Glossary

13.2 Propagators and transition amplitudes

2 min read•july 25, 2024

Quantum propagators are the mathematical wizards of particle evolution. They describe how quantum states change over time, connecting initial and final positions through a sum of all possible paths.

Propagators have practical applications in calculating transition probabilities and solving differential equations. By linking to Green's functions and utilizing Fourier transforms, they provide powerful tools for analyzing quantum systems and predicting particle behavior.

Propagator Fundamentals

Definition of propagator

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  • mathematically represents quantum evolution describing probability amplitude for particle travel between two spacetime points
  • Path integral formulation connects propagator by summing all possible paths between initial and final states aligning with Feynman's quantum mechanics interpretation
  • Mathematical expression K(xf,tf;xi,ti)=⟨xf∣e−iH(tf−ti)/ℏ∣xi⟩K(x_f, t_f; x_i, t_i) = \langle x_f | e^{-iH(t_f-t_i)/\hbar} | x_i \rangle encapsulates quantum state evolution
  • Properties include unitarity preserving probability and composition rule for sequential propagation (scattering, tunneling)

Propagator calculations for particles

  • derived using path integral yields K(xf,tf;xi,ti)=m2πiℏ(tf−ti)eim(xf−xi)2/2ℏ(tf−ti)K(x_f, t_f; x_i, t_i) = \sqrt{\frac{m}{2\pi i\hbar (t_f-t_i)}} e^{im(x_f-x_i)^2/2\hbar(t_f-t_i)}
  • calculated via operator method results in K(xf,tf;xi,ti)=mω2πiℏsin⁡(ω(tf−ti))eimω2ℏsin⁡(ω(tf−ti))[(xf2+xi2)cos⁡(ω(tf−ti))−2xfxi]K(x_f, t_f; x_i, t_i) = \sqrt{\frac{m\omega}{2\pi i\hbar \sin(\omega(t_f-t_i))}} e^{i\frac{m\omega}{2\hbar \sin(\omega(t_f-t_i))}[(x_f^2+x_i^2)\cos(\omega(t_f-t_i))-2x_fx_i]}
  • These expressions enable analysis of quantum behavior in various systems (particle in a box, potential well)

Propagator Applications

Propagator and Green's function connection

  • solves inhomogeneous differential equations relating to propagator through time-ordered equivalence
  • Relationship expressed as G(xf,tf;xi,ti)=−iθ(tf−ti)K(xf,tf;xi,ti)G(x_f, t_f; x_i, t_i) = -i\theta(t_f-t_i)K(x_f, t_f; x_i, t_i) links two fundamental concepts
  • Spectral representation expands Green's function in energy eigenstates revealing system's energy structure
  • bridges time-domain and energy-domain representations facilitating different analysis approaches

Propagator in transition probabilities

  • calculated using ⟨ψf∣U(tf,ti)∣ψi⟩=∫dxfdxiψf∗(xf)K(xf,tf;xi,ti)ψi(xi)\langle \psi_f | U(t_f, t_i) | \psi_i \rangle = \int dx_f dx_i \psi_f^*(x_f) K(x_f, t_f; x_i, t_i) \psi_i(x_i) quantifies quantum state changes
  • Probability determined by squaring transition amplitude magnitude P=∣⟨ψf∣U(tf,ti)∣ψi⟩∣2P = |\langle \psi_f | U(t_f, t_i) | \psi_i \rangle|^2
  • Applications include analyzing scattering processes and calculating decay rates in particle physics
  • Time-dependent utilizes propagator expansion for weak interactions enabling approximations in complex systems

Key Terms to Review (18)

