Glossary

4.4 Expectation values and the Ehrenfest theorem

3 min read•august 9, 2024

Quantum mechanics gets real with expectation values and observables. These concepts help us predict what we'll measure in experiments, connecting the weird quantum world to our everyday experiences.

The bridges the gap between quantum and classical physics. It shows how quantum particles behave like classical objects on average, helping us understand the fuzzy line between these two realms.

Expectation Values and Observables

Understanding Expectation Values

Top images from around the web for Understanding Expectation Values

  • Hamiltonian Simulation by Qubitization – Quantum View original

    Is this image relevant?

  • Hamiltonian Simulation by Qubitization – Quantum View original

    Is this image relevant?

1 of 2

Top images from around the web for Understanding Expectation Values

  • Hamiltonian Simulation by Qubitization – Quantum View original

    Is this image relevant?

  • Hamiltonian Simulation by Qubitization – Quantum View original

    Is this image relevant?

1 of 2
  • represents the average outcome of measuring an in a quantum system
  • Calculated using the formula ⟨A⟩=∫−∞∞ψ∗Aψdx\langle A \rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
  • Provides statistical information about quantum systems
  • Depends on the wave function and the associated with the observable
  • Crucial for predicting experimental results in quantum mechanics
  • Can be time-dependent for systems not in stationary states

Observables and Operators in Quantum Mechanics

  • Observable refers to a physical quantity that can be measured in a quantum system
  • Examples include position, momentum, energy, and angular momentum
  • Each observable corresponds to a Hermitian operator in quantum mechanics
  • Operators act on wave functions to produce new wave functions
  • Hermitian operators ensure real-valued expectation values
  • Eigenvalues of operators represent possible measurement outcomes
  • Measurement process collapses the wave function into an eigenstate of the measured observable

Mathematical Properties of Operators

  • Linear operators obey the superposition principle
  • Commutator of two operators defined as [A,B]=AB−BA[A, B] = AB - BA
  • Non-commuting operators lead to uncertainty relations ()
  • Commonly used operators include position operator x^=x\hat{x} = x and p^=−iℏddx\hat{p} = -i\hbar \frac{d}{dx}
  • Energy operator (Hamiltonian) plays a central role in the Schrödinger equation

Ehrenfest Theorem and Time Evolution

Ehrenfest Theorem Fundamentals

  • Ehrenfest theorem connects quantum expectation values to classical equations of motion
  • Describes the time evolution of expectation values for position and momentum
  • Stated mathematically as: ddt⟨x⟩=⟨p⟩m\frac{d}{dt}\langle x \rangle = \frac{\langle p \rangle}{m} ddt⟨p⟩=−⟨dVdx⟩\frac{d}{dt}\langle p \rangle = -\langle \frac{dV}{dx} \rangle
  • Demonstrates that expectation values follow classical trajectories on average
  • Applies to systems with slowly varying potentials and wave packets with small spread

Time Evolution of Expectation Values

  • Time-dependent Schrödinger equation governs the evolution of quantum states
  • Expectation values change over time for non-stationary states
  • Rate of change of an expectation value given by: ddt⟨A⟩=1iℏ⟨[A,H]⟩+⟨∂A∂t⟩\frac{d}{dt}\langle A \rangle = \frac{1}{i\hbar}\langle [A, H] \rangle + \langle \frac{\partial A}{\partial t} \rangle
  • Commutator term represents quantum effects
  • Partial derivative term accounts for explicit time dependence of the operator
  • Time evolution preserves normalization of the wave function

Classical Limit and Correspondence Principle

Understanding the Classical Limit

  • occurs when quantum effects become negligible
  • Typically observed for large quantum numbers or macroscopic systems
  • Wave packets become highly localized in position and momentum space
  • Uncertainty principle becomes less significant relative to measured quantities
  • Quantum interference effects diminish
  • Schrödinger equation approaches the classical Hamilton-Jacobi equation
  • Examples include high-energy particles and large-scale objects (planets, baseballs)

Correspondence Principle and its Implications

  • Correspondence principle states that quantum mechanics must reproduce classical physics in appropriate limits
  • Formulated by as a guiding principle in developing quantum theory
  • Ensures consistency between quantum and classical descriptions
  • Manifests in various ways:
    • Energy levels become continuous in the limit of large quantum numbers
    • Angular momentum quantization approaches classical values for large l
    • Wave packets follow classical trajectories for macroscopic objects
  • Helps in understanding the transition between quantum and classical regimes
  • Provides a framework for interpreting quantum results in familiar classical terms

Key Terms to Review (17)

