16.2 Symmetry groups and their representations
3 min read•august 7, 2024
Symmetry groups and their representations are crucial tools in physics for understanding the underlying structure of physical systems. They help us classify and analyze the symmetries of molecules, crystals, and quantum systems, providing insights into their properties and behavior.
allows us to study these symmetry groups using linear algebra techniques. By representing abstract group elements as matrices, we can perform calculations and derive important physical results, such as and energy level splittings in .
Symmetry Groups
Types of Symmetry Operations
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- Symmetry operation is a transformation that leaves an object invariant and unchanged in appearance
- Includes , , , and (combination of rotation and reflection)
- Each symmetry operation has an associated symmetry element (point, line, or plane) about which the operation is performed
- The leaves the object unchanged and is always present in any
- Successive application of symmetry operations forms a mathematical group structure
Classification of Symmetry Groups
- Point group describes the symmetry of a molecule or crystal based on its and operations
- Includes finite groups such as (water molecule) and (octahedral coordination)
- Space group describes the symmetry of a crystal structure, including both point group symmetry and translational symmetry
- There are 230 unique in three dimensions, classified by their and point group
- consists of all proper rotations about a fixed point, forming a subgroup of the
- Includes continuous groups such as (rotations in 2D) and (rotations in 3D)
Translational and Permutation Symmetry
- describes the symmetry of a crystal under translations by lattice vectors
- Includes infinite discrete groups such as the and the
- describes the symmetry of a system under permutations of its components
- Includes finite groups such as the (all permutations of objects) and the (even permutations of objects)
- is important in quantum mechanics for describing identical particles (bosons and fermions)
Representation Theory Fundamentals
Representation Theory Basics
- Representation theory studies abstract algebraic structures (groups, algebras, etc.) by representing their elements as linear transformations of vector spaces
- Provides a concrete way to analyze and manipulate abstract objects using matrices and linear algebra
- A representation of a group is a from to the general linear group of a vector space
- The dimension of the representation is the dimension of the vector space
- Representations can be used to classify and understand the symmetries of physical systems
Irreducible and Reducible Representations
- An irreducible representation (irrep) is a representation that cannot be decomposed into a direct sum of smaller representations
- Irreps are the building blocks of representation theory and provide a complete description of a group's structure
- A reducible representation can be decomposed into a direct sum of irreps
- Every representation can be uniquely decomposed into a direct sum of irreps (up to isomorphism)
- The number of times each irrep appears in the decomposition is given by its
Character Tables and Their Applications
- The character of a representation is the trace of the matrix for each group element
- The character table of a group summarizes the of all its irreps
- Each row corresponds to an irrep, and each column corresponds to a conjugacy class of group elements
- provide a compact way to analyze the symmetry properties of a system
- They can be used to determine the selection rules for transitions, the () of orbitals, and the vibrational modes of molecules
- Example: The character table of the point group (water molecule) has four irreps (, , , ) and four conjugacy classes (, , , )
Key Terms to Review (38)
Alternating Group: The alternating group is a mathematical concept that consists of all even permutations of a finite set. These groups play a crucial role in the study of symmetry, particularly in understanding how different objects can be transformed without altering their essential properties. The alternating group is particularly important in various areas of mathematics, including group theory and combinatorics, as it provides insights into the structure of symmetrical objects.
C2v: c2v is a specific point group in the classification of molecular symmetries, characterized by two vertical symmetry axes and a horizontal mirror plane. It is crucial for understanding the symmetry properties of molecules, particularly in the context of their vibrational modes and electronic transitions. The c2v point group includes symmetry operations such as rotations and reflections that help in analyzing molecular shapes and spectroscopic behavior.
Character tables: Character tables are mathematical tools used to summarize the properties of symmetry groups, particularly in the study of group representations. They provide a compact way to understand how group elements can be represented through matrices, highlighting the irreducible representations and their corresponding characters. This structure is essential for analyzing the symmetries present in various physical systems, linking algebraic concepts to geometric and physical applications.
Characters: In the context of symmetry groups and their representations, characters are complex-valued functions associated with the irreducible representations of a group. They provide important information about how group elements act on various mathematical objects, particularly in physics and chemistry, where they help in analyzing symmetry properties and simplifying calculations.
Dimension of representation: The dimension of representation refers to the number of independent parameters or degrees of freedom associated with a particular representation of a symmetry group. It indicates how many basis elements are needed to describe the actions of the group on a vector space. This concept is essential for understanding how symmetries influence physical systems, as it helps identify the possible states that a system can occupy under the influence of those symmetries.
Gl(v): The term gl(v) refers to the general linear group of all invertible linear transformations on a finite-dimensional vector space v. This concept is fundamental in understanding how symmetry and transformations operate within the framework of vector spaces, connecting directly to the representations of symmetry groups in various physical contexts.
Group theory: Group theory is a branch of mathematics that studies the algebraic structures known as groups, which consist of a set of elements combined with an operation that satisfies specific properties. In physics, group theory is crucial for understanding symmetry and its implications in various physical systems, allowing the classification of particles and the analysis of physical phenomena through their symmetries.
