Glossary

11.4 WKB approximation and semiclassical methods

3 min read•july 25, 2024

The is a powerful tool in quantum mechanics, bridging classical and quantum worlds. It finds approximate solutions to the Schrödinger equation when the potential varies slowly compared to the particle's wavelength, offering insights into complex systems.

WKB shines in semiclassical scenarios, providing intuitive understanding of quantum phenomena in classical terms. It's particularly useful for systems where exact solutions are elusive, like complex atoms and molecules, making it a valuable technique in various physics fields.

WKB Approximation Fundamentals

WKB approximation in semiclassical mechanics

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  • WKB approximation finds approximate solutions to linear differential equations applies to time-independent Schrödinger equation
  • Semiclassical approach bridges classical and quantum mechanics used when de Broglie wavelength is small compared to system's characteristic length
  • Assumes slowly varying potential compared to particle wavelength applies in high-energy limit or short-wavelength approximation
  • Developed independently by Wentzel, Kramers, and Brillouin in 1926 provides intuitive understanding of quantum phenomena in classical terms
  • Useful for systems where exact solutions are difficult or impossible to obtain (complex atoms, molecules)

Application of WKB to Schrödinger equation

  • General form of WKB solution: ψ(x)≈Aexp⁡(±iℏ∫xp(x′)dx′)\psi(x) \approx A \exp(\pm \frac{i}{\hbar} \int^x p(x') dx') where p(x)=2m(E−V(x))p(x) = \sqrt{2m(E-V(x))} is classical momentum
  • Apply WKB approximation:
    1. Identify classically allowed and forbidden regions
    2. Write general solution in each region
    3. Match solutions at classical
  • Approximation for wave function:
    • Classically allowed region: ψ(x)≈Ap(x)cos⁡(1ℏ∫xp(x′)dx′−π4)\psi(x) \approx \frac{A}{\sqrt{p(x)}} \cos(\frac{1}{\hbar} \int^x p(x') dx' - \frac{\pi}{4})
    • Classically forbidden region: ψ(x)≈B∣p(x)∣exp⁡(±1ℏ∫x∣p(x′)∣dx′)\psi(x) \approx \frac{B}{\sqrt{|p(x)|}} \exp(\pm \frac{1}{\hbar} \int^x |p(x')| dx')
  • Higher-order corrections obtained by including additional terms in WKB series expansion improve accuracy for complex potentials

Connection Formulas and Applications

Connection formulas and quantization conditions

  • Connection formulas link solutions across classical turning points ensure continuity of wave function and its derivative
  • : ∫x1x2p(x)dx=(n+12)πℏ\int_{x_1}^{x_2} p(x) dx = (n + \frac{1}{2})\pi\hbar where x1x_1 and x2x_2 are classical turning points and nn is non-negative integer
  • semiclassical analog of WKB quantization condition: ∮pdq=2πℏ(n+12)\oint p dq = 2\pi\hbar(n + \frac{1}{2})
  • Applications include determining energy level spacing in potential wells (harmonic oscillator) and calculating tunneling probabilities through potential barriers (alpha decay)

Validity of WKB in potential landscapes

  • Validity conditions: ∣dλdx∣≪1|\frac{d\lambda}{dx}| \ll 1, where λ=hp(x)\lambda = \frac{h}{p(x)} is local de Broglie wavelength equivalent to ∣dVdx∣≪∣2(E−V)3/2ℏ2m∣|\frac{dV}{dx}| \ll |\frac{2(E-V)^{3/2}}{\hbar\sqrt{2m}}|
  • WKB approximation fails near classical turning points where p(x)=0p(x) = 0 requires special treatment (Airy function approximation)
  • Accuracy varies in different potential landscapes:
    • Smooth, slowly varying potentials: generally good accuracy (Morse potential)
    • Rapidly varying potentials: poor accuracy (square well)
    • Discontinuous potentials: invalid near discontinuities (step potential)
  • Compares well with exact solutions for harmonic oscillator at high quantum numbers and Coulomb potential for high angular momentum states

Other semiclassical methods in quantum mechanics

  • Semiclassical propagator (Van Vleck-Gutzwiller propagator) connects initial and final states using classical trajectories
  • Periodic orbit theory relates quantum energy levels to classical periodic orbits useful in studying quantum chaos (stadium billiard)
  • Eikonal approximation high-energy limit of WKB approximation applied in scattering theory (nuclear physics)
  • Semiclassical coherent states minimum uncertainty wave packets bridge classical and quantum descriptions of phase space
  • generalizes WKB approximation applicable to multi-dimensional problems (molecular vibrations)
  • Applications span atomic and molecular physics, condensed matter physics, quantum optics, and chemical reaction dynamics

Key Terms to Review (16)

