Fundamentals of Quantum Mechanics

Working note — Part of my personal research notebook. Entries may be incomplete and are revised over time.

• physical system: In QM, a physical system typically consists of one or more quantum particles (e.g., electrons, protons, atoms) or a collection of such particles.
  • single electron in a hydrogen atom.
  • system of two entangled particles (e.g., an electron and a positron).
  • particle in a potential well (e.g., a harmonic oscillator).
  • In relativistic QM (e.g., the Dirac equation), the physical system may also include antiparticles and phenomena like particle creation and annihilation.
• state of the physical system: state of a physical system in QM is described by a wave function $\ket{\psi}$ (or $\psi(x,t)$ in the position representation) or a state vector in a Hilbert space. This state encodes all the probabilistic information about the system’s properties.
  • The system is described by its wave function or state vector, which encodes all the information about the system’s properties.
  • states in QM:
    • stationary state (eigenstate of the Hamiltonian) with definite energy
    • superposition state, where the system is in a combination of multiple states.
    • entangled state, where the properties of two or more particles are correlated.
• Physical law: the fundamental principles and equations that govern the behavior of quantum systems. These laws are derived from the postulates of quantum mechanics and the specific form of the Hamiltonian or wave equation.
  • Key physical laws in QM include:
    • Schrödinger equation (non-relativistic QM) or the Klein-Gordon and Dirac equations (relativistic QM), which describe the time evolution of the state.
    • uncertainty principle, which states that certain pairs of physical properties (e.g., position and momentum) cannot be simultaneously measured with arbitrary precision.
    • superposition principle, which allows quantum states to be combined linearly.
    • Born rule, which provides the probabilistic interpretation of the wave function.
    • commutation relations between operators (e.g., $[x,p]=i\hbar$), which define the algebraic structure of quantum observables.
  • In relativistic QM, additional laws include:
    • Lorentz invariance, which ensures that the equations of motion are consistent with special relativity.
    • the existence of antiparticles and the possibility of particle creation and annihilation.