Hexagonal Lattice: A hexagonal lattice is a two-dimensional arrangement of points in a repeating pattern where each point is surrounded by six nearest neighbors, forming a hexagon. This structure exhibits unique symmetrical properties that are important in various physical systems, including crystal structures, molecular arrangements, and materials science. The symmetry and arrangement play a crucial role in determining the physical properties of materials formed with this lattice structure.
Homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups, that respects the operations defined on those structures. In the context of group theory, this means if you take two elements from one group and apply the group operation to them, the result will correspond to the result of applying the operation to their images in another group. This concept is crucial for understanding how different groups relate to each other, particularly in terms of symmetry and representation.
Identity operation: The identity operation is a fundamental concept in mathematics and physics that refers to an operation that leaves an object unchanged. In the context of symmetry groups, it represents the element that corresponds to doing nothing to the system, effectively preserving its original state. This operation is crucial for understanding how symmetry transformations affect physical systems and their representations, acting as a reference point in the study of group behavior.
Improper Rotations: Improper rotations refer to a specific type of symmetry operation that combines a rotation about an axis with a reflection across a plane perpendicular to that axis. This concept is crucial in understanding how certain geometric objects behave under symmetry transformations, especially in the context of molecular structures and crystal lattices. By studying improper rotations, one can gain insight into the complete symmetry properties of physical systems and their representations in various contexts.
Integer lattice: An integer lattice is a regular grid of points in multi-dimensional space where each point has integer coordinates. These lattices play a crucial role in various mathematical contexts, particularly in geometry and number theory, as they help model structures with symmetrical properties and facilitate the study of symmetry groups and their representations.
Inversions: Inversions refer to a specific type of symmetry operation where an object is transformed by reflecting it through a point, effectively reversing its coordinates. This transformation is essential in the study of symmetry groups as it allows for the classification of various symmetrical structures and their properties in physical systems, impacting their representations and behaviors.
Irreducible representations: Irreducible representations are representations of a group that cannot be decomposed into simpler representations. This concept is central to understanding how symmetry groups act on different spaces and is key in connecting group theory to various physical phenomena, such as particle physics and crystallography. Irreducible representations help to categorize and analyze the behaviors of systems under symmetry transformations.
Lattice System: A lattice system refers to a structured arrangement of points or nodes in space, typically described by symmetry and periodicity. These systems are crucial for understanding the geometric and symmetry properties of crystals, as they help classify the different types of crystal structures based on their symmetry groups and representations. The lattice system establishes a framework that enables the analysis of physical properties of materials through their symmetrical properties.
Multiplicity: Multiplicity refers to the number of times a particular symmetry operation can be performed without changing the overall state of a system. In the context of symmetry groups and their representations, it is a critical concept that helps determine how many equivalent states or representations exist for a given symmetry. Understanding multiplicity allows for the classification of particles and their behavior under various transformations, revealing deeper insights into the structure of physical systems.
Oh: In the context of symmetry groups and their representations, 'oh' refers to a specific type of symmetry group known as the octahedral group. This group is essential in describing the symmetries of an octahedron, which includes various rotations and reflections that map the shape onto itself. The octahedral group is a key example in the study of three-dimensional symmetry and is closely related to other polyhedral groups such as tetrahedral and cubic groups.
Orthogonal Group: The orthogonal group, denoted as O(n), consists of all n x n orthogonal matrices, which are matrices whose columns and rows are orthonormal vectors. These matrices preserve the Euclidean norm and the inner product, making them essential in understanding transformations that maintain geometric properties, particularly in relation to symmetry groups and their representations.
Permutation group: A permutation group is a mathematical structure that consists of a set of permutations, which are bijective functions that rearrange the elements of a set. These groups play a vital role in understanding symmetry, as they can represent the ways in which a geometric object can be transformed without altering its essential properties. In the context of symmetry groups and their representations, permutation groups help characterize the symmetries of various objects and systems by showcasing how they can be manipulated or transformed.
Permutation symmetry: Permutation symmetry refers to the invariance of a system's properties or outcomes when the elements within that system are rearranged or permuted. This concept is crucial in understanding the behavior of particles and fields in physics, where the fundamental characteristics remain unchanged regardless of how these elements are ordered, highlighting the underlying symmetries in physical systems.
Point Groups: Point groups are mathematical groups that describe the symmetry of a geometric object, focusing on its rotational and reflectional symmetries. These groups help categorize shapes based on their symmetry properties and allow for a systematic classification of molecular structures in chemistry and physics. Point groups play a crucial role in understanding how symmetry affects physical properties and behavior in various scientific fields.
Quantum mechanics: Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy at very small scales, typically at the level of atoms and subatomic particles. It introduces concepts such as wave-particle duality, quantization of energy levels, and the uncertainty principle, which challenge classical notions of determinism and locality.
Reducible representations: Reducible representations refer to representations of a symmetry group that can be expressed as a direct sum of two or more irreducible representations. This means that the associated vector space can be decomposed into smaller, simpler components that exhibit distinct transformation properties under the action of the symmetry group. Understanding reducible representations is crucial for analyzing complex systems where symmetries play a significant role.
