Quantum Mechanics Basics

• Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles.
• It is also known as quantum physics or quantum theory.
• Quantum mechanics provides a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter.

Quantum Theory Basics

• Quantum theory evolved in response to the failure of classical physics to explain certain phenomena, such as black-body radiation and the photoelectric effect.
• Key concepts include quantization, wave-particle duality, the uncertainty principle, and quantum entanglement.
• Quantization refers to the idea that certain physical properties, such as energy, can only take on discrete values.
• The energy of a photon is quantized and is proportional to its frequency, described by the equation E = hf, where E is energy, h is Planck's constant, and f is frequency.
• Quantum mechanics is probabilistic, meaning that it predicts the probabilities of different outcomes rather than deterministic outcomes.
• The state of a quantum system is described by a wave function, which evolves in time according to the Schrödinger equation.
• Measurements on a quantum system cause the wave function to "collapse" into one of the possible states, with a probability determined by the wave function.
• Quantum mechanics incorporates the concept of spin, an intrinsic form of angular momentum possessed by elementary particles.
• Particles with half-integer spin (e.g., electrons) are called fermions and obey the Pauli exclusion principle, while particles with integer spin (e.g., photons) are called bosons and do not.

Wave-particle Duality

• Wave-particle duality is the concept that every elementary particle or quantum entity exhibits the properties of both particles and waves.
• The double-slit experiment is a classic demonstration of wave-particle duality.
• In this experiment, particles (such as electrons) are fired at a barrier with two slits.
• The particles pass through the slits and are detected on a screen behind the barrier.
• Surprisingly, an interference pattern is observed on the screen.
• This pattern is characteristic of waves, even though the particles are detected as discrete entities.
• Wave-particle duality is described mathematically by the de Broglie relation, which relates the wavelength (λ) of a particle to its momentum (p): λ = h/p, where h is Planck's constant.
• The wave nature of particles is significant at the atomic and subatomic levels but becomes less apparent at macroscopic scales due to the smallness of the de Broglie wavelength.

Quantum Entanglement

• Quantum entanglement is a phenomenon in which two or more quantum particles become linked together in such a way that they share the same fate, no matter how far apart they are in space.
• When one particle's state is measured, the state of the other particle is instantly determined, regardless of the distance between them.
• This correlation occurs even if the particles are separated by vast distances.
• It appears to violate the principle of locality, which states that an object is only directly influenced by its immediate surroundings.
• Entanglement is one of the key resources in quantum information processing, enabling quantum computation, quantum cryptography, and quantum teleportation.
• Einstein famously called entanglement "spooky action at a distance" because it seemed to contradict the speed limit imposed by the theory of relativity.
• Quantum entanglement does not allow for faster-than-light communication, as the measurement outcome on one particle is random and cannot be used to transmit information.

Quantum Superposition

• Quantum superposition is the principle that a quantum system can exist in multiple states simultaneously until a measurement is made.
• The system is in a combination of all possible states, with a certain probability associated with each state.
• This is often illustrated using the thought experiment of Schrödinger's cat, where a cat in a box is both dead and alive until the box is opened and the cat's state is observed.
• Mathematically, a superposition is represented as a linear combination of the possible states of the system.
• Upon measurement, the system collapses into one of the possible states, with the probability of collapsing into a particular state determined by the square of the amplitude of the corresponding term in the superposition.
• Superposition is a key feature that distinguishes quantum mechanics from classical mechanics.

Quantum Tunneling

• Quantum tunneling is a phenomenon in which a particle can pass through a potential energy barrier that it classically cannot surmount.
• In classical physics, a particle must have enough energy to overcome a barrier to pass through it.
• However, in quantum mechanics, there is a non-zero probability that a particle can tunnel through the barrier, even if its energy is less than the barrier's height.
• The probability of tunneling depends on the width and height of the barrier, as well as the particle's energy.
• Quantum tunneling is essential in nuclear fusion in stars, radioactive decay, and certain electronic devices such as tunnel diodes.
• The probability of tunneling decreases exponentially with increasing barrier width and height.
• Scanning Tunneling Microscopy (STM) utilizes quantum tunneling to image surfaces at the atomic level.