Quantum Numbers, Wave Functions, and Hydrogen Atomic Orbitals

Course Information and Announcements

• Administrative Updates:

  • An Exam Information Module link is located at the bottom of the HOME page.

  • Online learning resources, specifically Visualizing Atomic Orbitals, have been added to Canvas.

  • Online simulation tools cited for orbital exploration:

  • Hydrogen Atom Orbital Viewer

  • The Orbitron: 4f atomic orbitals radial distribution function

• Lecture Overview:

  • Date: September 14, 2026 (Lecture #7)

  • Key Topics:

  • Quantum numbers practice problems and relationships.

  • Section 8-7: Interpreting and Representing the Orbitals of the Hydrogen Atom.

  • Section 8-8: Electron Spin: The Fourth Quantum Number.

Fundamental Rules of Quantum Numbers and Shells

• Principal Shells ($n$):

  • The principal quantum number $n$ specifies the main energy shell and overall size of the orbital.

  • Allowed values: n = 1, 2, 3, 4, \r\ndots

• Subshells ($\\ell$):

  • The angular momentum quantum number $\ell$ defines the shape of the subshell.

  • Allowed values for a given $n$: $\ell = 0, 1, 2, \dots, (n - 1)$

  • Subshell letter designations:

  • $\ell = 0$: $s$ subshell

  • $\ell = 1$: $p$ subshell

  • $\ell = 2$: $d$ subshell

  • $\ell = 3$: $f$ subshell

• Orbitals per Subshell ($m_\ell$):

  • The magnetic quantum number $m_\ell$ specifies the spatial orientation of an orbital.

  • Allowed values for a given $\ell$: $m_\ell = -\ell, \dots, 0, \dots, +\ell$

  • Each subshell contains $(2\ell + 1)$ orbitals:

  • $s$ subshell ($\ell = 0$): 1 orbital ($m_\ell = 0$

  • $p$ subshell ($\ell = 1$): 3 orbitals ($m_\ell = 0, \pm 1$

  • $d$ subshell ($\ell = 2$): 5 orbitals ($m_\ell = 0, \pm 1, \pm 2$

  • $f$ subshell ($\ell = 3$): 7 orbitals ($m_\ell = 0, \pm 1, \pm 2, \pm 3$

• Electron Spin ($m_s$):

  • The electron spin quantum number $m_s$ describes the intrinsic spin orientation of the electron.

  • Allowed values: $m_s = +\frac{1}{2}$ or $m_s = -\frac{1}{2}$

Worked Examples: Quantum Numbers and Orbital Designations

• EXAMPLE 8-10: Applying Relationships Among Quantum Numbers

  • Question 1: Can an orbital have the quantum numbers $n = 2$, $\ell = 2$, and $m_\ell = 2$?

  • Answer: No.

  • Explanation: For $n = 2$, the maximum allowed value of $\ell$ is $n - 1 = 1$. Thus, $\ell$ can only be $0$ or $1$. A value of $\ell = 2$ is invalid.

  • Question 2: Can an orbital have the quantum numbers $n = 3$, $\ell = 0$, and $m_\ell = 0$?

  • Answer: Yes.

  • Explanation: For $n = 3$, allowed $\ell$ values are $0, 1, 2$. For $\ell = 0$, $m_\ell$ must be $0$. This set represents a valid $3s$ orbital.

  • Question 3: For an orbital with $n = 3$ and $m_\ell = 1$, what is (are) the possible value(s) of $\ell$?

  • Answer: $\ell = 1$ or $\ell = 2$

  • Explanation: For $n = 3$, $\ell$ can be $0, 1, 2$. Because $m_\ell = 1$, $\ell$ must be at least $1$ (since $|m_\ell| \le \ell$). Therefore, $\ell$ can be $1$ (a $3p$ orbital) or $2$ (a $3d$ orbital).

• EXAMPLE 8-11: Relating Orbital Designations and Quantum Numbers

  • Question: Write an orbital designation corresponding to the quantum numbers $n = 4$, $\ell = 2$, $m_\ell = 0$.

