9.4 Bound states and scattering states

Bound states and scattering states are fundamental concepts in quantum mechanics, crucial for understanding atomic structures and particle interactions. Bound states describe confined particles with discrete energy levels, while scattering states represent unbound particles with continuous energy spectra.

This topic explores the mathematical formulation, physical examples, and experimental observations of these states. It delves into quantum tunneling, spectral decomposition, and scattering theory, highlighting their applications in molecular binding and particle collisions.

Definition of bound states

• Bound states form a fundamental concept in quantum mechanics describing particles confined within a potential well
• These states play a crucial role in understanding atomic and molecular structures in spectral theory
• Bound states exhibit discrete energy levels and localized wavefunctions, key features in spectroscopic analysis

Energy levels of bound states

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• Characterized by quantized, discrete energy values determined by the potential well shape
• Negative energies relative to the continuum indicate binding to the potential
• Energy spectrum follows $E_n = -\frac{13.6 \text{ eV}}{n^2}$ for hydrogen-like atoms (n = principal quantum number)
• Higher energy levels correspond to excited states, with the ground state having the lowest energy

Wavefunctions for bound states

• Describe the spatial probability distribution of finding a particle in a specific state
• Exhibit exponential decay outside the potential well, ensuring localization
• Characterized by nodes and antinodes, with the number of nodes increasing for higher energy states
• Satisfy the time-independent Schrödinger equation $H\psi = E\psi$, where H is the Hamiltonian operator

Normalization of bound states

• Ensures the total probability of finding the particle anywhere in space equals 1
• Achieved by multiplying the wavefunction by a normalization constant
• Expressed mathematically as $\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$
• Allows for meaningful comparison between different bound states and calculation of expectation values

Properties of scattering states

• Scattering states describe unbound particles interacting with a potential in quantum mechanics
• These states are essential for understanding particle collisions and interactions in spectral theory
• Unlike bound states, scattering states extend to infinity and have positive energies

Continuous energy spectrum

• Scattering states possess a continuous range of allowed energies above the potential well
• Energy values are not quantized, allowing for any positive energy value
• Described by the energy-momentum relation $E = \frac{\hbar^2k^2}{2m}$, where k is the wave number
• Continuous spectrum results from the infinite spatial extent of scattering states

Asymptotic behavior of scattering states

• Wavefunctions approach plane waves far from the scattering center
• Asymptotic form given by $\psi(r) \sim e^{ikz} + f(\theta, \phi)\frac{e^{ikr}}{r}$ for 3D scattering
• Incoming plane wave represented by $e^{ikz}$, scattered spherical wave by $f(\theta, \phi)\frac{e^{ikr}}{r}$
• Scattering amplitude f(θ, φ) contains information about the interaction potential

Normalization of scattering states

• Cannot be normalized in the same way as bound states due to their infinite spatial extent
• Utilize delta-function normalization $\int_{-\infty}^{\infty} \psi_k^*(x)\psi_{k'}(x) dx = \delta(k-k')$
• Ensures orthogonality between states with different momenta
• Allows for proper treatment in spectral decompositions and scattering calculations

Comparison of states

• Understanding the differences between bound and scattering states is crucial in spectral theory
• These distinctions impact the analysis of quantum systems and their observable properties
• Comparison provides insights into the nature of quantum confinement and free particle behavior

Bound vs scattering states

• Bound states localized within a potential well, scattering states extend to infinity
• Energy levels discrete for bound states, continuous for scattering states
• Bound state wavefunctions decay exponentially, scattering states oscillate at large distances
• Normalization methods differ, with bound states using standard normalization and scattering states using delta-function normalization
• Bound states associated with stable atomic or molecular configurations, scattering states with particle collisions

Discrete vs continuous spectra

• Discrete spectra arise from bound states, characterized by sharp spectral lines (atomic emission spectra)
• Continuous spectra result from scattering states, appearing as broad bands (blackbody radiation)
• Transition between discrete and continuous spectra occurs at the ionization threshold
• Rydberg states bridge the gap between discrete and continuous spectra, with closely spaced energy levels approaching the continuum

Mathematical formulation

• Mathematical framework of quantum mechanics provides a rigorous description of bound and scattering states
• Formalism allows for precise calculations of observable quantities and predictions of experimental outcomes
• Spectral theory utilizes this mathematical foundation to analyze quantum systems and their properties

Hamiltonian operators

• Represent the total energy of the system, including kinetic and potential energy terms
• For a particle in a potential V(x), the Hamiltonian takes the form $H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)$
• Eigenfunctions of the Hamiltonian correspond to stationary states (bound or scattering)
• Time evolution of quantum states governed by the Schrödinger equation $i\hbar\frac{\partial\psi}{\partial t} = H\psi$

Boundary conditions

• Determine the allowed solutions for wavefunctions in a given physical system
• For bound states, require wavefunctions to vanish at infinity $\lim_{x\to\pm\infty} \psi(x) = 0$
• Scattering states must satisfy asymptotic conditions, approaching plane waves at large distances
• Continuity and smoothness conditions imposed at potential discontinuities ensure physical solutions

Eigenvalue equations

• Describe the relationship between Hamiltonian operators and stationary states
• Take the form $H\psi = E\psi$, where E represents the energy eigenvalue
• For bound states, yield discrete eigenvalues corresponding to allowed energy levels
• Scattering states associated with continuous eigenvalues above the potential well
• Solving eigenvalue equations central to determining energy spectra and wavefunctions in quantum systems

Physical examples

• Concrete examples of bound and scattering states in physical systems illustrate the practical applications of spectral theory
• These examples demonstrate how theoretical concepts manifest in observable phenomena
• Understanding these systems crucial for interpreting experimental results and designing new quantum devices

Bound states in atoms

• Electrons in atomic orbitals represent prototypical bound states
• Energy levels described by principal quantum number n, angular momentum l, and magnetic quantum number m
• Hydrogen atom energy levels given by $E_n = -\frac{13.6 \text{ eV}}{n^2}$, with n = 1, 2, 3, ...
• Atomic spectra result from transitions between bound states, producing characteristic emission or absorption lines
• Multi-electron atoms exhibit more complex energy level structures due to electron-electron interactions

Scattering states in nuclear physics

• Describe interactions between nucleons or between nuclei and incident particles
• Neutron scattering used to probe nuclear structure and properties
• Cross-sections for nuclear reactions depend on the energy of incident particles
• Resonance scattering occurs when incident particle energy matches a quasi-bound state of the compound nucleus
• Analysis of scattering data provides information about nuclear forces and internal structure of nuclei

Quantum tunneling

• Quantum phenomenon allowing particles to penetrate potential barriers classically forbidden
• Demonstrates the wave-like nature of matter in quantum mechanics
• Plays a crucial role in various physical processes and technological applications

Tunneling through potential barriers

• Occurs when a particle encounters a potential barrier higher than its kinetic energy
• Wavefunction decays exponentially inside the barrier but remains non-zero
• Transmission probability depends on barrier height, width, and particle energy
• Described mathematically by solving the Schrödinger equation for a step or rectangular potential barrier
• Applications include scanning tunneling microscopy, nuclear fusion in stars, and quantum computing

Connection to bound states

• Tunneling enables transitions between bound states in double-well potentials
• Explains phenomena like ammonia molecule inversion and hydrogen bonding in DNA
• Contributes to alpha decay in radioactive nuclei, treated as tunneling through a Coulomb barrier
• Tunneling between coupled quantum dots creates artificial molecules with tunable properties
• Understanding tunneling essential for designing quantum devices and interpreting molecular spectra

Spectral decomposition

• Mathematical technique for expressing quantum states in terms of energy eigenstates
• Provides a powerful framework for analyzing and solving quantum mechanical problems
• Crucial for understanding the relationship between discrete and continuous spectra in spectral theory

Discrete and continuous spectra

• Discrete spectra arise from bound states with quantized energy levels
• Continuous spectra associated with scattering states and unbound particles
• Spectral decomposition allows representation of arbitrary states as superpositions of energy eigenstates
• For discrete spectra, decomposition takes the form $\psi = \sum_n c_n \psi_n$, where ψn are energy eigenstates
• Continuous spectra represented by integrals over energy eigenstates $\psi = \int c(E) \psi_E dE$

Completeness relations

• Express the idea that energy eigenstates form a complete basis for the Hilbert space
• For discrete spectra, completeness relation given by $\sum_n |\psi_n\rangle\langle\psi_n| = 1$
• Continuous spectra completeness relation $\int |E\rangle\langle E| dE = 1$
• Allow expansion of arbitrary operators in terms of energy eigenstates
• Crucial for calculating expectation values and transition probabilities in quantum systems

