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 → Statistical mechanics and thermodynamics ↓


Quantum statistical mechanics


Quantum statistical mechanics is an essential topic in physics that combines quantum mechanics and statistical mechanics principles to describe the behavior of systems containing large numbers of particles. It provides a comprehensive framework for understanding how quantum systems behave in thermodynamic equilibrium. In this lesson, we will discuss in depth the key principles, formulas, and examples that illustrate the fascinating world of quantum statistical mechanics.

Fundamental concepts

To understand quantum statistical mechanics, we first need to revisit some fundamental concepts in quantum mechanics and statistical mechanics. Quantum mechanics describes the physical properties of nature on the scale of atoms and subatomic particles. Statistical mechanics, on the other hand, provides a bridge between microscopic laws and macroscopic phenomena, dealing with the behavior of systems with a large number of components.

Quantum states

In quantum mechanics, the state of a system is represented by a wave function, usually denoted as Ψ (Psi). Each system or particle is described by a set of quantum numbers that determine its state. The wave function contains all measurable information about a system.

|Ψ|^2 = probability density

The probability density |Ψ|^2 gives the probability of finding a particle at a particular location in space. The wave function is subject to the Schrödinger equation, a fundamental equation of quantum mechanics that governs how a quantum state evolves over time.

Schrödinger equation

The time-dependent Schrödinger equation is written as:

iħ ∂Ψ/∂t = HΨ

where i is the imaginary unit, ħ is the reduced Planck constant, ∂Ψ/∂t is the partial derivative of the wave function with respect to time, and H is the Hamiltonian operator, representing the total energy of the system.

From classical to quantum statistical mechanics

In classical statistical mechanics, systems are described in terms of groups – large collections of systems in thermal equilibrium. Quantum statistical mechanics extends this further by incorporating principles of quantum mechanics. The main difference is that where classical mechanics treats energy states as continuous, quantum mechanics treats them as discrete, which fundamentally changes the way entropy and partition functions are calculated.

Quantum ensembles

In quantum statistical mechanics, the states of a system are described by quantum ensembles. The most common ensembles are microcanonical, canonical, and grand canonical ensembles.

Microcanonical ensembles

The microcanonical ensemble is used to describe an isolated system with fixed energy, volume, and number of particles. In this ensemble, each state has equal probability, which reflects the constrained configuration of the system:

W = Ω(E, V, N)

where Ω(E, V, N) is the number of quantum states available to the system at energy E, volume V and particle number N

Canonical ensemble

The canonical group describes a system in thermal equilibrium at a certain temperature T The probability of a system being in a state with energy E_i is given by the Boltzmann distribution:

P(E_i) = e^(-E_i/kT) / Z

where k is the Boltzmann constant, and Z is the partition function defined as:

Z = Σ e^(-E_i/kT)

Grand canonical ensemble

The grand canonical ensemble is used when the system can exchange energy and particles with a reservoir. Here the probability is determined by both the temperature and the chemical potential:

P(E_i, N_i) = e^(-(E_i - μN_i)/kT) / Ξ

where μ is the chemical potential and Ξ is the grand partition function:

Ξ = Σ e^(-(E_i - μN_i)/kT)

Density matrix and quantum statistics

In quantum statistical mechanics, the density matrix or density operator ρ is used to describe the statistical state of a quantum system. It contains all the information about the statistical behavior of the system, which is important for distinguishing between mixed and pure states:

ρ = Σ P_i |Ψ_i⟩⟨Ψ_i|

where P_i is the probability of the system being in the quantum state |Ψ_i⟩.

Applications and examples

Quantum statistical mechanics provides essential tools for understanding a variety of physical systems, from ideal gases to complex quantum fluids such as Bose–Einstein condensates.

Bose–Einstein condensate

Predicting the behaviour and properties of Bose-Einstein condensates (BECs) is a notable application. BECs form when particles known as bosons are cooled to temperatures close to absolute zero, causing them to occupy a single quantum state, a notable area of study in quantum physics. Here, bosons exhibit a macroscopic quantum phenomenon.

Fermi–Dirac statistics

Another example involves Fermi-Dirac statistics, which describes the distribution of fermions - electron-like particles that obey the Pauli exclusion principle. The statistics of fermions plays an important role in understanding the electronic configurations of atoms, the behavior of electrons in metals, and quantum gases.

Conclusion

Quantum statistical mechanics provides an important framework to combine quantum mechanics and statistical theories to analyze the behavior of systems containing many interacting particles. It helps in exploring many quantum phenomena encountered in various applications in physics, chemistry, and beyond. Through applications such as ensembles, the Schrödinger equation, density matrices, and Bose-Einstein condensates and Fermi-Dirac statistics, quantum statistical mechanics provides a deeper understanding of how quantum systems behave on a macroscopic, observable scale.


Graduate → 3.2


U
username
0%
completed in Graduate

← Prev (3.1.4)
Critical events and phase transitions
Next (3.2.1) →
Density Matrix and Ensemble Theory

Comments 



Quantum statistical mechanics

https://www.physicsflow.com/g/3.2?pfal=en