Albert Einstein: Albert Einstein was a theoretical physicist best known for developing the theory of relativity, which revolutionized the understanding of space, time, and energy. His contributions to quantum mechanics and the photoelectric effect laid the groundwork for modern physics, making him a pivotal figure in both classical and quantum theories.
Classical Limit: The classical limit refers to the conditions under which quantum mechanical systems behave like classical systems, typically as Planck's constant approaches zero or when quantum numbers become very large. This transition indicates that quantum effects become negligible, allowing the predictions of classical physics to emerge from quantum mechanics. Understanding the classical limit helps in bridging the gap between the two domains, showing how classical mechanics can be derived from quantum principles in certain situations.
Determinism: Determinism is the philosophical concept that every event or state of affairs, including human actions, is determined by preceding events in accordance with the natural laws. In the context of quantum mechanics, this idea is challenged as phenomena at microscopic scales often exhibit inherent randomness, leading to questions about predictability and control over systems.
Ehrenfest Theorem: The Ehrenfest Theorem is a fundamental result in quantum mechanics that establishes a connection between quantum mechanical expectation values and classical mechanics. It shows how the time evolution of expectation values of observables in quantum systems corresponds to their classical counterparts, providing a bridge between quantum and classical physics. This theorem implies that under certain conditions, the average behavior of quantum systems can be described using classical equations of motion.
Expectation Value: The expectation value, denoted as ⟨a⟩, is a fundamental concept in quantum mechanics that represents the average value of a physical quantity 'a' in a given quantum state described by the wave function ψ. This average is calculated using the integral $$⟨a⟩ = ∫ψ^* a ψ \, dx$$, where ψ* is the complex conjugate of ψ. This concept connects to important principles like the Ehrenfest theorem, which links quantum mechanics to classical mechanics by showing how the expectation values of observables evolve over time.
Heisenberg Uncertainty Principle: The Heisenberg 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 in measuring quantum systems, emphasizing that the more accurately one property is known, the less accurately the other can be determined. It challenges classical notions of measurement and reality, connecting deeply with various quantum phenomena and measurement issues.
Hilbert Space: Hilbert space is a fundamental concept in quantum mechanics, representing a complete inner product space that provides the framework for quantum states and operators. It allows for the mathematical description of quantum systems using vectors and enables the calculation of probabilities, expectation values, and dynamics through linear algebra. The properties of Hilbert space make it essential for understanding various phenomena in quantum mechanics, including state representation, observables, and interpretations.
Linear Algebra: Linear algebra is a branch of mathematics focused on vector spaces and the linear mappings between them, involving concepts such as vectors, matrices, and systems of linear equations. It plays a crucial role in quantum mechanics by providing the mathematical framework for understanding state vectors and operators, which are essential for calculating expectation values and applying the Ehrenfest theorem.
Mean Value Theorem: The Mean Value Theorem is a fundamental concept in calculus that states if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point where the derivative of the function is equal to the average rate of change over that interval. This theorem connects to expectation values and the Ehrenfest theorem as it helps illustrate the behavior of quantum mechanical observables and their expected values over time.
Measurement Problem: The measurement problem refers to the fundamental issue in quantum mechanics regarding how and when quantum systems transition from a superposition of states to a single definite outcome upon measurement. This problem highlights the discrepancy between the linear evolution of quantum states described by the wave function and the seemingly instantaneous 'collapse' to a particular state when an observation occurs, leading to profound implications for understanding reality.
Momentum operator: The momentum operator is a fundamental concept in quantum mechanics, represented as \\$ \hat{p} = -i\hbar \frac{d}{dx} \\$ in one dimension. It plays a crucial role in the formulation of quantum mechanics by relating to the measurement of momentum and influencing the behavior of wave functions. This operator is essential for understanding how different observables, like position and momentum, interact with one another through commutation relations, and it is integral to calculating expectation values, as well as illustrating the correspondence between classical and quantum mechanics.
Niels Bohr: Niels Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, particularly with his model of the hydrogen atom. His work established crucial principles that paved the way for modern quantum mechanics, influencing various topics related to wave-particle duality, measurement, and the behavior of particles in potential wells.
Observable: An observable is a physical quantity in quantum mechanics that can be measured, represented mathematically by a Hermitian operator. This concept connects to the framework of Hilbert space and state vectors, allowing us to describe the possible outcomes of measurements and their probabilities. Observables are crucial for determining expectation values and understanding how a quantum system evolves over time, as highlighted by the Ehrenfest theorem.
Operator: In quantum mechanics, an operator is a mathematical object that represents a physical observable or action that can be performed on a quantum state. Operators are essential for calculating expectation values and determining how quantum systems evolve over time, playing a crucial role in linking measurable quantities to the underlying wave functions of particles.
Probability Amplitude: Probability amplitude is a complex number associated with the likelihood of finding a quantum system in a particular state. This concept plays a crucial role in connecting the mathematical formulation of quantum mechanics to physical predictions, highlighting how it influences measurement outcomes, expectation values, and wave function interpretations.
Statistical Interpretation: Statistical interpretation refers to the understanding that quantum mechanics provides probabilities for different outcomes rather than certainties. In quantum mechanics, this idea is crucial as it connects the mathematical framework of wave functions and operators with observable quantities, illustrating that measurements yield results defined by probability distributions rather than definite values.
Wave Function Collapse: Wave function collapse refers to the process by which a quantum system transitions from a superposition of multiple possible states to a single, definite state upon measurement. This concept is crucial in understanding how quantum systems behave during observation, highlighting the distinction between the probabilities described by the wave function and the actual outcomes observed in experiments.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide