PX262: Quantum Mechanics and its applications, Part 1

Flashcards about Quantum Mechanics, covering topics like failures of classical physics, Schrödinger equation, postulates of quantum mechanics, ladder operators, angular momentum, 3D quantum mechanical systems, additional interactions in atoms, and photon emission.

Photoelectric effect

Observed when light strikes a metal surface, and electrons are emitted. The energy of the electrons depends on the frequency of the light. E = hf - φ where h is Planck's constant and φ is the work function.

Leads us too understand that photon energy is quantised and E=hf

Note that h(bar) = h/2π

Compton Scattering

The scattering of light on free electrons, resulting in a change in wavelength. This supports the idea that light behaves as a particle. we note that we can define the momentum of a photon p = h(bar) k

!Compton Shift Formula

The change in wavelength (\Delta \lambda) in Compton scattering is given by: \Delta \lambda = \lambda' - \lambda = \frac{h}{me c} (1 - \cos{\theta}), where \lambda is the initial wavelength, \lambda' is the final wavelength, h is Planck's constant, me is the mass of the electron, c is the speed of light, and \theta is the scattering angle.

Line Spectra of Light

Light emitted from atoms has only discrete, specific wavelengths. Frequencies for hydrogen atoms follow a specific pattern, implying that electrons in atoms can only be in specific, discrete energy levels.

De Broglie Waves

Particles can behave as waves, with assigned frequency and wave-vector. The double-slit experiment provides experimental evidence for this, showing interference patterns with electrons similar to light waves.

De Broglie Wavelength Formula

The de Broglie wavelength (\lambda) of a particle with momentum p is given by: \lambda = \frac{h}{p} = \frac{h}{mv}, where h is Planck's constant, m is the mass of the particle, and v is its velocity.

Wave-Particle Duality

Both particles and waves can behave as particles or waves depending on the experiment or phenomenon being described.

Uncertainty Principle

There is a fundamental limit to how well we can simultaneously know or measure position and momentum.

Heisenberg Uncertainty Principle Formula

The Heisenberg uncertainty principle is mathematically expressed as: \Delta x \Delta p \geq \frac{\hbar}{2}, where \,Delta x is the uncertainty in position, \Delta p is the uncertainty in momentum, and \hbar is the reduced Planck constant (\hbar = \frac{h}{2\pi}).

Schrödinger Equation

The main equation to describe quantum systems, relating the time evolution of a wave function to the particle's potential energy.

Time-Dependent Schrödinger Equation (General Form)

The general form of the time-dependent Schrödinger equation is: i\hbar \frac{\partial}{\partial t} \Psi(r, t) = \hat{H} \Psi(r, t), where i is the imaginary unit, \,hbar is the reduced Planck constant, \,Psi(r, t) is the wave function, and \hat{H} is the Hamiltonian operator.

Time-Independent Schrödinger Equation

Used to solve time-independent potentials, separating the wave function into spatial and temporal components.

Time-Independent Schrödinger Equation Formula

The time-independent Schrödinger equation is given by: \hat{H} \psi(r) = E \psi(r), where \,hat{H} is the Hamiltonian operator, \psi(r) is the time-independent wave function, and E is the energy eigenvalue.

Boundary Conditions

Conditions that a wave function must satisfy to be a valid solution, including continuity, single-valuedness, and finite integral over all space.

Particle in Infinite Square Well

A system in which a particle is confined to a finite region with infinite potential energy outside that region. Leads to discrete energy levels.

Energy Levels in Infinite Square Well Formula

For a particle in an infinite square well of width L, the energy levels are given by: E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}, where n is the quantum number (n = 1, 2, 3,…), \,hbar is the reduced Planck constant, and m is the mass of the particle.

Quantum Tunneling

A quantum mechanical effect where a particle can pass through a potential barrier even if its energy is less than the barrier height.

Transmission Probability in Quantum Tunneling Formula

The transmission probability T through a potential barrier of height V0 and width L for a particle with energy E is approximately: T \approx e^{-2\kappa L}, where \kappa = \sqrt{\frac{2m(V0 - E)}{\hbar^2}}.

One-Dimensional Harmonic Oscillator

A system where potential energy is proportional to the square of the displacement. It serves as a model to approximate various systems.

Potential Energy of Harmonic Oscillator Formula

The potential energy V(x) for a one-dimensional harmonic oscillator is given by: V(x) = \frac{1}{2} m \omega^2 x^2, where m is the mass of the particle and \omega is the angular frequency.