Terms and States (Electronic Spectroscopy and Excited States) Lecture Notes

This flashcard set covers the historical development, quantum mechanical foundations, mathematical operators, and term symbol calculations in electronic spectroscopy as detailed in Professor Paul Walton's lecture notes.

Rydberg constant ($R_H$)

A constant used in the mathematical expression for the spectrum of the hydrogen atom, valued at $109677.6\,cm^{-1}$.

Lemma 1

In the atomic world, every state has a fixed energy.

Lemma 2

Electrons can be modelled as waves, which can be added to each other (superposition).

Statistical Interpretation (Born)

The principle that while the wave function $\psi$ is not observable, $\psi^* \psi = |\psi^2|$ is an observable quantity representing the probability of locating a particle in a given volume.

Time-Dependent Wave Equation

Postulated by Schrödinger in 1926 as $i\hbar \frac{\partial\Psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t)$. It is used as a 'recipe' to calculate wavefunctions or energies.

Complex Numbers

Expressions in the form of $a + ib$ (where $i = \sqrt{-1}$) used in wave equations to represent two independent properties, such as space and time.

Free Wave Function

A valid solution to the wave equation ($Ae^{i(kx-\omega t)}$) that has not yet been bound to the Coulombic attraction of a nucleus.

Standing Waves

Naturally quantised waves represented by solutions to the time-independent wave equation where the probability distribution is constant in time.

Spherical Harmonics

Three-dimensional equivalents of standing waves that represent solutions to the wave equation with all boundary conditions applied ($\psi(r, \theta, \phi) = R(r)Y(\theta, \phi)$).

Orbital Angular Momentum

The result of an electron rotating around an axis (e.g., the z-axis), which induces a magnetic field at the centre of rotation.

Balanced State

A collection of microstates where the sum of individual quantum numbers adds to zero (e.g., the combination of $m_l = 1$ and $m_l = -1$).

Spin-Orbit Coupling

The interaction of the two magnetic fields associated with an electron in an orbital with non-zero $m_l$ values, involving orbital momentum ($l$) and spin ($s$).

Russell-Saunders (RS) Coupling Scheme

A method for multi-electron atoms that first maximally adds all individual $l$ vectors to give $L$ and all $s$ vectors to give $S$, then adds $L$ and $S$ to find the overall quantum number $J$.

Atomic Term Symbol

A symbol in the form ${^{2S+1}}L$ used to represent the total orbital ($L$) and spin ($S$) angular momenta of an atom.

Spin Multiplicity

The value calculated as $2S+1$ in a term symbol, categorizing states as singlet, doublet, triplet, etc.

Hund’s Rules

Guidelines used to identify the ground state configuration, stating it must have maximum spin ($S$) and maximum orbital angular momentum ($L$).

Normal Multiplet

A state where the number of electrons in the subshell is less than half full; its ground state takes the lowest value of $J$.

Inverted Multiplet

A state where the number of electrons in the subshell is more than half full; its ground state takes the highest value of $J$.

Selection Rule (1-Electron Atom)

A rule stating that allowed electronic transitions must obey $\Delta l = \pm 1$.

Microstates

The multiple ways electrons can be arranged within a given electron configuration (e.g., there are 15 microstates for a $p^2$ configuration).