Glossary

9.4 Potential wells, barriers, and tunneling

2 min read•july 25, 2024

Quantum mechanics reveals fascinating particle behavior in potential wells and barriers. The governs these systems, yielding standing waves in wells and allowing tunneling through barriers. These concepts are crucial for understanding quantum phenomena.

Transmission and reflection coefficients describe how particles interact with barriers, while bound and characterize different energy behaviors. These ideas form the foundation for understanding quantum systems and their applications in various fields.

Potential Wells and Barriers

Solutions for Schrödinger equation

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  • Time-independent Schrödinger equation −ℏ22md2ψdx2+V(x)ψ=Eψ-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi governs quantum behavior
  • confines particle between rigid walls yields standing wave solutions ψn(x)=2Lsin⁡(nπxL)\psi_n(x) = \sqrt{\frac{2}{L}}\sin(\frac{n\pi x}{L})
  • allows penetration into classically forbidden regions produces transcendental equation for energy levels
  • V(x)=V0V(x) = V_0 for 0<x<L0 < x < L requires matching solutions at boundaries (particle in a box)

Concept of quantum tunneling

  • enables particles to penetrate classically forbidden barriers
  • decreases exponentially with barrier width and height
  • e−2γe^{-2\gamma} quantifies tunneling probability where γ=2m(V0−E)Lℏ\gamma = \frac{\sqrt{2m(V_0-E)L}}{\hbar}
  • Applications include alpha decay, scanning tunneling microscopy (STM), and nuclear fusion in stars

Transmission and reflection coefficients

  • Incident, reflected, and transmitted waves describe particle behavior at barriers
  • j=ℏ2mi(ψ∗dψdx−ψdψ∗dx)j = \frac{\hbar}{2mi}(\psi^*\frac{d\psi}{dx} - \psi\frac{d\psi^*}{dx}) measures probability flow
  • T=jtransmittedjincidentT = \frac{j_{\text{transmitted}}}{j_{\text{incident}}} gives probability of particle passing through barrier
  • Reflection coefficient R=jreflectedjincidentR = \frac{j_{\text{reflected}}}{j_{\text{incident}}} gives probability of particle bouncing back
  • requires R+T=1R + T = 1
  • T=4E(V0−E)4E(V0−E)+V02sinh⁡2(κL)T = \frac{4E(V_0-E)}{4E(V_0-E) + V_0^2\sinh^2(\kappa L)} where κ=2m(V0−E)ℏ2\kappa = \sqrt{\frac{2m(V_0-E)}{\hbar^2}}
  • T=11+(mZℏ2k)2T = \frac{1}{1 + (\frac{mZ}{\hbar^2k})^2} where ZZ is barrier strength

Properties of quantum states

  • have discrete energy levels and normalized ∫−∞∞∣ψ(x)∣2dx=1\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1
  • Scattering states exhibit continuous energy spectrum and oscillatory behavior
  • E=ℏ2k22mE = \frac{\hbar^2k^2}{2m} connects particle energy and wavenumber
  • Wavefunctions display even ψ(−x)=ψ(x)\psi(-x) = \psi(x) or odd ψ(−x)=−ψ(x)\psi(-x) = -\psi(x) parity
  • Nodes (zero amplitude) and antinodes (maximum amplitude) characterize standing waves
  • applies to slowly varying potentials providing semiclassical solutions
  • occurs in double barrier structures enhancing transmission at specific energies

Key Terms to Review (18)

