cartesian partial derivatives in spherical coordinates are therefore (Gasiorowicz 1974, pp. 10: Plane Polar and Spherical Coordinates, Mathematical Methods in Chemistry (Levitus), { "10.01:_Coordinate_Systems" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10.02:_Area_and_Volume_Elements" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10.03:_A_Refresher_on_Electronic_Quantum_Numbers" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10.04:_A_Brief_Introduction_to_Probability" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10.05:_Problems" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Before_We_Begin" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Complex_Numbers" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_First_Order_Ordinary_Differential_Equations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Second_Order_Ordinary_Differential_Equations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Power_Series_Solutions_of_Differential_Equations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Fourier_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Calculus_in_More_than_One_Variable" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Exact_and_Inexact_Differentials" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Plane_Polar_and_Spherical_Coordinates" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_Operators" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_Partial_Differential_Equations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_Determinants" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "14:_Vectors" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15:_Matrices" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "16:_Formula_Sheets" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "showtoc:no", "authorname:mlevitus", "differential volume element", "differential area element", "license:ccbyncsa", "licenseversion:40", "source@https://www.public.asu.edu/~mlevitus/chm240/book.pdf" ], https://chem.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FPhysical_and_Theoretical_Chemistry_Textbook_Maps%2FMathematical_Methods_in_Chemistry_(Levitus)%2F10%253A_Plane_Polar_and_Spherical_Coordinates%2F10.02%253A_Area_and_Volume_Elements, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), 10.3: A Refresher on Electronic Quantum Numbers, source@https://www.public.asu.edu/~mlevitus/chm240/book.pdf, status page at https://status.libretexts.org.