⚛️Intro to Quantum Mechanics I Unit 6 – 1D Potential Wells and Barriers

Key Concepts

• Quantum mechanics describes the behavior of particles at the atomic and subatomic scales
• Potential wells are regions of space where a particle is trapped due to a potential energy minimum
• Infinite potential wells have infinitely high walls, confining the particle completely
• Finite potential wells have walls of finite height, allowing the particle to escape if it has enough energy
• Potential barriers are regions of high potential energy that a particle must overcome to pass through
• Quantum tunneling is the phenomenon where a particle can pass through a potential barrier even if its energy is lower than the barrier height
• Quantum states are the discrete energy levels that a particle can occupy in a potential well or barrier system
• Wave functions describe the probability distribution of a particle's position and momentum in a quantum system

Mathematical Foundations

• Schrödinger equation is the fundamental equation of quantum mechanics, describing the time-dependent behavior of a quantum system
  • Time-independent Schrödinger equation: $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi$
  • $\hbar$ is the reduced Planck's constant, $m$ is the particle mass, $V(x)$ is the potential energy, and $E$ is the total energy
• Wave functions $\psi(x)$ are complex-valued functions that describe the quantum state of a particle
  • Probability density is given by $|\psi(x)|^2$, representing the probability of finding the particle at position $x$
• Normalization condition ensures that the total probability of finding the particle somewhere in space is equal to 1
  • $\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$
• Boundary conditions determine the allowed energy levels and wave functions in a potential well or barrier system
• Expectation values of physical quantities (position, momentum, energy) are calculated using the wave function
  • $\langle A \rangle = \int_{-\infty}^{\infty} \psi^*(x) \hat{A} \psi(x) dx$, where $\hat{A}$ is the operator corresponding to the physical quantity

Potential Wells: Infinite and Finite

• Infinite potential well is a 1D system where a particle is confined between two infinitely high potential walls
  • Potential energy: $V(x) = 0$ for $0 < x < L$, and $V(x) = \infty$ elsewhere
  • Allowed energy levels: $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$, where $n = 1, 2, 3, ...$
  • Wave functions: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi x}{L})$ for $0 < x < L$, and $\psi_n(x) = 0$ elsewhere
• Finite potential well is a 1D system where a particle is confined between two potential walls of finite height
  • Potential energy: $V(x) = 0$ for $0 < x < L$, and $V(x) = V_0$ elsewhere
  • Allowed energy levels are found by solving the Schrödinger equation and applying boundary conditions
  • Wave functions have both sinusoidal (inside the well) and exponential (outside the well) components
• Quantum harmonic oscillator is a special case of a potential well with a quadratic potential energy function
  • Potential energy: $V(x) = \frac{1}{2}m\omega^2x^2$, where $\omega$ is the angular frequency
  • Allowed energy levels: $E_n = (n + \frac{1}{2})\hbar\omega$, where $n = 0, 1, 2, ...$

Potential Barriers: Tunneling Effect

• Potential barrier is a region of high potential energy that a particle must overcome to pass through
  • Classically, a particle can only pass through a barrier if its energy is greater than the barrier height
• Quantum tunneling allows a particle to pass through a potential barrier even if its energy is lower than the barrier height
  • Probability of tunneling depends on the barrier height, width, and the particle's energy
• Transmission coefficient $T$ quantifies the probability of a particle tunneling through a barrier
  • $T = \frac{|A_t|^2}{|A_i|^2}$, where $A_i$ and $A_t$ are the amplitudes of the incident and transmitted waves
• Tunneling current in scanning tunneling microscopy (STM) is an application of quantum tunneling
  • STM uses the tunneling current between a sharp probe tip and a sample surface to map the surface topography with atomic resolution
• Alpha decay is another example of quantum tunneling, where an alpha particle escapes from an atomic nucleus by tunneling through the potential barrier

