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 ↓


Time-dependent Schrödinger equation


Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature on the scale of atoms and subatomic particles. One of the main formulas of quantum mechanics is the Schrödinger equation, developed by Erwin Schrödinger in the 1920s. This equation is a mathematical description of the quantum state of a system, and it sets the basis for understanding quantum phenomena.

Introduction to the Schrödinger equation

The Schrödinger equation can be thought of as a way to calculate the behavior of particles in quantum mechanics over time. There are two main forms of the Schrödinger equation: time-dependent and time-independent equations. Here, we focus on the time-dependent Schrödinger equation, which deals with the evolution of quantum states over time.

The math behind the equation

The time-dependent Schrödinger equation is expressed as:

iħ ∂ψ/∂t = Hψ
iħ ∂ψ/∂t = Hψ

In this equation:

  • i is the imaginary unit, which satisfies ( i^2 = -1 ).
  • ħ (h-bar) is the reduced Planck constant, which relates the energy of the photon to its frequency.
  • ψ (psi) is the wave function, which contains all the information about the system.
  • H is the Hamiltonian operator, which represents the total energy (kinetic + potential) of the system.
  • ∂ψ/∂t denotes the partial derivative of ψ with respect to time.

The wave function ψ is central to quantum mechanics because it describes the state of a system. The square of its absolute value, ( |ψ(x, t)|^2 ), gives the probability density of finding a particle at position ( x ) and time ( t ).

Understanding the components

Hamiltonian operator

The Hamiltonian H is an operator that includes the kinetic and potential energy of the particle. For a single particle, the Hamiltonian is often written as:

H = - (ħ² / 2m) ∇² + V(x)
H = - (ħ² / 2m) ∇² + V(x)

Where:

  • m is the mass of the particle.
  • ∇² is the Laplace operator, which acts on the spatial coordinates of the wave function.
  • V(x) is the potential energy function.

Wave function

Conceptually, the wave function can be viewed as an oscillating wave that carries information about the position, momentum, and energy of a particle. It does not provide a definite path as in classical mechanics, but rather provides a probability of where the particle might be at any given moment.

ψ(x, t)

Above is a simple depiction of the wave function ψ(x, t) along a one-dimensional axis. The wave-like appearance is characteristic of quantum phenomena.

Solving the time-dependent Schrödinger equation

Solving the time-dependent Schrödinger equation involves determining how the wave function ψ evolves over time given the initial conditions and the potential energy landscape. For most practical purposes, especially in high dimensions or complex systems, numerical methods such as finite element methods are used to solve these equations.

Analytical solution in free space

In cases where there is no potential energy (i.e., free particles), the potential V(x) = 0, and the equation simplifies considerably. It becomes:

iħ ∂ψ/∂t = - (ħ² / 2m) ∇² ψ
iħ ∂ψ/∂t = - (ħ² / 2m) ∇² ψ

This can be solved using separation of variables, where the wave function ψ(x, t) is expressed as the product of a spatial component and a temporal component:

ψ(x, t) = φ(x)T(t)
ψ(x, t) = φ(x)T(t)

Substitution in the Schrödinger equation and separation of variables gives the solution for the free particle in terms of plane waves:

ψ(x, t) = A e^(i(kx - ωt))
ψ(x, t) = A e^(i(kx - ωt))

Where:

  • A is the amplitude of the wave.
  • k is the wave number, which is related to momentum.
  • ω is the angular frequency, which is related to the energy of the particle.

Examples and applications

A particle in a box

Consider a particle confined in a one-dimensional box (or well) of length L and whose potential walls are infinitely high. In such a system, the particle is free to move around inside the box but cannot exist outside it.

There are marginal situations

  • ψ(0, T) = 0
  • ψ(l, t) = 0

The energy levels of the particle are quantized, and the wave functions are stationary waves:

ψ_n(x, t) = √(2/L) sin(nπx/L) e^(-iE_n t/ħ)
ψ_n(x, t) = √(2/L) sin(nπx/L) e^(-iE_n t/ħ)

where ( E_n = n²π²ħ²/(2mL²) ) are the quantized energy levels, and n is an integer.

Quantum Harmonic Oscillator

The quantum harmonic oscillator is another classic problem where the potential energy is ( V(x) = 1/2 mω²x² ). The time-dependent solution for this system involves Hermite polynomials and can be expressed as:

ψ_n(x, t) = N_n e^((-mωx²)/(2ħ)) H_n(√(mω/ħ) x) e^(-iE_n t/ħ)
ψ_n(x, t) = N_n e^((-mωx²)/(2ħ)) H_n(√(mω/ħ) x) e^(-iE_n t/ħ)

Where:

  • N_n is a normalization factor.
  • H_n is the Hermite polynomial.
  • E_n are quantized energy levels.

Understanding such systems is important because they serve as models for a wide range of physical phenomena, including molecular vibrations and quantum field theory.

Conceptual shift in quantum mechanics

The implications of the time-dependent Schrödinger equation are very profound and represent a significant departure from the deterministic view of classical mechanics of particle motion. In quantum physics, deterministic trajectories are replaced by probabilistic outcomes.

The phenomenon of superposition, where a quantum system can exist in multiple states simultaneously, emerges from these foundations. For example, a particle described by a combination of wave functions can be in multiple energy states simultaneously, a phenomenon routinely demonstrated by experiments such as the double-slit experiment.

Conclusion

The time-dependent Schrödinger equation is integral to our understanding of quantum systems and describes how wave functions evolve over time. It bridges the gap between theoretical physics and experimental results by providing a comprehensive framework to study a variety of quantum phenomena.

This equation exemplifies the beautiful interrelationship between mathematics and nature, which remains fundamental to the exploration of modern physics, from quantum chemistry to cutting-edge research in quantum computing.


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Time-dependent Schrödinger equation

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