  • Answer: $4d$

  • Explanation: $n = 4$ specifies the 4th shell, and $\ell = 2$ specifies a $d$ subshell. Combined, the orbital designation is $4d$.

• EXAMPLE 8-12: Choosing an Appropriate Combination of Quantum Numbers

  • Task: Identify the single correct set of quantum numbers $(n, \ell, m_\ell, m_s)$ from the list and state its orbital designation:

  • $(2, 1, 0, 0)$: Incorrect ($m_s$ cannot be $0$).

  • $(2, 0, 1, \frac{1}{2})$: Incorrect (if $\ell = 0$, $m_\ell$ must be $0$).

  • $(2, 2, 0, \frac{1}{2})$: Incorrect (if $n = 2$, $\ell$ cannot be $2$).

  • $(2, -1, 0, \frac{1}{2})$: Incorrect ($\ell$ cannot be negative).

  • $(2, 1, 0, -\frac{1}{2})$: Correct! Represents a $2p$ orbital.

• PRACTICE EXAMPLE A: Evaluating Invalid Quantum Number Sets

  • $(3, 2, -2, 1)$: Wrong ($m_s$ must be $\pm \frac{1}{2}$, not $1$).

  • $(3, 1, -2, \frac{1}{2})$: Wrong (for $\ell = 1$, $m_\ell$ must be $-1, 0, +1$; $-2$ is invalid).

  • $(3, 0, 0, \frac{1}{2})$: Valid ($3s$ orbital).

  • $(2, 3, 0, \frac{1}{2})$: Wrong (if $n = 2$, $\ell$ cannot be $3$).

  • $(1, 0, 0, -\frac{1}{2})$: Valid ($1s$ orbital).

  • $(2, -1, -1, 1)$: Wrong ($\ell$ cannot be $-1$ and $m_s$ cannot be $1$).

• PRACTICE EXAMPLE B: Identifying Specific Errors in Quantum Sets

  • $(2, 1, 1, 0)$: Error is $m_s = 0$ (must be $\pm \frac{1}{2}$).

  • $(1, 1, 0, \frac{1}{2})$: Error is $\ell = 1$ when $n = 1$ (max $\ell = 0$).

  • $(3, -1, 1, -\frac{1}{2})$: Error is $\ell = -1$ (angular quantum number must be $\ge 0$).

  • $(0, 0, 0, -\frac{1}{2})$: Error is $n = 0$ (principal quantum number $n$ must be an integer $\ge 1$).

  • $(2, 1, 2, \frac{1}{2})$: Error is $m_\ell = 2$ when $\ell = 1$ (max $m_\ell = 1$).

End-of-Chapter Problems on Quantum Numbers

• Problem 58: Acceptable Values for Missing Quantum Numbers

  • a. $n = 3$, $\ell = ?$, $m_\ell = 2$, $m_s = +\frac{1}{2}$

  • Solution: $\ell = 2$. (Since $m_\ell = 2$, $\ell$ must be at least $2$; for $n = 3$, max $\ell = 2$).

  • b. $n = ?$, $\ell = 2$, $m_\ell = 1$, $m_s = -\frac{1}{2}$

  • Solution: $n \ge 3$ (any integer $3, 4, 5, \dots$).

  • c. $n = 4$, $\ell = 2$, $m_\ell = 0$, $m_s = ?$

  • Solution: $m_s = +\frac{1}{2}$ or $m_s = -\frac{1}{2}$.

  • d. $n = ?$, $\ell = 0$, $m_\ell = ?$, $m_s = ?$

  • Solution: $n = 1, 2, 3, \dots$; $m_\ell = 0$; $m_s = \pm \frac{1}{2}$.

• Problem 60: Evaluation of Statements for $n = 4$ and $m_\ell = 2$

  • Statement a: "The electron is in the fourth principal shell."

  • Correct. $n = 4$ directly defines the 4th principal shell.

  • Statement b: "The electron may be in a d orbital."

  • Correct. For $n = 4$, $\ell$ can equal $2$ ($d$ orbital), which accommodates $m_\ell = 2$

  • Statement c: "The electron may be in a p orbital."