Classical limit: The classical limit refers to the transition from quantum mechanics to classical mechanics, where quantum effects become negligible and classical descriptions of physical systems take precedence. This concept is crucial in understanding how macroscopic behaviors emerge from quantum systems, bridging the gap between two fundamental theories of physics. In this context, the classical limit helps in interpreting phenomena through familiar frameworks, such as position and momentum, while still utilizing the foundational principles of quantum mechanics.
Correspondence Principle: The correspondence principle is a concept that asserts that the predictions of quantum mechanics must converge to those of classical mechanics in the appropriate limit, typically as quantum numbers become large. This principle bridges the gap between the two frameworks, ensuring that quantum theory aligns with classical physics in scenarios where classical mechanics is expected to hold true, thus providing a consistent understanding across different physical regimes.
David Bohm: David Bohm was a theoretical physicist known for his significant contributions to quantum mechanics and his philosophical interpretations of the theory. His work, particularly on the causal interpretation of quantum mechanics, challenges traditional views and connects deeply with concepts like nonlocality and the role of the observer in quantum systems.
Feynman Path Integral: The Feynman Path Integral is a formulation of quantum mechanics that sums over all possible paths a particle can take between two points, assigning a probability amplitude to each path based on its action. This approach connects quantum mechanics to classical mechanics through the principle of least action and provides a framework for understanding various phenomena in physics, including propagators and statistical mechanics.
Fourier Transform: The Fourier transform is a mathematical operation that transforms a function of time (or space) into a function of frequency, essentially breaking down signals into their constituent frequencies. This tool is crucial for analyzing physical systems, as it connects time-domain and frequency-domain representations, enabling various applications in physics, engineering, and applied mathematics.
Free particle propagator: The free particle propagator is a mathematical tool used in quantum mechanics that describes how a quantum state evolves over time for a free particle, meaning one that is not subject to any forces. It essentially represents the amplitude for a particle to travel from one point in space and time to another. This propagator plays a critical role in connecting initial and final states in calculations involving transition amplitudes, showcasing the dynamics of particles in quantum systems.
Green's Function: A Green's function is a mathematical tool used to solve inhomogeneous differential equations, providing a way to express the solution in terms of the source or forcing function. It serves as a fundamental solution that relates the effects of point sources to the resulting field, making it crucial for analyzing various physical systems in classical and quantum mechanics.
Harmonic oscillator propagator: The harmonic oscillator propagator is a mathematical function that describes how the state of a quantum harmonic oscillator evolves over time. This propagator is essential for calculating transition amplitudes between different states in quantum mechanics, allowing for the analysis of how a system behaves under various conditions.
Operator Formalism: Operator formalism is a mathematical framework used in quantum mechanics where physical quantities are represented as operators acting on state vectors in a Hilbert space. This approach allows for the systematic handling of quantum states and observables, highlighting the importance of linear transformations and eigenvalues in determining measurable properties of a system. By employing operator formalism, one can derive propagators and transition amplitudes, as well as understand the implications of quantum measurement and decoherence.
Perturbation theory: Perturbation theory is a mathematical approach used to find an approximate solution to a problem that cannot be solved exactly, by starting with a known exact solution and adding small changes, or perturbations, to it. This method is particularly valuable in systems where the Hamiltonian can be expressed as a sum of an exactly solvable part and a small perturbative term, allowing for analysis of how these small changes affect the system's behavior and solutions.
Propagator: A propagator is a mathematical object that describes how a physical system evolves over time, linking initial and final states in quantum mechanics and field theory. It acts as a transition amplitude, providing a way to calculate the probability of a particle moving from one point to another, while encapsulating the effects of interactions in the system. Propagators play a crucial role in connecting quantum states with physical observables and can be represented visually through Feynman diagrams.
Quantum Field Theory: Quantum Field Theory (QFT) is a fundamental framework in physics that combines classical field theory, special relativity, and quantum mechanics to describe how particles interact and propagate. It provides a way to understand particle physics through the concepts of fields and their quantization, leading to a better grasp of forces and particles at the quantum level. This theory forms the backbone of modern particle physics and connects closely with concepts like Green's functions, propagators, and transition amplitudes, illustrating how particles move and interact in a quantized field.
Quantum superposition: Quantum superposition is the principle that a quantum system can exist in multiple states simultaneously until it is measured. This concept highlights that particles, like electrons, can be in various configurations or positions at once, leading to interference effects and a range of potential outcomes. The idea of superposition is crucial for understanding complex phenomena in quantum mechanics, influencing various concepts like transition amplitudes, mixed states, and information processing.
Richard Feynman: Richard Feynman was a renowned American theoretical physicist known for his contributions to quantum mechanics and quantum electrodynamics. His innovative ideas, such as the path integral formulation, revolutionized our understanding of particle physics and made complex concepts more accessible through intuitive visuals like Feynman diagrams.
Scattering theory: Scattering theory is a framework in quantum mechanics that describes how particles (such as photons or electrons) interact with each other and scatter as a result of these interactions. It provides a way to analyze and predict the outcomes of these interactions, particularly focusing on the transition amplitudes, which describe the probability of a particle transitioning from one state to another after scattering.
Time-evolution operator: The time-evolution operator is a mathematical construct that describes how a quantum state changes over time. It encapsulates the dynamics of the system according to the Schrödinger equation, allowing us to relate the state of a quantum system at one time to its state at another. This operator plays a crucial role in connecting quantum mechanics to observable phenomena through propagators and transition amplitudes.
Transition Amplitude: Transition amplitude refers to the probability amplitude for a quantum system to transition from one state to another, typically expressed in terms of wave functions or state vectors. This concept is fundamental in quantum mechanics, where it connects the mathematical framework of Dirac notation and matrix representations to physical processes like particle interactions and propagators. Transition amplitudes provide essential information about how likely a system is to evolve from an initial state to a final state over a given time interval.
Uncertainty Principle: The uncertainty principle is a fundamental concept in quantum mechanics that states it is impossible to simultaneously know both the exact position and exact momentum of a particle. This principle highlights the intrinsic limitations of measurement and introduces a fundamental limit to precision, connecting deeply to the behavior of observables and the wave function of quantum systems.
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