Scattering theory

• Branch of quantum mechanics dealing with particle collisions and interactions
• Provides a framework for analyzing and predicting outcomes of scattering experiments
• Connects theoretical descriptions of scattering states to observable quantities

Cross sections

• Measure the probability of a specific scattering outcome
• Defined as the ratio of scattered particles to incident particle flux
• Differential cross-section $\frac{d\sigma}{d\Omega} = |f(\theta, \phi)|^2$ relates to scattering amplitude
• Total cross-section obtained by integrating over all angles $\sigma = \int \frac{d\sigma}{d\Omega} d\Omega$
• Depend on the interaction potential and incident particle energy

Phase shifts

• Describe the change in phase of scattered waves relative to incident waves
• Determined by solving the radial Schrödinger equation for each angular momentum component
• Related to scattering amplitude through partial wave expansion $f(\theta) = \frac{1}{2ik}\sum_l (2l+1)(e^{2i\delta_l}-1)P_l(\cos\theta)$
• Provide information about the strength and nature of the scattering interaction
• Analysis of phase shifts allows reconstruction of the scattering potential

S-matrix formalism

• Describes the relationship between incoming and outgoing scattering states
• S-matrix elements given by $S_{fi} = \delta_{fi} - 2\pi i \delta(E_f - E_i)T_{fi}$, where T is the transition matrix
• Unitarity of S-matrix ensures conservation of probability in scattering processes
• Poles of S-matrix in complex energy plane correspond to bound states and resonances
• Provides a unified description of bound states, scattering states, and resonances in quantum systems

Experimental observations

• Experimental techniques for studying bound and scattering states in quantum systems
• Bridge between theoretical predictions and observable phenomena
• Crucial for validating quantum mechanical models and discovering new physical effects

Spectroscopic measurements

• Probe energy level structure of bound states through absorption and emission spectra
• Techniques include optical spectroscopy, X-ray spectroscopy, and nuclear magnetic resonance
• Atomic spectra reveal discrete energy levels, confirming quantum mechanical predictions
• Molecular spectroscopy provides information about rotational, vibrational, and electronic states
• High-resolution spectroscopy enables precise measurements of energy level splittings and transition frequencies

Scattering experiments

• Investigate interactions between particles and target systems
• Include electron scattering, neutron scattering, and particle collider experiments
• Measure cross-sections, angular distributions, and energy spectra of scattered particles
• Rutherford scattering experiment historically crucial in discovering atomic structure
• Modern scattering experiments probe subatomic structure and fundamental interactions

Applications in quantum mechanics

• Practical applications of bound and scattering state concepts in various areas of physics and chemistry
• Demonstrate the wide-ranging impact of spectral theory in understanding and manipulating quantum systems
• Crucial for developing new technologies and advancing scientific knowledge

Molecular binding

• Describes formation of stable molecular structures through quantum mechanical interactions
• Potential energy curves determine equilibrium bond lengths and dissociation energies
• Vibrational and rotational energy levels arise from quantization of molecular motion
• Molecular orbitals formed by linear combinations of atomic orbitals (LCAO method)
• Understanding molecular binding essential for predicting chemical reactivity and designing new materials

Particle collisions

• Study of interactions between subatomic particles in high-energy physics
• Scattering theory used to analyze collision outcomes and cross-sections
• Particle accelerators create controlled environments for studying fundamental interactions
• Discovery of new particles (Higgs boson) through analysis of collision data
• Quantum chromodynamics describes strong interactions in hadron-hadron collisions

Numerical methods

• Computational techniques for solving quantum mechanical problems involving bound and scattering states
• Essential for systems too complex for analytical solutions
• Enable accurate predictions and comparisons with experimental data

Variational techniques for bound states

• Approximate methods for finding upper bounds on ground state energies
• Based on minimizing the expectation value of the Hamiltonian $E[\psi] = \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}$
• Trial wavefunctions with adjustable parameters optimized to minimize energy
• Rayleigh-Ritz method extends variational approach to excited states
• Applications include electronic structure calculations in atoms and molecules

Computational approaches for scattering

• Numerical solutions of Schrödinger equation for scattering potentials
• Finite difference methods discretize space and solve coupled equations
• Partial wave analysis computes phase shifts for each angular momentum component
• R-matrix theory combines inner region (complex interactions) with outer region (asymptotic behavior)
• Monte Carlo methods simulate particle trajectories for complex scattering geometries