  • Incorrect. A $p$ orbital has $\ell = 1$, which limits $m_\ell$ to $-1, 0, +1$. It cannot have $m_\ell = 2$

  • Statement d: "The electron must have $m_s = +\frac{1}{2}$."

  • Incorrect. The spin quantum number $m_s$ can be either $+\frac{1}{2}$ or $-\frac{1}{2}$.

Wave Functions of Hydrogen-Like Atoms

• Mathematical Definition of Atomic Orbitals:

  • Orbitals are wave functions ($\psi$) that serve as mathematical solutions to the Schrödinger wave equation.

  • For a single-electron atom or ion ($H, He^+, Li^{2+}$), the total wave function in spherical polar coordinates is factorized into a radial component and an angular component:     $\psi(r, \theta, \phi) = R(r) Y(\theta, \phi)$

• Components of the Wave Function:

  • $R(r)$: Radial Wave Function. Depends only on distance $r$ from the nucleus. Determines how electron probability density varies with radial distance and provides information on orbital size.

  • $Y(\theta, \phi)$: Angular Wave Function. Depends on angles $\theta$ and $\phi$. Independent of the principal quantum number $n$; all orbitals of a given subshell type ($s, p, d, f$) share identical angular behavior and shape regardless of shell level.

• Key Parameters and Constants:

  • $Z$: Atomic number (number of protons in the nucleus).

  • $a_0$: Bohr radius ($a_0 = 52.92\,\text{pm}$), corresponding to the radius of the lowest-energy orbit in the Bohr model.

  • Scaled distance parameter:     $\sigma = \frac{2 Z r}{n a_0}$

• Explicit Mathematical Functions (Table 8.2):

  • Angular Parts $Y(\theta, \phi)$:

  • $Y(s) = \left(\frac{1}{4\pi}\right)^{1/2}$

  • $Y(p_x) = \left(\frac{3}{4\pi}\right)^{1/2} \frac{x}{r} = \left(\frac{3}{4\pi}\right)^{1/2} \sin(\theta) \cos(\phi)$

  • $Y(p_y) = \left(\frac{3}{4\pi}\right)^{1/2} \frac{y}{r} = \left(\frac{3}{4\pi}\right)^{1/2} \sin(\theta) \sin(\phi)$

  • $Y(p_z) = \left(\frac{3}{4\pi}\right)^{1/2} \frac{z}{r} = \left(\frac{3}{4\pi}\right)^{1/2} \cos(\theta)$

  • $Y(d_{z^2}) = \left(\frac{5}{16\pi}\right)^{1/2} \frac{3z^2 - r^2}{r^2} = \left(\frac{5}{16\pi}\right)^{1/2} (3\cos^2(\theta) - 1)$

  • $Y(d_{x^2-y^2}) = \left(\frac{15}{16\pi}\right)^{1/2} \frac{x^2 - y^2}{r^2} = \left(\frac{15}{16\pi}\right)^{1/2} \sin^2(\theta) \cos(2\phi)$

  • $Y(d_{xy}) = \left(\frac{15}{4\pi}\right)^{1/2} \frac{xy}{r^2} = \left(\frac{15}{16\pi}\right)^{1/2} \sin^2(\theta) \sin(2\phi)$

  • $Y(d_{xz}) = \left(\frac{15}{4\pi}\right)^{1/2} \frac{xz}{r^2} = \left(\frac{15}{4\pi}\right)^{1/2} \sin(\theta) \cos(\theta) \cos(\phi)$

  • $Y(d_{yz}) = \left(\frac{15}{4\pi}\right)^{1/2} \frac{yz}{r^2} = \left(\frac{15}{4\pi}\right)^{1/2} \sin(\theta) \cos(\theta) \sin(\phi)$

  • Radial Parts $R_{n\ell}(r)$:

  • $R_{1s} = 2 \left(\frac{Z}{a_0}\right)^{3/2} e^{-\sigma/2}$

  • $R_{2s} = \frac{1}{2\sqrt{2}} \left(\frac{Z}{a_0}\right)^{3/2} (2 - \sigma) e^{-\sigma/2}$