Quantum States and Energy Levels

• Quantum states are the discrete energy levels that a particle can occupy in a potential well or barrier system
• Ground state is the lowest energy state, while excited states are higher energy states
• Quantum number $n$ labels the energy levels, with $n = 1$ corresponding to the ground state
• Degeneracy occurs when multiple quantum states have the same energy
  • Degeneracy can be caused by symmetry in the potential energy function
• Fermi energy is the highest occupied energy level in a system of fermions (particles with half-integer spin) at absolute zero temperature
  • Fermi energy depends on the particle density and the dimensionality of the system
• Quantum confinement occurs when the size of a system is comparable to the de Broglie wavelength of the particles
  • Quantum confinement leads to the quantization of energy levels and the modification of electronic and optical properties (quantum dots, nanowires)

Applications and Real-World Examples

• Quantum dots are nanoscale semiconductor structures that exhibit quantum confinement effects
  • Quantum dots have discrete energy levels and size-dependent optical properties, making them useful for applications in displays, solar cells, and biomedical imaging
• Quantum well lasers are based on the confinement of electrons and holes in a thin semiconductor layer (quantum well)
  • Quantum well lasers have lower threshold currents and higher efficiency compared to bulk semiconductor lasers
• Scanning tunneling microscopy (STM) uses quantum tunneling to image surfaces with atomic resolution
  • STM has been used to study the atomic structure of materials, molecular adsorption, and chemical reactions on surfaces
• Josephson junctions are based on the tunneling of Cooper pairs (bound electron pairs) through a thin insulating barrier between two superconductors
  • Josephson junctions are used in superconducting quantum interference devices (SQUIDs) for sensitive magnetic field measurements and in quantum computing as qubits
• Quantum cascade lasers are based on the transitions between energy levels in a series of coupled quantum wells
  • Quantum cascade lasers emit light in the mid-infrared to terahertz range and are used in gas sensing, spectroscopy, and imaging applications

Problem-Solving Strategies

• Identify the type of potential well or barrier system (infinite well, finite well, barrier)
• Write down the Schrödinger equation for the system and the corresponding potential energy function
• Apply boundary conditions to determine the allowed energy levels and wave functions
  • For infinite wells, use the condition that the wave function must be zero at the walls
  • For finite wells and barriers, ensure the continuity of the wave function and its derivative at the boundaries
• Normalize the wave functions to satisfy the normalization condition
• Calculate the transmission coefficient for potential barrier problems
  • Use the ratio of the transmitted and incident wave amplitudes
• Check the units and the limiting cases (e.g., infinite barrier height, zero barrier width) to verify the results
• Use symmetry arguments to simplify the problem, if applicable
  • For symmetric potential wells, the wave functions can be classified as even or odd functions

Common Misconceptions and FAQs

• Misconception: Quantum tunneling violates the conservation of energy
  • Explanation: Quantum tunneling is a consequence of the wave-particle duality and the Heisenberg uncertainty principle. The particle does not gain energy during tunneling; it has a probability of being found on the other side of the barrier.
• Misconception: A particle in a potential well can have any energy value
  • Explanation: The energy levels in a potential well are quantized, meaning that the particle can only have specific discrete energy values determined by the Schrödinger equation and the boundary conditions.
• FAQ: What is the difference between a potential well and a potential barrier?
  • Answer: A potential well is a region of low potential energy surrounded by regions of higher potential energy, confining the particle. A potential barrier is a region of high potential energy that the particle must overcome to pass through.
• FAQ: Can a particle tunnel through any potential barrier?
  • Answer: In principle, a particle can tunnel through any finite potential barrier. However, the probability of tunneling decreases exponentially with increasing barrier height and width. For sufficiently high and wide barriers, the tunneling probability becomes negligible.
• FAQ: What is the significance of the ground state in a potential well?
  • Answer: The ground state is the lowest energy state that a particle can occupy in a potential well. It has the lowest quantum number (n = 1) and the highest probability density near the center of the well. Understanding the ground state is crucial for describing the behavior of quantum systems at low temperatures.