⚛️Quantum Mechanics Unit 4 – One–Dimensional Quantum Systems

Key Concepts and Foundations

• Quantum mechanics describes the behavior of matter and energy at the atomic and subatomic scales
• Particles exhibit both wave-like and particle-like properties (wave-particle duality)
  • Electrons can diffract through a double-slit experiment demonstrating their wave nature
  • Photons can collide with electrons and transfer momentum demonstrating their particle nature
• The Heisenberg uncertainty principle states that the more precisely the position of a particle is determined, the less precisely its momentum can be known, and vice versa
  • Mathematically expressed as $\Delta x \Delta p \geq \frac{\hbar}{2}$, where $\hbar$ is the reduced Planck's constant
• The wavefunction $\Psi(x,t)$ is a complex-valued function that contains all the information about a quantum system
  • The probability of finding a particle at a specific location is given by the square of the absolute value of the wavefunction $|\Psi(x,t)|^2$
• Operators in quantum mechanics correspond to observable quantities and act on the wavefunction to extract information
  • The position operator $\hat{x}$ returns the position of a particle when acting on the wavefunction
  • The momentum operator $\hat{p} = -i\hbar \frac{\partial}{\partial x}$ returns the momentum of a particle when acting on the wavefunction
• The expectation value of an operator $\langle \hat{A} \rangle$ represents the average value of the corresponding observable quantity in a given quantum state

Mathematical Framework

• The wavefunction $\Psi(x,t)$ is a solution to the Schrödinger equation and contains all the information about a quantum system
• The Schrödinger equation is an eigenvalue problem where the Hamiltonian operator $\hat{H}$ acts on the wavefunction to yield the energy eigenvalues $E_n$ and eigenstates $\psi_n(x)$
  • The time-independent Schrödinger equation is given by $\hat{H}\psi_n(x) = E_n\psi_n(x)$
  • The time-dependent Schrödinger equation is given by $i\hbar \frac{\partial}{\partial t}\Psi(x,t) = \hat{H}\Psi(x,t)$
• The Hamiltonian operator $\hat{H}$ consists of the kinetic energy operator $\hat{T}$ and the potential energy operator $\hat{V}$
  • In one dimension, the Hamiltonian is given by $\hat{H} = \hat{T} + \hat{V} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)$
• The normalization condition ensures that the total probability of finding the particle somewhere in space is equal to 1
  • Mathematically, $\int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx = 1$
• Orthogonality of eigenstates means that different eigenstates are independent and do not overlap
  • Mathematically, $\int_{-\infty}^{\infty} \psi_m^*(x)\psi_n(x) dx = \delta_{mn}$, where $\delta_{mn}$ is the Kronecker delta
• The expectation value of an operator $\hat{A}$ is calculated using the integral $\langle \hat{A} \rangle = \int_{-\infty}^{\infty} \Psi^*(x,t) \hat{A} \Psi(x,t) dx$

The Schrödinger Equation in 1D

• The time-independent Schrödinger equation in one dimension is given by $-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$
  • $\hbar$ is the reduced Planck's constant, $m$ is the mass of the particle, $V(x)$ is the potential energy, and $E$ is the total energy
• The solutions to the Schrödinger equation depend on the form of the potential energy $V(x)$
  • For a free particle ($V(x) = 0$), the solutions are plane waves $\psi(x) = Ae^{ikx} + Be^{-ikx}$, where $k = \sqrt{2mE}/\hbar$
  • For a particle in a box ($V(x) = 0$ inside the box and $V(x) = \infty$ outside), the solutions are standing waves $\psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi x}{L})$ with energy eigenvalues $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$
• The boundary conditions determine the allowed energy levels and wavefunctions for a given potential
  • For a particle in a box, the wavefunction must vanish at the walls ($\psi(0) = \psi(L) = 0$)
• The probability density $|\psi(x)|^2$ gives the probability of finding the particle at a specific location $x$
  • For a particle in a box, the probability density is $|\psi_n(x)|^2 = \frac{2}{L} \sin^2(\frac{n\pi x}{L})$
• The energy eigenvalues and eigenstates can be found by solving the Schrödinger equation for a given potential $V(x)$
  • Analytical solutions exist for simple potentials like the infinite square well, harmonic oscillator, and hydrogen atom
  • Numerical methods are required for more complex potentials

