Introduction to Quantum Mechanics: Key Definitions

Flashcards containing key terminology and fundamental definitions from the textbook 'Introduction to Quantum Mechanics' by David J. Griffiths.

Schrödinger equation

The fundamental equation of motion in quantum mechanics that determines the time evolution of the wave function: $iℏ \frac{∂Ψ}{∂t} = БΨ$.

Born's statistical interpretation

The principle that the square of the magnitude of the wave function, $|Ψ(x, t)|^2$, is the probability density for finding the particle at point $x$ at time $t$.

Normalization

The requirement that the total probability of finding a particle in all of space is 1: $∫_{-∞}^{+∞} |Ψ(x, t)|^2 dx = 1$.

Expectation value

The average value of a series of measurements of an observable made on an ensemble of identically prepared systems, given by ⟨Q⟩ = ∫ Ψ^* ̑ Q Ψ dx.

Heisenberg uncertainty principle

The physical limit stating that the product of the standard deviations of position and momentum cannot be less than half of the reduced Planck constant: $σ_x σ_p ≥ \frac{ℏ}{2}$.

Stationary state

A state with a definite total energy $E$, characterized by a wave function where the time dependence is a simple phase factor: $Ψ(x, t) = ψ(x)e^{-iEt/ℏ}$.

Hamiltonian operator

The operator corresponding to the total energy (kinetic plus potential) of a system: $Б = -\frac{ℏ^2}{2m} ∇^2 + V$.

Orthonormal

A property of a set of functions that are both normalized and mutually orthogonal, satisfying $⟨ψ_m | ψ_n⟩ = δ_{mn}$.

Hilbert space

The complete inner product space composed of all square-integrable functions ($L^2$) in which quantum mechanical wave functions live.

Hermitian operator

An operator that is equal to its own adjoint ($Q = Q^†$), ensuring that its eigenvalues are real and its eigenfunctions belonging to distinct eigenvalues are orthogonal.

Commutator

An algebraic operation between two operators defined as $[A, B] = AB - BA$; if it is nonzero, the observables are incompatible.

Spin

An intrinsic form of angular momentum ($S$) possessed by elementary particles, independent of their motion in space.

Fermions

Particles with half-integer spin (such as electrons) that obey the Pauli exclusion principle and require antisymmetric total wave functions.

Bosons

Particles with integer spin (such as a photons) that do not obey the exclusion principle and require symmetric total wave functions.

Pauli exclusion principle

The principle stating that two identical fermions cannot occupy the same quantum state simultaneously.

Variational principle

A technique for approximating the ground-state energy of a system by minimizing the expectation value of the Hamiltonian for a trial wave function.

WKB approximation

A method (Wentzel, Kramers, Brillouin) for finding approximate solutions to the time-independent Schrödinger equation in cases where the potential varies slowly compared to the wavelength.

Fine structure

Corrections to the Bohr energy levels of hydrogen caused by relativistic effects and spin-orbit coupling, which are smaller by a factor of $α^2$.

Zeeman effect

The splitting of atomic energy levels when an atom is placed in an external magnetic field.

Adiabatic theorem

The theorem stating that if a Hamiltonian changes gradually from an initial form to a final form, a particle starting in the $n^{th}$ eigenstate will remain in the corresponding $n^{th}$ state.