Reflections: Reflections are transformations in geometry that produce a mirror image of a shape across a specific line, known as the line of reflection. This concept is central to understanding symmetry groups, as reflections help define the symmetries of shapes and patterns. By examining how objects behave under reflections, one can gain insights into their symmetrical properties and group representations, which are crucial for various physical applications.
Representation theory: Representation theory is a branch of mathematics that studies how groups can be represented through linear transformations of vector spaces. This concept connects algebraic structures with geometrical and physical ideas, enabling the exploration of symmetries in various systems. By understanding how a group's elements correspond to matrices or operators, representation theory becomes a powerful tool in analyzing the symmetry properties of physical systems and quantum mechanics.
Rotation Group: The rotation group is a mathematical structure that describes all possible rotations in a given space, particularly focusing on how objects can be rotated about a fixed point without changing their shape. This concept is central to understanding symmetry, as it captures the ways in which objects can be transformed while retaining their essential properties. By studying the rotation group, one can analyze how these transformations interact and form a group under composition.
Rotations: Rotations refer to the movement of an object around a central point or axis, resulting in a change in orientation while maintaining its shape and size. In the context of symmetry groups, rotations play a crucial role as they help define the symmetrical properties of objects, including their invariance under certain transformations. Understanding rotations is essential for analyzing how different representations relate to symmetry and structure.
Salcs: Salcs, or symmetry-adapted linear combinations of atomic orbitals, are combinations of atomic orbitals that transform according to the symmetry properties of a molecule or crystal. This concept is crucial in understanding how molecular orbitals are formed from atomic orbitals in a way that respects the symmetry of the overall system, allowing for simplified calculations and interpretations in quantum chemistry and solid-state physics.
Selection rules: Selection rules are criteria that determine the allowed transitions between quantum states based on symmetries and conservation laws. These rules arise from the fundamental principles of quantum mechanics and group theory, dictating which physical processes are permitted during transitions such as electronic excitations, vibrations, or rotational motions. By identifying which transitions are allowed, selection rules help to predict the outcomes of spectroscopic experiments and the behavior of systems under various interactions.
Sn: The term 'sn' refers to the symmetric group on n elements, which is a mathematical structure that consists of all possible permutations of a finite set of n elements. Each permutation can be thought of as a rearrangement of these elements, and the symmetric group captures the essence of symmetry in various mathematical contexts, especially in group theory and representation theory.
So(2): The notation so(2) represents the Lie algebra of the special orthogonal group SO(2), which consists of all the infinitesimal rotations in a two-dimensional space. This algebra is crucial in understanding the symmetries and representations related to rotations, especially in physics and geometry. It plays a vital role in the study of angular momentum and provides a mathematical framework for analyzing physical systems exhibiting rotational symmetry.
So(3): so(3) is the Lie algebra associated with the rotation group SO(3), which describes all possible rotations in three-dimensional space. This mathematical structure is fundamental in understanding symmetries and conservation laws in physics, particularly in mechanics and quantum mechanics. The elements of so(3) can be represented by skew-symmetric matrices, which play a key role in the analysis of angular momentum and rotational dynamics.
Space Groups: Space groups are mathematical classifications that describe the symmetry of crystal structures in three-dimensional space. They combine the concepts of point groups and translational symmetry, allowing for the systematic categorization of all possible crystal arrangements and their properties. Understanding space groups is essential for analyzing how the atomic arrangement in a material influences its physical properties, such as conductivity and optical behavior.
Symmetric group: The symmetric group is a mathematical concept that refers to the group of all permutations of a finite set, where each permutation is a rearrangement of the elements in that set. This group plays a crucial role in understanding symmetry and the algebraic structures of various mathematical and physical systems, as it encapsulates the ways in which objects can be transformed while preserving their inherent properties.
Symmetry elements: Symmetry elements are specific points, lines, or planes in a geometric figure or molecule where symmetry can be observed. These elements play a crucial role in defining the symmetry properties of a system, which can reveal information about its behavior and characteristics. Understanding symmetry elements helps in classifying molecular structures and understanding their physical properties through the corresponding symmetry groups.
Symmetry group: A symmetry group is a mathematical concept that describes the set of transformations that can be applied to an object while preserving its structure. These transformations include operations such as rotations, reflections, and translations, and they play a crucial role in understanding the inherent properties of the object, particularly in physics and chemistry where symmetry helps explain physical phenomena.
Symmetry-adapted linear combinations: Symmetry-adapted linear combinations are specific combinations of atomic or molecular orbitals that respect the symmetry properties of a system. By grouping orbitals according to their symmetry characteristics, these combinations simplify the analysis of systems in quantum mechanics and provide insights into their electronic structure and behavior. This method is vital for understanding how symmetry influences molecular interactions and energy states.
Translation group: The translation group consists of all transformations that shift points in a given space by a constant vector, effectively moving every point in that space without changing its structure. This concept is essential in understanding symmetry, as it describes how objects can be moved around while retaining their original properties. The translation group is a fundamental example of a symmetry group, showcasing how physical systems behave under specific transformations.