Potential Wells and Barriers

• Potential wells are regions where the potential energy is lower than the surrounding areas, creating a "well" that can trap particles
  • An infinite square well has $V(x) = 0$ inside the well and $V(x) = \infty$ outside, confining the particle within the well
  • A finite square well has a finite depth $V_0$, allowing for the possibility of quantum tunneling
• Potential barriers are regions where the potential energy is higher than the surrounding areas, creating a "barrier" that particles must overcome
  • A square potential barrier has a constant potential energy $V_0$ over a finite region and zero potential energy elsewhere
• The solutions to the Schrödinger equation for potential wells and barriers depend on the energy of the particle relative to the potential
  • For an infinite square well, the energy eigenstates are discrete and given by $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$, where $L$ is the width of the well
  • For a finite square well, there are both bound states (discrete energy levels) and scattering states (continuous energy spectrum)
• The transmission and reflection coefficients determine the probability of a particle passing through or being reflected by a potential barrier
  • The transmission coefficient $T$ is the ratio of the transmitted probability current to the incident probability current
  • The reflection coefficient $R$ is the ratio of the reflected probability current to the incident probability current
• The continuity of the wavefunction and its derivative at the boundaries of the potential well or barrier leads to the matching conditions
  • These conditions are used to determine the unknown coefficients in the wavefunction solutions

Quantum Tunneling

• Quantum tunneling is the phenomenon where a particle can pass through a potential barrier that it classically could not surmount
  • This is a consequence of the wave-particle duality and the Heisenberg uncertainty principle
• The probability of a particle tunneling through a barrier depends on the barrier height, width, and the particle's energy
  • The transmission coefficient $T$ decreases exponentially with increasing barrier width and height
  • Higher energy particles have a greater probability of tunneling through a given barrier
• The tunneling current in a scanning tunneling microscope (STM) is an example of quantum tunneling
  • In an STM, electrons tunnel between a sharp conducting tip and a sample surface, allowing for high-resolution imaging of the surface
• Alpha decay is another example of quantum tunneling
  • An alpha particle (helium nucleus) tunnels through the potential barrier of the nuclear force to escape the nucleus
• Quantum tunneling has applications in various fields, including:
  • Scanning tunneling microscopy (STM) for surface analysis
  • Tunneling diodes and transistors in electronics
  • Josephson junctions in superconducting devices
  • Quantum computing and information processing
• The WKB (Wentzel-Kramers-Brillouin) approximation is a method for calculating the transmission coefficient for a given potential barrier
  • The WKB approximation is valid when the potential varies slowly compared to the wavelength of the particle

Harmonic Oscillator

• The quantum harmonic oscillator is a model system that describes a particle subject to a quadratic potential energy $V(x) = \frac{1}{2}m\omega^2x^2$
  • $m$ is the mass of the particle, and $\omega$ is the angular frequency of the oscillator
• The Schrödinger equation for the harmonic oscillator is given by $-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + \frac{1}{2}m\omega^2x^2\psi(x) = E\psi(x)$
• The energy eigenvalues for the quantum harmonic oscillator are discrete and evenly spaced, given by $E_n = \hbar\omega(n + \frac{1}{2})$, where $n = 0, 1, 2, ...$
  • The ground state energy $E_0 = \frac{1}{2}\hbar\omega$ is a consequence of the Heisenberg uncertainty principle
• The eigenstates of the quantum harmonic oscillator are given by $\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-m\omega x^2/2\hbar} H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right)$
  • $H_n(x)$ are the Hermite polynomials, which are orthogonal polynomials
• The probability density for the eigenstates of the harmonic oscillator is given by $|\psi_n(x)|^2$
  • The probability density has $n$ nodes and is symmetric about $x = 0$
• The harmonic oscillator has applications in various fields, including:
  • Modeling molecular vibrations and phonons in solids
  • Describing the electromagnetic field in quantum optics
  • Quantum field theory, where particles are treated as excitations of underlying harmonic oscillator fields
• The creation and annihilation operators, denoted by $\hat{a}^{\dagger}$ and $\hat{a}$, respectively, are used to raise or lower the energy eigenstates of the harmonic oscillator
  • $\hat{a}^{\dagger}\psi_n(x) = \sqrt{n+1}\psi_{n+1}(x)$ and $\hat{a}\psi_n(x) = \sqrt{n}\psi_{n-1}(x)$

