PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Quantum mechanics ↓


Schrödinger Equation


The Schrödinger equation is a fundamental concept in quantum mechanics that describes how the quantum state of a physical system changes over time. It was formulated by Austrian physicist Erwin Schrödinger in 1925, and provides a way to predict how a quantum system evolves. This equation is as important to quantum mechanics as Newton's laws are to classical mechanics. In this exposition, we will dive deep into the details of the Schrödinger equation, its implications, and its role in the quantum world. We will also discuss several illustrative examples to better understand its meaning and implications.

Understanding the Schrödinger equation

At its core, the Schrödinger equation is a mathematical expression that describes the time evolution of the wave function of a system. The wave function, often represented by the Greek letter psi (ψ), contains all the information about a quantum system. For a single non-relativistic particle, the time-dependent Schrödinger equation is written as:

iħ ∂ψ/∂t = Hψ

Where:

  • i is the imaginary unit.
  • ħ (h-bar) is the reduced Planck constant (about 1.0545718 × 10^−34 Js).
  • ∂ψ/∂t is the partial derivative of the wave function with respect to time, which shows how the wave function changes with time.
  • H is the Hamiltonian operator, corresponding to the total energy of the system.

Wave function

The wave function is a crucial part of the Schrödinger equation and is central to understanding quantum mechanics. It is a complex-valued function that encodes the probability amplitude of different outcomes in a quantum experiment. The square of the absolute value of the wave function |ψ(x, t)|² gives the probability density of finding a particle at a particular point in space at a specific time.

Mathematical form of the wave function

The wave function for a one-dimensional system can be expressed as:

ψ(x, t) = A e^(i(kx - ωt))

Where:

  • A denotes the amplitude of the wave function.
  • e is the Euler number, which is approximately 2.71828.
  • k is the wave number, which is related to the momentum of the particle.
  • ω is the angular frequency.

Example of a wave function

Consider a particle in a box, which is a fundamental example in quantum mechanics. The wave function can be represented as a sine wave, which is analogous to a standing wave, something like this:

ψ(x) = A sin(nπx/L)

where n is an integer indicating the energy level, and L is the length of the box. The particle is most likely to be found in positions corresponding to the peaks of the wave.

Time-dependent vs. time-independent Schrödinger equation

There are two main forms of the Schrödinger equation: the time-dependent and the time-independent Schrödinger equation. The time-dependent form was given first. The time-independent Schrödinger equation is used when we are not directly concerned about the time evolution, but want to find the stationary states (energy eigenstates) of the system.

Time-independent Schrödinger equation

Hψ = Eψ

Where:

  • E is the energy eigenvalue corresponding to the state ψ.

Example

One of the most famous applications of the time-independent Schrödinger equation is the "particle in a one-dimensional box" model. In this simplified system, a particle is confined to moving in a straight line between two impenetrable barriers. The solutions are sinusoidal waves of the following type:

ψ_n(x) = sqrt(2/L) sin(nπx/L)

These represent the stationary states of the particles, and their energy levels are given as follows:

E_n = n²π²ħ²/(2mL²)

Pictorial example

Consider a box with different energy levels represented as standing waves:

n=1 n=2

The blue wave probably represents the ground state (n=1) and is just a single hump. The red wave (n=2) would have nodes at the ends and a node in the middle, representing the first excited state.

Quantum tunneling

An interesting phenomenon that emerges from the Schrödinger equation is quantum tunneling. This is a situation in which particles can cross barriers that they should not cross due to the probabilistic nature of the quantum world.

Imagine a particle approaching a barrier. Classically, if the particle does not have enough energy, it is completely reflected. However, quantum physics, as modeled by the Schrödinger equation, allows a non-zero probability that the particle will tunnel through the barrier, even though classically it seems forbidden.

Conclusion

The Schrödinger equation plays a key role in quantum mechanics, which describes how physical systems evolve at the quantum level. It allows physicists to predict the probabilities of finding particles in different states, describe phenomena such as quantum tunneling, and provide insights through the concept of the wave function, which ultimately shapes our understanding of the quantum world.

Simply put, the Schrödinger equation is the key to unlocking the mysteries of the microscopic universe. Though the math may seem daunting, it ultimately unravels the mysteries of nature at the smallest scales.


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Schrödinger Equation

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