Quantum Mechanics Study Notes

Regions of high curvature (peaks and troughs) in a wave can create complex interference patterns.

A wave function described by a sinusoidal wave on a string.

A parameter in a wave function that indicates the amount of spatial variation in the function.

Describes how fast the wave oscillates. It's related to frequency by 𝜔 = 2πf.

A particle not subject to external forces or confinement, able to move freely in its environment.

A particle confined to a specific region in space due to a potential well or other constraining forces.

Energy levels of a bound particle are specific and separate, not continuous.

The wave equation describes how waves vary and propagate.

A model describing a particle confined to a finite region of space, often represented by a cubical box with infinite potential walls.

A mathematical technique used to solve partial differential equations by breaking them down into simpler ordinary differential equations.

Constraints imposed on the solutions of a differential equation at the boundaries of the system, ensuring physically realistic behavior.

Integer values that describe the allowed energy levels and properties of a particle in a quantum system. In a 3D box, there are three quantum numbers (nx, ny, nz).

The phenomenon where multiple distinct quantum states have the same energy level.

The allowed energy levels for a particle in a 3D box are quantized and determined by the equation: E = (h^2/(8mL^2)) * (n_x^2 + n_y^2 + n_z^2), where n_x, n_y, n_z are the quantum numbers and L is the side length of the box.

Points in a wave function where the probability of finding the particle is zero.

The symmetry of the cubical box leads to degeneracy because different combinations of quantum numbers can result in the same energy level.

Energy levels are quantized (specific values) and given by E_n = (n + 1/2)ℏω, where n is the quantum number, ℏ is reduced Planck's constant, and ω is the angular frequency, demonstrating a constant energy spacing.

The lowest energy state, occurring when n = 0, possessing a non-zero minimum energy.

The time-independent Schrödinger equation, including potential term (kx^2), defines the wave functions (possible states) of the harmonic oscillator.

As position approaches infinity (very large or small), the wave function must approach zero, ensuring it never increases or exceeds infinity.

Wave functions must decay to zero as the displacement x gets larger to fulfil boundary conditions. Only some energy levels result in such functions

Areas beyond certain energy thresholds, where wave functions should approach zero. Classical physics ideas do not always describe these scenarios correctly.

The classical limit of a quantum system is when quantization effects become negligible, and the quantum system behaves like a classical system.

Equation that describes the relationship between energy and wave function of a quantum mechanical system.

Waves that propagate through space, carrying energy and momentum. They have a specific wavelength and frequency that determine their speed.

A phenomenon where a particle can pass through a potential barrier even if its energy is less than the barrier's height. This is a strictly quantum mechanical effect and has no classical counterpart.

States that it's impossible to simultaneously know both a particle's position and momentum with perfect accuracy. The more precisely one is determined, the less precisely the other can be known.

Discrete, specific values that the energy of a particle can take on within a quantum system. These levels are often quantized, meaning they can only take on certain values.

A fundamental equation in quantum mechanics used to describe the time evolution of a quantum system. It predicts the behavior of particles in quantum systems.

A region in space where a particle experiences an attractive force, causing it to be confined within the well. The well can have infinite or finite depth.

Patterns in chemical behavior of elements across the periodic table based on their electron configurations.

Horizontal rows in the periodic table. Elements in a period have the same number of electron shells.

Vertical columns in the periodic table. Elements in a group have similar electron configurations in their outermost shell, leading to similar chemical properties.

Group 18 elements with completely filled outer shells (e.g., helium, neon). They are generally unreactive.

Group 1 elements with one valence electron in an s-orbital (e.g., lithium, sodium). They are highly reactive metals.

Group 2 elements with two valence electrons in an s-orbital (e.g., beryllium, magnesium). They are reactive metals, but less so than alkali metals.

Group 17 elements with seven valence electrons (e.g., fluorine, chlorine). They are highly reactive nonmetals.

Elements (Z = 21 to Z = 30) that fill 3d and 4s subshells, showing similar properties.

The interaction between an electron's spin and orbital angular momenta, causing a splitting of energy levels in atoms.

Two closely spaced spectral lines emitted by sodium atoms due to spin-orbit coupling, located at 589.0 nm and 589.6 nm.

The combined angular momentum of an electron's orbital and spin angular momenta, represented by the quantum number 'j'.

When an electron's orbital and spin angular momenta are aligned (in the same direction), resulting in j = l + s.

When an electron's orbital and spin angular momenta are opposite (in opposite directions), resulting in j = l - s.

A system used to represent atomic energy levels, using terms like 2P1/2 and 2P3/2, where the superscript indicates spin multiplicity, the letter represents the orbital angular momentum (l), and the subscript represents the total angular momentum (j).

Small energy shifts in atomic energy levels resulting from spin-orbit coupling, magnetic effects, and relativistic corrections.