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 ↓


Quantum harmonic oscillator and coherent states


The quantum harmonic oscillator is one of the most important model systems in quantum mechanics. It is a model that describes a particle subject to a restoring force proportional to its displacement from an equilibrium position. It is an idealized system in which the particle oscillates back and forth, much like a weight on a spring or a pendulum. In the quantum realm, it serves as the basis for understanding more complex systems including fields and particles in advanced wave mechanics.

Basics of Quantum Harmonic Oscillator

The classical harmonic oscillator can be represented by the following second-order differential equation:

m * d²x/dt² = -k * x

where m is the mass of the particle, k is the spring constant, and x is the displacement from equilibrium.

Quantum mechanical description

To obtain the quantum mechanical description, we use the Schrödinger equation. The Hamiltonian for the harmonic oscillator is given as:

H = p²/(2m) + (1/2) * m * ω² * x²

where p is the momentum operator, ω is the angular frequency, and x is the position operator.

In quantum mechanics, we solve the time-independent Schrödinger equation:

Hψ = Eψ

where ψ is the wave function and E is the energy eigenvalue.

Energy levels of a quantum harmonic oscillator

Solving the Schrödinger equation for a harmonic oscillator, we find that the energies are quantized and given by:

E_n = (n + 1/2)ħω

where n is a non-negative integer (quantum number), ħ is the reduced Planck constant, and ω is the angular frequency. In these energy levels, we see the presence of zero-point energy, (1/2)ħω, which means that the oscillator always has some minimum energy, even at the ground state (n=0).

Wave functions of the quantum harmonic oscillator

The wave functions or state functions for harmonic oscillators are Hermite polynomials multiplied by a Gaussian factor. They can be expressed as:

ψ_n(x) = N_n * H_n(ξ) * exp(-ξ²/2)

where H_n(ξ) are the Hermite polynomials, ξ = (mω/ħ)¹⁄² * x, and N_n is a standardization factor. The Hermite polynomials H_n(x) can be indexed by the quantum number n.

Generalization and orthogonality

The wave functions are perpendicular to each other and can be generalized as follows:

∫ψ*_n(x)ψ_m(x) dx = δ_nm

where δ_nm is the Kronecker delta (1 if n=m, 0 otherwise).

Example of Hermite polynomials

The first few Hermite polynomials are:

H_0(x) = 1
H_1(x) = 2x
H_2(x) = 4x² - 2
H_3(x) = 8x³ - 12x

Coherent state

Coherent states are a special type of quantum state of a harmonic oscillator. They are particularly interesting because they more closely resemble classical states than energy eigenstates.

Definition of coherent states

Coherent states are defined as eigenstates of the annihilation operator a:

a |α⟩ = α |α⟩

where α is a complex number and |α⟩ is a coherent state. The annihilation operator a is related to the position and momentum operators as follows:

a = (mω/2ħ)¹⁄²(x + (i/ω)p)

Properties of coherent states

Coherent states have several important properties:

  1. Normalization: Coherent states are normalized such that ⟨α|α⟩ = 1.
  2. Minimal uncertainty: they satisfy the minimal uncertainty relation, making them as close to classical states as possible.
  3. Overlap and completeness: The overlap between two consistent states is given by: ⟨β|α⟩ = exp(−|β|²/2) exp(−|α|²/2) exp(β*α).

Visualization using circles in the complex plane

α_actual α_Image

In the complex plane, a coherent state can be viewed as a point (α), where the real and imaginary parts represent different aspects of the oscillatory motion.

Time evolution of coherent states

Another fascinating property of coherent states is their time evolution. Under the evolution determined by the harmonic oscillator Hamiltonian:

H = ħω(a†a + 1/2)

The coherent state |α⟩ evolves as follows:

|α(t)⟩ = |α e^(iωt)⟩

This means that the state rotates in the complex plane, but its shape and size remain unchanged, indicating the stability of coherent states against time evolution.

Physical applications and significance

Quantum harmonic oscillators and coherent states have many applications in physics:

  1. Quantum optics: Coherent states model laser light, which exhibits properties similar to classical electromagnetic waves.
  2. Quantum field theory: The fundamental particles and fields in quantum field theory use the concepts of harmonic oscillators.
  3. Molecular Physics: Vibrational modes of molecules are analyzed using quantum harmonic oscillators.

Example - Simple Pendulum Model

Imagine a pendulum swinging back and forth. When its amplitude is small, its behavior can be approximated by a harmonic oscillator. Energy levels will be quantized according to the quantum harmonic oscillator solution. Each quantized state can be viewed as a different swing or motion type with a specific energy.

Conclusion

Quantum harmonic oscillators and coherent states provide important insights into quantum mechanics and bridge the gap between quantum and classical physics. Understanding these concepts is essential for advanced studies in quantum mechanics, making them important for theoretical physics and beyond.


Graduate → 4.1.3


U
username
0%
completed in Graduate

← Prev (4.1.2)
Path integral formulation
Next (4.2) →
Angular momentum and spin

Comments 



Quantum harmonic oscillator and coherent states

https://www.physicsflow.com/g/4.1.3?pfal=en