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.
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.
Delta function barrier transmission: Delta function barrier transmission refers to the quantum mechanical phenomenon where a particle can pass through a potential barrier described by a delta function potential. This concept is critical for understanding tunneling effects, which occur when particles penetrate barriers that would be classically insurmountable. The delta function serves as an idealized model of a very thin and high potential barrier, highlighting the unusual behavior of particles in quantum mechanics.
Energy-momentum relation: The energy-momentum relation is a fundamental equation in physics that connects an object's energy and momentum, expressed as $$E^2 = p^2c^2 + m^2c^4$$. This relationship reveals how energy and momentum are interlinked, especially in the realm of particle physics and quantum mechanics, highlighting the role of mass and the speed of light as constants in these interactions.
Finite square well: A finite square well is a potential energy function used in quantum mechanics that describes a region where a particle experiences a lower potential energy compared to its surroundings, confined within a finite depth and width. This concept helps in understanding the behavior of particles in various potential scenarios, allowing for the analysis of bound states and tunneling effects.
Gamow Factor: The Gamow factor is a crucial term in quantum mechanics that quantifies the probability of tunneling through a potential barrier. It arises from the quantum mechanical description of particle behavior in potential wells and barriers, emphasizing how particles can penetrate barriers that would be classically forbidden. This factor plays a significant role in processes like nuclear fusion, where particles need to overcome energy barriers to interact and fuse.
Infinite square well: The infinite square well is a fundamental quantum mechanical model that describes a particle confined to a one-dimensional box with infinitely high potential walls. This model illustrates how a particle can only occupy specific energy levels due to the restrictions imposed by the potential well, demonstrating key principles of quantization and wave functions in quantum mechanics.
Potential Barrier: A potential barrier is a region in space where the potential energy is higher than the surrounding areas, creating an obstacle for particles attempting to pass through. In quantum mechanics, potential barriers play a crucial role in phenomena like tunneling, where particles can seemingly bypass these barriers due to their wave-like nature. Understanding potential barriers is essential when studying particle behavior in potential wells and their interactions with various forces.
Probability Current Density: Probability current density is a vector quantity in quantum mechanics that describes the flow of probability associated with a quantum state. It quantifies how the probability density of finding a particle changes with respect to time and space, providing insights into the dynamics of particles, especially in scenarios involving potential wells, barriers, and tunneling phenomena. This concept is crucial for understanding how particles behave under quantum constraints, illustrating their movement and interactions within different potential landscapes.
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.
Rectangular barrier transmission: Rectangular barrier transmission refers to the quantum mechanical phenomenon where a particle can pass through a potential energy barrier that it classically shouldn't be able to cross. This occurs due to the wave-like nature of particles in quantum mechanics, allowing for tunneling effects, particularly when the energy of the particle is less than the height of the barrier. This concept connects deeply with potential wells and barriers as it highlights how quantum mechanics defies classical intuition.
Resonant Tunneling: Resonant tunneling refers to the quantum mechanical phenomenon where a particle can pass through a potential barrier due to the presence of energy states that align with the barrier's characteristics. This occurs in a potential well scenario, where specific energy levels lead to enhanced tunneling probabilities, often resulting in a peak in the transmission coefficient. It highlights the significance of quantum superposition and interference effects in determining the behavior of particles at the quantum scale.
Scattering states: Scattering states refer to the quantum mechanical states of a particle that interact with a potential barrier or well, resulting in the particle being deflected or transmitted rather than captured. These states are crucial in understanding how particles behave when they encounter obstacles, revealing fundamental principles of quantum behavior, such as tunneling and reflection.
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.
Transmission Coefficient: The transmission coefficient is a measure of the probability that a particle will pass through a potential barrier rather than being reflected. This concept is crucial in understanding quantum phenomena such as tunneling, where particles can penetrate barriers despite classically not having enough energy. It provides insights into the behavior of particles in potential wells and barriers, as well as in relativistic frameworks where traditional intuition may not apply.
Tunneling probability: Tunneling probability refers to the likelihood that a particle can pass through a potential barrier, even when its energy is less than the height of the barrier. This phenomenon is a core aspect of quantum mechanics, illustrating how particles can exist in states that classically would be forbidden, leading to important implications in fields like quantum mechanics and semiconductor physics.
Wavefunctions: Wavefunctions are mathematical functions that describe the quantum state of a particle or system of particles. They provide a comprehensive representation of all possible states and their probabilities, capturing information about position, momentum, and other physical properties. In the context of potential wells, barriers, and tunneling, wavefunctions are essential for understanding how particles behave when they encounter varying potentials, revealing phenomena such as quantization and tunneling.
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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