  • $R_{3s} = \frac{1}{9\sqrt{3}} \left(\frac{Z}{a_0}\right)^{3/2} (6 - 6\sigma + \sigma^2) e^{-\sigma/2}$

  • $R_{2p} = \frac{1}{2\sqrt{6}} \left(\frac{Z}{a_0}\right)^{3/2} \sigma e^{-\sigma/2}$

  • $R_{3p} = \frac{1}{9\sqrt{6}} \left(\frac{Z}{a_0}\right)^{3/2} (4 - \sigma) \sigma e^{-\sigma/2}$

  • $R_{3d} = \frac{1}{9\sqrt{30}} \left(\frac{Z}{a_0}\right)^{3/2} \sigma^2 e^{-\sigma/2}$

Radial Functions, Radial Nodes, and Distance Behavior

• Exponential Decay and Size Trends:

  • All radial wave functions decay exponentially toward zero as distance $r \rightarrow \infty$

  • Higher principal quantum numbers $n$ extend farther from the nucleus, corresponding to larger orbital dimensions.

• Behavior at the Nucleus ($r = 0$):

  • For $s$ orbitals ($\ell = 0$), the radial function $R(r)$ has a nonzero value at the nucleus ($r = 0$).

  • For non-$s$ orbitals ($\ell \neq 0$), the radial function equals zero at the nucleus ($R(0) = 0$).

• Radial Nodes:

  • A radial node is a spherical surface at a specific distance $r$ from the nucleus where the radial function $R(r) = 0$ and changes sign.

  • Formula for Number of Radial Nodes:     $\text{Radial Nodes} = n - \ell - 1$

Angular Functions, Angular Nodes, and Phase

• Symmetry and Geometry:

  • $s$ Orbitals ($\ell = 0$): Angular function is constant ($Y = (4\pi)^{-1/2}$), producing a completely spherical probability distribution with zero angular nodes.

  • $p$ Orbitals ($\ell = 1$): Possess directional orientation along $x, y, z$ axes ($p_x, p_y, p_z$) and contain 1 angular node (a planar surface passing through the nucleus).

  • $d$ Orbitals ($\ell = 2$): Contain 2 angular nodes (two intersecting nodal planes or a nodal cone).

• Formula for Number of Angular Nodes:   $\text{Angular Nodes} = \ell$

• Formula for Total Nodes:   $\text{Total Nodes} = \text{Radial Nodes} + \text{Angular Nodes} = (n - \ell - 1) + \ell = n - 1$

• Wave Phase/Sign:

  • The sign ($+\text{ or }-$) of the angular and wave functions indicates the algebraic phase of the wave amplitude (often color-coded red and blue in 3D visualizations).

Detailed Structural Characteristics of Atomic Orbitals

• $s$ Subshell ($\ell = 0$):

  • Geometry: Spherical shell symmetry.

  • Visualizations: Rendered using wave function magnitude $\psi$, contour density maps, isosurface boundary plots (typically enclosing 95% probability), "foggy" electron density distributions, and probability density plots $\psi^2$

  • Nodal Analysis:

  • $1s$ ($n=1$): 0 radial nodes, 0 angular nodes (total = 0).

  • $2s$ ($n=2$): 1 radial node, 0 angular nodes (total = 1).

  • $3s$ ($n=3$): 2 radial nodes, 0 angular nodes (total = 2).

• $p$ Subshell ($\ell = 1$):

  • Geometry: Dumbbell / peanut shape with three orthogonal orientations ($p_x, p_y, p_z$).

  • Nodal Planes:

  • $p_x$ orbital: Nodal plane is the $yz$ plane.

  • $p_y$ orbital: Nodal plane is the $xz$ plane.

  • $p_z$ orbital: Nodal plane is the $xy$ plane.

  • Nodal Analysis:

  • $2p$ ($n=2, \ell=1$): 0 radial nodes, 1 angular node (total = 1).

