100 days of Quantum Computing

Day-8 The Map of Quantum Physics

Between 1800 and 1900 AD

1800 to 1900

1. Blackbody Radiation:
2. Photoelectric Effect:
3. Atomic Spectra:
4. Stability of Atoms:
5. Matter Waves (de Broglie Wavelength):
6. Quantum Tunneling (Pre-Quantum Understanding):
7. Lack of Explanation for Atomic Sizes:
8. Lack of Explanation for Chemical Bonding:
9. Failure of Classical Mechanics at Small Scales:

These mysteries paved the way for the development of quantum mechanics in the early 20th century, which revolutionised our understanding of the microscopic world.

The double-slit experiment is a fundamental experiment in quantum mechanics that demonstrates the wave-particle duality of particles, such as electrons or photons. It was first performed with light by Thomas Young in 1801 and later extended to particles in the 20th century. The experiment is a crucial illustration of the strange and counterintuitive behavior exhibited by particles at the quantum level.

Quantum Interference: Quantum interference occurs when the probability amplitudes of different quantum states combine, leading to constructive or destructive interference.

Wave-Particle Duality: Particles such as electrons and photons exhibit both wave-like and particle-like properties. This duality is a fundamental concept in quantum mechanics.

Statement of the Born Rule:

For a quantum system described by a wavefunction ΨΨ, the probability density P(x) of finding a particle at position x is given by the square of the magnitude of the wavefunction at that position:The Born Rule is a fundamental part of the mathematical formalism of quantum mechanics and is essential for making predictions about the behavior of particles at the quantum level.

Schrödinger Equation:

The Schrödinger Equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It was formulated by the Austrian physicist Erwin Schrödinger in 1925. The equation is central to understanding the behavior of particles at the quantum level, such as electrons in atoms.

Key Concepts:

1. Wavefunction (ΨΨ or ψ):The solution to the Schrödinger Equation represents the wavefunction, which describes the probability amplitude of finding a particle in a particular state.
2. Probability Density: The square of the magnitude of the wavefunction (∣Ψ∣2∣Ψ∣2 or ∣∣2∣ψ∣2) gives the probability density of finding the particle in a specific position.
3. Quantization of Energy: The solutions to the Schrödinger Equation yield quantized energy levels for a given system.
4. Hamiltonian Operator (^H^): Represents the total energy operator for the system.
5. Uncertainty Principle: The Schrödinger Equation is a deterministic equation for the evolution of the wavefunction, but it inherently involves probabilities, reflecting the probabilistic nature of quantum mechanics.

The Schrödinger Equation is a cornerstone of quantum mechanics and has been essential in explaining the behavior of particles on the quantum scale. It is a key tool in predicting and understanding the properties of matter and energy at the atomic and subatomic levels.

Dirac Equation

The Dirac Equation is a fundamental equation in quantum mechanics and relativistic quantum field theory. It was formulated by the physicist Paul Dirac in 1928. Unlike the Schrödinger Equation, which describes non-relativistic particles, the Dirac Equation incorporates special relativity and provides a description of relati

t is time, α and β are 4x4 matrices (Dirac matrices), p is the momentum operator, c is the speed of light, m is the rest mass of the electron.

Key Concepts:

1. Relativistic Effects: The Dirac Equation accounts for relativistic effects and describes the behavior of electrons moving at speeds close to the speed of light.
2. Spinors: The wavefunction ψ in the Dirac Equation is a four-component spinor, representing the electron’s spin and its associated quantum states.
3. Prediction of Antimatter: The Dirac Equation predicted the existence of antimatter, as it allows for solutions corresponding to negative energy states. This prediction was experimentally confirmed with the discovery of the positron.
4. Quantization of Angular Momentum: Like the Schrödinger Equation, the Dirac Equation leads to the quantization of angular momentum, with the addition of spin angular momentum.
5. Incorporation of Spin: The inclusion of spin in the Dirac Equation was a significant advancement, explaining fine details of the hydrogen spectrum.
6. Spin-Orbit Interaction: The Dirac Equation naturally includes the spin-orbit interaction, providing a more accurate description of the interaction between an electron’s orbital motion and its intrinsic spin.

The Dirac Equation has been crucial in the development of quantum field theory and the understanding of relativistic quantum mechanics. It laid the foundation for the formulation of quantum electrodynamics (QED) and has broader applications in describing particles with spin in relativistic settings.

Reference :

2. https://www.youtube.com/watch?v=QI13S04w8dM&list=PLUl4u3cNGP60QlYNsy52fctVBOlk-4lYx