Quantum Mechanics: Wavefunction and Probability

The fundamental equation in quantum mechanics describing the time evolution of a quantum system. It relates the time derivative of the wavefunction to the system's Hamiltonian operator.

A mathematical function that describes the state of a quantum system. It assigns a complex number to every point in space and time, providing the complete information about the system.

The energy operator in quantum mechanics. It acts on the wavefunction to determine its corresponding energy.

The derivative of the wavefunction with respect to time, keeping position constant.

The reduced Planck constant (h/2π). It appears in many fundamental quantum mechanics equations.

The superposition principle in quantum mechanics states that any linear combination of solutions to the Schrödinger equation is also a solution.

The de Broglie relation states that the momentum of a particle is proportional to its wavevector. It establishes the wave-particle duality of matter.

The Einstein relation states that the energy of a particle is proportional to its angular frequency. It forms the basis of the energy-frequency relationship in quantum mechanics.

The TISE is second-order in spatial derivatives, meaning the wavefunction must be continuous and smooth. This also implies that the potential cannot be too pathological, allowing for infinite potentials and discontinuous potentials, but not worse.

The probability of finding a particle between two points (x and x+dx) at a given time (t) is given by the square of the wavefunction's magnitude integrated over that region.

The probability density function for position is defined as the square of the absolute value of the wavefunction, representing the probability of finding the particle at a given position.

The integral of the probability density over all space must equal 1, indicating that the particle exists somewhere in the universe.

The probability of finding a particle between points a and b is calculated by integrating the squared magnitude of the wavefunction over that region.

The probability of finding the particle somewhere in the universe remains constant, regardless of time or other particle interactions.

Quantum particles tend to avoid regions of infinite potential due to the mathematical nature of the Schrödinger equation resulting in a zero probability of finding the particle in such regions.

A particle's behavior in quantum mechanics is governed by the probability of finding it in a certain region, not by its definite position like in classical mechanics.

The probability current density describes the flow of probability in quantum mechanics. It is a vector quantity that indicates the direction and magnitude of the probability flux.

The probability current density for an incident wave is proportional to the wave number kI and the square of the amplitude of the incident wave.

The probability current density for the reflected wave is proportional to the wave number kII and the square of the amplitude of the reflected wave, but with a negative sign due to the wave traveling in the opposite direction.

The probability current density for the transmitted wave is proportional to the wave number kII and the square of the amplitude of the transmitted wave.

The probability of reflection is the ratio of the reflected wave's probability current density to the incident wave's probability current density.

The probability of transmission is the ratio of the transmitted wave's probability current density to the incident wave's probability current density, adjusted for the different wave numbers.

The reflection and transmission coefficients depend on the incident and transmitted wave numbers, which are related to the particle's energy and the potential barrier's height.

The intrinsic angular momentum of a particle, quantized in units of ħ/2 and having two possible values: +ħ/2 and -ħ/2.

An experiment where a beam of silver atoms is passed through an inhomogeneous magnetic field, demonstrating the quantization of spin.

The measurement of a particle's spin along a specific direction can only yield one of two discrete values, +ħ/2 or -ħ/2.

If the spin of a particle is measured along one axis and then measured again along the same axis, the result will be the same.

If the spin is known along one direction (e.g., z), a measurement along a perpendicular direction (e.g., x) is completely unpredictable, with equal probability for either +ħ/2 or -ħ/2.

Even if the spin is measured along the same axis multiple times with consistent results, measuring it along another axis and then back to the original axis will introduce uncertainty in the original measurement.

The state of a spin-1/2 particle is represented by a vector in a two-dimensional complex vector space.

Spin observables (quantities that can be measured) are represented by 2x2 Hermitian matrices with eigenvalues corresponding to the spin values.

A quantum phenomenon where a particle penetrates a potential barrier even if its energy is less than the barrier's height. This is possible due to the wave nature of particles, allowing them to 'tunnel' through the barrier with a certain probability.

A type of potential where the particle is confined to a specific region, usually a line or box, and cannot exist outside of it. The potential is infinitely high outside the region, ensuring the particle remains trapped.

A mathematical solution to the time-independent Schrödinger equation (TISE) that describes the allowed energy states of a system. Each eigenfunction corresponds to a specific energy level.

The value of energy that a particle can have in a given potential. These energies are quantized, meaning they can only take on discrete values.

A process that ensures the probability of finding the particle somewhere in space is always equal to 1. It involves scaling the wavefunction to satisfy this condition.

A type of potential that is finite (has a specific value) within a certain region and drops to zero outside. This allows for the possibility of the particle escaping the potential, making it more realistic than the infinite potential well.

A state where the particle remains bound within the potential, meaning it doesn't have enough energy to escape. This is a common situation in atoms and molecules.

The time evolution of a quantum system is described by theTDSE. It establishes a relation between the time derivative of the wavefunction and the system's Hamiltonian operator.

A constant which represents the total energy of the system in the TISE.

A fundamental equation in quantum mechanics that gives the relationship between the energy of a system and its wavefunction. It is a time-independent version of the Schrödinger equation.

The wavefunction at time t is determined by the wavefunction at time 0 and the energy of the system. It describes how the wavefunction changes with time.

The probability of finding a particle between points a and b is calculated by integrating the squared magnitude of the wavefunction over that region.