  • $3p$ ($n=3, \ell=1$): 1 radial node, 1 angular node (total = 2).

• $d$ Subshell ($\ell = 2$):

  • Geometry: Five degenerate orbitals ($d_{xy}, d_{xz}, d_{yz}, d_{x^2-y^2}, d_{z^2}$). Four exhibit a four-lobed "cloverleaf" or double-peanut structure, while $d_{z^2}$ consists of two lobes along the z-axis surrounded by a donut-shaped ring (torus) in the xy plane.

  • Angular Nodes: Every $d$ orbital has exactly 2 angular nodes.

  • Nodal Analysis:

  • $3d$ ($n=3, \ell=2$): 0 radial nodes, 2 angular nodes (total = 2).

  • $4d_{xy}$ ($n=4, \ell=2$): 1 radial node, 2 angular nodes (total = 3).

• $f$ Subshell ($\ell = 3$):

  • Geometry: Seven degenerate orbitals with multi-lobed / flower-like shapes.

  • Designations: $4f_{y^3-3yx^2}$, $4f_{5yz^2-yr^2}$, $4f_{5xz^2-3xr^2}$, $4f_{5z^3-3zr^2}$, $4f_{x^3-3xy^2}$, $4f_{zx^2-zy^2}$, $4f_{xyz}$

  • Nodal Analysis:

  • Angular nodes = $\ell = 3$

  • For $4f$ ($n=4, \ell=3$): Radial nodes = $4 - 3 - 1 = 0$

Solved Practice Problems: Nodal Structure and Orbital Identification

• Practice Problem 1: Identifying Orbitals from Node Counts

  • a. 1 radial node and 1 angular node:

  • Angular nodes = 1 $\implies \ell = 1$ ($p$ orbital).

  • Radial nodes = $n - \ell - 1 = n - 1 - 1 = 1 \implies n = 3$

  • Identification: $3p$ orbital.

  • b. 0 radial nodes and 2 angular nodes:

  • Angular nodes = 2 $\implies \ell = 2$ ($d$ orbital).

  • Radial nodes = $n - 2 - 1 = 0 \implies n = 3$

  • Identification: $3d$ orbital.

  • c. 2 radial nodes and 3 angular nodes:

  • Angular nodes = 3 $\implies \ell = 3$ ($f$ orbital).

  • Radial nodes = $n - 3 - 1 = 2 \implies n = 6$

  • Identification: $6f$ orbital.

  • d. 2 radial nodes and 1 angular node:

  • Angular nodes = 1 $\implies \ell = 1$ ($p$ orbital).

  • Radial nodes = $n - 1 - 1 = 2 \implies n = 4$

  • Identification: $4p$ orbital.

  • e. 5 radial nodes and 0 angular nodes:

  • Angular nodes = 0 $\implies \ell = 0$ ($s$ orbital).

  • Radial nodes = $n - 0 - 1 = 5 \implies n = 6$

  • Identification: $6s$ orbital.

• Practice Problem 2: Orbital Identification from Cross-Sectional Contour Maps

• First Map Analysis ($3p_z$):

  • Features 1 radial node circle (dashed line) and 1 angular nodal plane perpendicular to the axis.

  • Result: $n = 3$, $\ell = 1$

• Second Map Analysis ($4d_{xy}$):

  • Features 4 lobes in the xy plane, 2 angular nodal lines, and 1 radial nodal circle.

  • Result: $n = 4$, $\ell = 2$

  • Practice Problem 3: Wave Function vs. Probability Density Analysis

• Question: Does a direct plot of wavefunction $\psi$ convey the probability of finding a matter wave? If not, sketch the appropriate plot, estimate the most probable location, and count the nodes.

• Analysis:

  • A raw wavefunction plot $\psi$ does not directly give the probability. Probability density is given by $|\psi|^2$ (or radial probability density $4\pi r^2 |\psi|^2$).

  • Most Probable Location: Corresponds to the global maximum peak of the $|\psi|^2$ plot.

  • Nodes: Locations where $\psi = 0$ (where the wave function crosses the horizontal zero axis).