Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Quantum mechanics ↓


Advanced wave mechanics


Advanced wave mechanics is a part of quantum mechanics that delves deep into the understanding of the wave properties of particles. Unlike classical mechanics where Newtonian principles are used, quantum mechanics relies on the wave and particle nature of matter. This duality is quintessential for understanding quantum systems. This document attempts to provide a thorough but understandable explanation of advanced wave mechanics for individuals immersed in undergraduate physics.

The concept of wave-particle duality

In classical physics, particles and waves are distinct entities. However, quantum mechanics shows that particles such as electrons exhibit both wave and particle characteristics. This duality is fundamental to understanding quantum systems. For example, consider an electron. In some experiments, it behaves like a particle; in others, it exhibits wave-like properties such as interference and diffraction.

Visualize the wave nature of a particle with the following model:

Here, the overlapping wave circles symbolize electrons behaving as waves, which cross each other and form an interference pattern.

Schrödinger equation

The wave function, ψ, is fundamental in quantum mechanics because it describes the quantum state of a system. The evolution of this wave function over time is governed by the Schrödinger equation. In its time-dependent form, it is expressed as:

        iℏ&partial;ψ/&partial;t = -(ℏ^2/2m)∇^2ψ + Vψ

Here:

  • i is the imaginary unit.
  • ℏ is the reduced Planck constant.
  • ψ is the wave function.
  • t is the time.
  • m is the mass of the particle.
  • V is the potential energy.

This equation represents the probability of finding a particle in a certain quantum state. To solve the Schrödinger equation, boundary and initial conditions are usually required.

Probability explanation

The wave function itself is not directly observable. Its squared modulus, |ψ|^2, provides the probability density function for the position of a quantum particle. For example, if ||ψ(x,t)|^2 is high at a particular point, this indicates a high probability of finding the particle at that position at time t.

Let us consider an example:

In this visualization, the blue curve represents the wave function, while the peaks and red dots indicate areas with a high probability of finding the particle.

Stationary states and energy quantization

For many systems, solutions to the time-independent Schrödinger equation give stationary states where the energy levels are quantized. This occurs in systems such as electrons in an atom, where only discrete energy levels are allowed.

Consider a simple case: a particle in a one-dimensional box of length L The marginal conditions are:

        ψ(0) = ψ(L) = 0

The solutions are as follows:

        ψn(x) = sqrt(2/L) sin(nπx/L)

The corresponding quantized energies are:

        En = (n^2π^2ℏ^2)/(2mL^2)

where n is a quantum number (1, 2, 3, ...).

A visual representation of this quantization could be like this:

The solid and dashed lines represent two different energy states of the particle in the box.

Quantum tunneling

In classical mechanics, if a particle does not have enough energy to cross the barrier, it simply cannot. However, quantum mechanics introduces the concept of tunneling, where there is a possibility that the particle can cross the energy barrier.

For example, consider a particle approaching a potential barrier of height V0. Even if the energy of the particle E is less than V0, there is still a finite probability of finding the particle on the other side of the barrier.

        ψ(x) = A e^(kx) + B e^(-kx), k^2 = 2m(V0-E)/ℏ^2

Harmonic oscillator

The quantum harmonic oscillator is another important system studied in wave mechanics because many physical systems can be approximated as harmonic oscillators. Its wave functions are described by Hermite polynomials, and its energy levels are quantized.

The Schrödinger equation for a harmonic oscillator with mass m and angular frequency ω is:

        En = ℏω(n + 1/2)

where n = 0, 1, 2, ...

The corresponding wave functions are given as:

        ψn(x) = (1/sqrt(2^nn!))(mω/ℏπ)^(1/4) exp(-mωx^2/(2ℏ)) Hn(sqrt(mω/ℏ) x)

Conclusion

Advanced wave mechanics provides profound insights into the behaviour of quantum systems. Understanding concepts such as wave-particle duality, the Schrödinger equation, and quantum tunnelling and quantisation lays the groundwork for even more complex quantum phenomena. The principles of advanced wave mechanics continue to prove indispensable as you explore various quantum systems and their applications.


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Advanced wave mechanics

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