Airy Functions: Airy functions are special functions that arise as solutions to the differential equation $$y'' - xy = 0$$, which is related to the behavior of quantum mechanical systems under certain conditions. They are particularly important in the context of the WKB approximation and semiclassical methods, as they provide asymptotic solutions to problems involving potential barriers and tunneling effects. Airy functions can be divided into two types: Ai(x) and Bi(x), which have different behaviors at positive and negative infinity.
Asymptotic Expansion: An asymptotic expansion is a mathematical expression that approximates a function as an argument approaches a limit, typically infinity. This concept is essential in simplifying complex problems, allowing us to describe the behavior of solutions in a more manageable way. By providing a series of terms that converge to the function in question, asymptotic expansions help reveal important features and characteristics of solutions within various physical contexts.
Bohr-Sommerfeld Quantization: Bohr-Sommerfeld quantization is a rule used in the WKB approximation that provides a method to quantize the energy levels of a classical system by applying quantization conditions to the action integral of the system's motion. This approach extends Bohr's original quantization ideas, allowing for systems with more complex motions, particularly in systems exhibiting periodic behavior. It forms a bridge between classical mechanics and quantum mechanics by relating classical orbits to discrete energy states.
Bound states: Bound states refer to quantum states in which a particle is confined to a particular region in space due to the presence of a potential well. In these states, the particle's energy is less than the potential energy outside the well, leading to discrete energy levels and a wave function that does not extend infinitely. This concept is crucial for understanding how particles behave in potential wells and under tunneling conditions.
Classical Action: Classical action is a fundamental concept in physics, defined as the integral of the Lagrangian over time, which provides a measure of the dynamics of a system. This quantity is central to both classical mechanics and quantum mechanics, linking the trajectories of particles to their respective path integrals. By applying the principle of least action, one can derive the equations of motion for a system, highlighting its importance in both semiclassical methods and the path integral formulation.
Eigenstates: Eigenstates are specific states of a quantum system that correspond to definite values (eigenvalues) of an observable, represented mathematically by operators acting on the state vector in Hilbert space. They play a crucial role in quantum mechanics as they help define the possible outcomes of measurements and can also be utilized in semiclassical approximations and symmetry operations.
Karl Friedrich Gauss: Karl Friedrich Gauss was a renowned German mathematician and physicist, known for his contributions to various fields including number theory, statistics, and astronomy. His work laid the foundation for many mathematical methods that are pivotal in both classical and quantum mechanics, influencing the development of the WKB approximation and semiclassical methods.
Niels Bohr: Niels Bohr was a Danish physicist known for his foundational contributions to understanding atomic structure and quantum mechanics, especially through the Bohr model of the atom. He introduced the idea that electrons exist in discrete energy levels and that they can transition between these levels by absorbing or emitting energy, which connects deeply with concepts like adiabatic invariants, wave functions, and angular momentum in quantum mechanics.
Phase integral method: The phase integral method is a semiclassical technique used to approximate the solutions of quantum mechanical problems, particularly in the context of the WKB approximation. It provides a way to analyze the behavior of wave functions in regions where classical mechanics can be applied, allowing for insights into tunneling and energy quantization. This method connects classical trajectories with quantum behavior by integrating the phase of the wave function along these paths.
Quantization condition: The quantization condition refers to the set of rules that determine the allowed energy levels in quantum systems, specifically arising from the constraints imposed by boundary conditions or physical principles. This concept is central to semiclassical methods, where classical mechanics is combined with quantum mechanics to describe systems that exhibit both behaviors. By enforcing these conditions, one can derive discrete energy values and other quantized properties, linking classical trajectories with quantum states.
Quantum tunneling: Quantum tunneling is a quantum mechanical phenomenon where a particle passes through a potential barrier that it classically should not be able to overcome due to insufficient energy. This occurs because particles, such as electrons, exhibit wave-like properties, allowing them to have a probability of being found on the other side of the barrier even when their energy is lower than the height of the barrier. This concept is crucial in understanding behaviors in potential wells and barriers, and it connects with various semiclassical approaches for approximating quantum systems.
Semiclassical approximation: The semiclassical approximation is a method that combines classical mechanics and quantum mechanics to analyze systems where quantum effects are important but can still be treated using classical concepts. This approach is particularly useful for problems where quantum behavior can be approximated by classical trajectories, allowing for a more manageable way to solve quantum mechanical problems using classical intuition.
Tunneling phenomena: Tunneling phenomena refers to the quantum mechanical process in which a particle passes through a potential energy barrier that it classically would not be able to surmount. This intriguing behavior arises from the principles of quantum mechanics, where particles can exhibit wave-like properties, allowing them to exist in multiple states simultaneously. Tunneling has profound implications in various fields, including nuclear fusion, semiconductor technology, and the stability of certain atomic structures.
Turning Points: Turning points are critical positions in a potential energy curve where the kinetic energy of a particle is zero, leading to a change in the direction of motion. They signify the boundaries of classically allowed regions and play a significant role in understanding how particles behave in quantum mechanics, particularly within the WKB approximation framework.
Wave-packet dynamics: Wave-packet dynamics refers to the behavior of wave packets, which are localized groups of waves that can propagate through space and time. These packets can represent particles in quantum mechanics, demonstrating how their position and momentum evolve according to the principles of wave behavior and uncertainty. Understanding wave-packet dynamics is crucial for grasping the underlying mechanisms in the WKB approximation and semiclassical methods, where classical trajectories intersect with quantum effects.
WKB Approximation: The WKB approximation is a mathematical method used to find approximate solutions to the Schrödinger equation in quantum mechanics, especially in situations where the potential varies slowly. This technique is particularly useful for analyzing problems involving potential wells and barriers, as it simplifies complex wave functions into more manageable forms. By treating the wave function as an exponential function, the WKB approximation helps in understanding phenomena such as tunneling and quantization of energy levels in bound states.
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