Glossary

14.1 Time evolution operator and Schrödinger picture

2 min read•july 25, 2024

Quantum mechanics uses the to predict how particles change over time. This mathematical tool connects a system's initial state to its future state, solving the and revealing quantum behavior.

The operator's properties, like unitarity and composition, help physicists understand quantum dynamics. It's key for grasping concepts like and quantum tunneling, making it essential for studying particle behavior in various systems.

Time Evolution Operator

Time evolution operator in quantum dynamics

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  • Time evolution operator U(t,t0)U(t, t_0) maps initial quantum state at t0t_0 to final state at tt
  • Describes quantum state changes over time formally solving time-dependent Schrödinger equation
  • Relates to through ∣ψ(t)⟩=U(t,t0)∣ψ(t0)⟩|\psi(t)\rangle = U(t, t_0)|\psi(t_0)\rangle
  • Functions as quantum propagator connecting states at different times
  • Enables prediction of future quantum system behavior (wave function collapse, quantum tunneling)

Schrödinger equation for state evolution

  • Time-dependent Schrödinger equation governs quantum state evolution: iℏ∂∂t∣ψ(t)⟩=H∣ψ(t)⟩i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle = H|\psi(t)\rangle
  • HH represents total system energy (kinetic + potential)
  • Time-independent Hamiltonians yield general solution U(t,t0)=e−iH(t−t0)/ℏU(t, t_0) = e^{-iH(t-t_0)/\hbar}
  • Energy eigenstate expansion gives ∣ψ(t)⟩=∑ncne−iEnt/ℏ∣En⟩|\psi(t)\rangle = \sum_n c_n e^{-iE_nt/\hbar}|E_n\rangle
  • Phase factors e−iEnt/ℏe^{-iE_nt/\hbar} determine time evolution of eigenstates

Solutions for time-dependent Schrödinger equations

  • Separation of variables technique solves many simple quantum systems
  • Stationary states evolve with constant energy, changing only in phase
  • Simple systems include:
    • Free particle (plane waves)
    • Particle in a box (standing waves)
    • Harmonic oscillator (Hermite polynomials)
  • Sudden perturbations cause instantaneous changes in quantum state
  • Adiabatic approximation applies for slowly varying Hamiltonians (quantum )

Properties of time evolution operator

  • Unitarity of U(t,t0)U(t, t_0) preserves norm and probability (U†U=UU†=IU^\dagger U = UU^\dagger = I)
  • Composition property combines successive evolutions: U(t2,t0)=U(t2,t1)U(t1,t0)U(t_2, t_0) = U(t_2, t_1)U(t_1, t_0)
  • Inverse relation allows backward evolution: U−1(t,t0)=U(t0,t)U^{-1}(t, t_0) = U(t_0, t)
  • Time-reversal symmetry explores behavior under t→−tt \rightarrow -t (T-symmetry)
  • Group structure of time evolution operators forms unitary group
  • Connects to conservation laws (energy conservation for time-independent Hamiltonians)

Key Terms to Review (18)