Applications and Examples

• The particle in a box model is used to describe the behavior of electrons in conjugated systems, such as polyenes and aromatic compounds
  • The energy levels and wavefunctions of the electrons can be calculated using the particle in a box model
• Quantum dots are nanoscale structures that confine electrons in three dimensions, creating a system analogous to the particle in a box
  • The energy levels and optical properties of quantum dots can be engineered by controlling their size and shape
• The scanning tunneling microscope (STM) relies on the principle of quantum tunneling to image surfaces with atomic resolution
  • The tunneling current between the STM tip and the sample surface depends on the local electronic structure of the surface
• The Josephson effect in superconductors is a manifestation of quantum tunneling
  • Cooper pairs (bound electron pairs) can tunnel through a thin insulating barrier between two superconductors, leading to a supercurrent
• The quantum harmonic oscillator model is used to describe the vibrational motion of atoms in molecules and solids
  • The vibrational energy levels and transitions can be measured using spectroscopic techniques like infrared and Raman spectroscopy
• The quantum harmonic oscillator is also used to model the electromagnetic field in quantum optics
  • The energy levels of the electromagnetic field are described by the number of photons in each mode, with the photon being the quantum of the field
• Quantum tunneling is exploited in various electronic devices, such as tunneling diodes and transistors
  • These devices rely on the ability of electrons to tunnel through potential barriers to control the flow of current
• Alpha decay is a radioactive decay process that occurs through quantum tunneling
  • The alpha particle (helium nucleus) tunnels through the potential barrier of the nuclear force to escape the nucleus

Problem-Solving Strategies

• Identify the type of potential energy function $V(x)$ in the problem (e.g., infinite square well, finite square well, harmonic oscillator)
• Write down the time-independent Schrödinger equation for the given potential: $-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$
• Determine the boundary conditions for the wavefunction based on the physical constraints of the system
  • For example, in an infinite square well, the wavefunction must vanish at the walls: $\psi(0) = \psi(L) = 0$
• Solve the Schrödinger equation for the given potential and boundary conditions to find the energy eigenvalues and eigenstates
  • For simple potentials like the infinite square well or harmonic oscillator, analytical solutions can be obtained
  • For more complex potentials, numerical methods such as the shooting method or the variational method may be required
• Normalize the wavefunctions to ensure that the total probability of finding the particle is equal to 1: $\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$
• Calculate the expectation values of relevant operators (e.g., position, momentum, energy) using the normalized wavefunctions: $\langle \hat{A} \rangle = \int_{-\infty}^{\infty} \psi^*(x) \hat{A} \psi(x) dx$
• For problems involving potential barriers and quantum tunneling, determine the transmission and reflection coefficients using the boundary conditions and continuity of the wavefunction and its derivative
• Interpret the results in terms of the physical properties of the system, such as the probability density, energy levels, and expectation values
• Use symmetry arguments and selection rules to simplify the problem when applicable
  • For example, in a symmetric potential, the eigenstates will have definite parity (even or odd)
• Check the units and orders of magnitude of the results to ensure they are physically reasonable