⟨φ|: The notation ⟨φ| represents a bra vector in the context of quantum mechanics, specifically within the framework of Dirac notation. It is used to denote a linear functional that acts on a ket vector |ψ⟩, providing a way to calculate inner products and probabilities in quantum systems. This concept is crucial for understanding the mathematical structure of quantum mechanics, especially when discussing state vectors and their evolution over time.
|ψ(t)⟩: |ψ(t)⟩ represents the state of a quantum system at time t, encapsulating all the information about the system's physical properties. This notation is part of the Schrödinger picture, where states evolve in time while operators remain fixed. The time-dependent state vector is crucial for understanding how quantum systems change over time, making it a fundamental concept in quantum mechanics and its mathematical framework.
Adiabatic Theorem: The adiabatic theorem states that a quantum system remains in its instantaneous eigenstate if the Hamiltonian changes slowly enough compared to the energy differences between its eigenstates. This principle is crucial in understanding how systems evolve over time, particularly in contexts involving adiabatic processes and changes in external conditions.
Conservation of Probability: Conservation of probability is a fundamental principle in quantum mechanics stating that the total probability of finding a quantum system within a certain state is always equal to one. This principle ensures that the probabilities calculated from a wave function remain consistent over time, reflecting the physical reality that a particle must exist somewhere in its defined space. It connects deeply to the behaviors observed in potential wells and barriers, as well as the time evolution of quantum states.
David Hilbert: David Hilbert was a German mathematician known for his foundational contributions to various fields, including algebra, number theory, and mathematical logic. His work laid the groundwork for many concepts in modern mathematics and influenced the development of quantum mechanics, particularly through his formulation of Hilbert spaces, which are essential in the mathematical formulation of quantum theory and the time evolution operator.
Erwin Schrödinger: Erwin Schrödinger was an influential Austrian physicist known for his pivotal contributions to quantum mechanics, particularly through the formulation of the Schrödinger equation, which describes how quantum systems evolve over time. His work laid the foundation for understanding wave-particle duality and the behavior of particles at a quantum level, connecting various essential aspects of quantum mechanics including time dependence and perturbation theory.
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.
Hamiltonian Operator: The Hamiltonian operator is a crucial mathematical entity in quantum mechanics that represents the total energy of a system, encompassing both kinetic and potential energy. It plays a vital role in both the time-dependent and time-independent Schrödinger equations, allowing the description of how quantum states evolve over time. This operator is often denoted as \( \hat{H} \) and is fundamental in the formulation of the time evolution operator and understanding the dynamics of quantum systems.
Heisenberg Picture: The Heisenberg Picture is one of the formulations of quantum mechanics, where the state vectors remain constant over time while the operators evolve with time. In this framework, the focus shifts from the state of a system to how observable quantities change over time, allowing a different perspective on the dynamics of quantum systems compared to the Schrödinger Picture. This formulation emphasizes the importance of observables and their time evolution, making it particularly useful in certain contexts.
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.
Schrödinger Equation: The Schrödinger Equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It plays a crucial role in understanding the behavior of particles at the quantum level, allowing us to connect wave functions to observable properties like energy and momentum.
Schrödinger Picture: The Schrödinger Picture is one of the formulations of quantum mechanics where the state vectors evolve in time while the operators remain constant. This contrasts with other formulations like the Heisenberg Picture, where operators evolve and state vectors remain static. In this picture, the dynamics of a quantum system can be described using the time evolution operator, which connects the system's state at one time to its state at another.
State Vector: A state vector is a mathematical representation of the quantum state of a physical system, encapsulating all the information necessary to describe the system's properties and behaviors. It is typically represented as a column vector in a complex vector space, and its components correspond to the probabilities of finding the system in various states upon measurement. The state vector is crucial in connecting concepts like linear transformations, operators, and the time evolution of quantum systems.
Stone's Theorem: Stone's Theorem, in the context of quantum mechanics, establishes a connection between the time evolution of quantum states and the structure of their underlying Hilbert space. It shows that if you have a self-adjoint operator representing an observable in quantum mechanics, the time evolution operator can be expressed in terms of that operator, demonstrating how quantum states evolve over time within the Schrödinger picture.
Superposition Principle: The superposition principle states that in a linear system, the net response at a given time or position is the sum of the individual responses from all influencing factors. This principle is central to understanding wave functions and states in quantum mechanics, where a particle can exist simultaneously in multiple states until measured, allowing for complex behaviors like interference and entanglement.
Time evolution operator: The time evolution operator is a mathematical construct that describes how the state of a quantum system changes over time. It encapsulates the dynamics of the system governed by the Schrödinger equation and can be used to relate the state of the system at one time to its state at another. Understanding this operator is crucial for analyzing phenomena like perturbation theory, transitioning between different representations, and describing mixed states in quantum mechanics.
Unitary Operator: A unitary operator is a linear operator on a Hilbert space that preserves the inner product, ensuring that the norm of a vector is maintained. This property makes it crucial for the description of quantum systems, as it guarantees the conservation of probabilities and allows for the transformation of states in a way that respects orthogonality. The connection between unitary operators and inner product spaces emphasizes their role in defining the geometric structure of quantum mechanics, while their use as time evolution operators highlights their importance in understanding how quantum states change over time.
Wave function collapse: Wave function collapse refers to the process by which a quantum system transitions from a superposition of states to a single definite state when a measurement is made. This phenomenon is crucial for understanding how quantum mechanics describes the behavior of particles and their properties during observation, highlighting the difference between quantum systems' predictions and classical outcomes.
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