Symmetry Ops Slides Symmetry Operations
Foundations of Group Theory in Physics
The Hexagonal Mystery
Why do snowflakes must have six-fold symmetry?
Is it a physical law, a geometric coincidence, or an inevitable mathematical consequence of the water molecule's structure?
❄️
Macroscopic Shape
💧
Molecular Symmetry
What is a Symmetry Operation?
Definition
An action that moves an object into a configuration that is indistinguishable from the original.
The center of mass remains fixed.
Distances and angles are preserved (Isometry).
Points are mapped to points of the same type.
C4
Example: A 90° rotation leaves a square invariant.
Element vs. Operation
Symmetry Element
The "Geometrical Entity"
A point, line, or plane with respect to which the operation is performed.
Example: Rotation Axis
Symmetry Operation
The "Action"
The actual movement that leaves the object invariant.
Example: Rotation by \( \theta = \frac{2\pi}{n} \)
The Geometric Toolkit
\( E \)
Identity
The operation of "doing nothing." Present in every object. Mathematically essential for group structure.
\( C_n \)
Proper Rotation
Rotation by \( 360/n \) degrees about an axis. \( C_2 \), \( C_3 \), \( C_4 \), etc.
\( \sigma \)
Reflection
Reflection through a plane. Vertical (\( \sigma_v \)), Horizontal (\( \sigma_h \)), or Dihedral (\( \sigma_d \)).
\( i \)
Inversion
Every point \( (x, y, z) \) is mapped to \( (-x, -y, -z) \) through a center of symmetry.
Mathematical Group Criteria
1
Closure
If \( A, B \in G \), then \( A \cdot B \in G \).
2
Associativity
\( (A \cdot B) \cdot C = A \cdot (B \cdot C) \).
3
Identity
There exists \( E \) such that \( A \cdot E = E \cdot A = A \).
4
Inverses
For every \( A \), there is an \( A^{-1} \) where \( A \cdot A^{-1} = E \).
“Symmetry operations of a molecule form a point group because they satisfy all four axioms.”
Improper Rotations \( S_n \)
A composite operation : Rotation by \( 360/n \) followed by reflection through a plane perpendicular to the rotation axis.
\( S_n = \sigma_h \cdot C_n \)
Note: The object may not have \( C_n \) or \( \sigma_h \) individually, but can still have \( S_n \)!
Reflected + Rotated
Polyhedron Analysis Worksheet Polyhedron Analysis
Introduction to Symmetry Operations & Group Theory
Name:
Date:
1
Element Identification
For each geometric shape below, list the primary symmetry elements present. Use standard notation (\(E, C_n, \sigma_v, \sigma_h, i, S_n\)).
A. Regular Square (\(D_{4h}\))
2D Shape
Rotation Axes (\(C_n\)):
Reflection Planes (\(\sigma\)):
B. Regular Tetrahedron (\(T_d\))
3D Shape
Rotation Axes (\(C_n\)):
Improper Rotations (\(S_n\)):
2
Multiplication Table for \(C_{3v}\)
Complete the group multiplication table for the \(C_{3v}\) point group (symmetry of ammonia). Operations include: \(E, C_3, C_3^2, \sigma_a, \sigma_b, \sigma_c\). Remember that the order of operations is right-to-left: \(AB\) means \(B\) happens first.
\( G \cdot G \) \( E \) \( C_3 \) \( C_3^2 \) \( \sigma_a \) \( \sigma_b \) \( \sigma_c \) \( E \) \( E \) \( C_3 \) \( C_3 \) \( C_3^2 \) \( E \) \( \sigma_c \) \( C_3^2 \) \( \sigma_a \) \( \sigma_b \) \( \sigma_c \) \( E \) \( \sigma_b \) \( \sigma_c \)
3
Mathematical Proofs
Prove that the set of symmetry operations for a square forms a group by addressing Closure and Inverses using specific examples from your table or analysis above.
Axiom: Closure (Explain how combining two rotations \(C_4\) results in another group member):
Axiom: Inverses (Determine the inverse operation for a reflection \(\sigma_v\)):
Symmetry Fundamentals Teacher Guide Symmetry Fundamentals
Teacher Guide • Lesson 1
Undergraduate Physics Sequence
Module: Groups & Conservation
Instructional Intent
This lesson transitions students from intuitive geometric understanding to rigorous algebraic classification. The primary goal is to ensure students can not only identify symmetry elements visually but also understand the mathematical structure (the Group) that governs these operations.
Key Vocabulary
• Isometry
• Symmetry Element
• Point Group
• Axiomatic Group
• Improper Rotation
Discussion Facilitation
The "Do Nothing" Identity (\(E\))
"Why do we need a mathematical operation that does literally nothing? What physical system fails if \(E\) isn't included in the group?"
Talking Points: Identity is required for the existence of inverses (returning to the start state) and for the algebraic completeness of the symmetry description.
Improper Rotations (\(S_n\))
"Consider the methane molecule (Tetrahedral). It has no \(C_4\) axis. It has no \(\sigma_h\) plane. Yet it has three \(S_4\) axes. How can a product exist if the components do not?"
Talking Points: Emphasize that \(S_n\) is a single geometric mapping. The 'rotation' and 'reflection' are conceptual steps to reach the final indistinguishable state.
Common Misconceptions
Operation vs Element
Students often confuse the axis (the line) with the rotation (the movement). Remind them that one element (e.g., \(C_3\) axis) can generate multiple operations (\(C_3^1, C_3^2\)).
Commutativity
Students often assume symmetry operations commute (\(AB = BA\)). Use the \(C_{3v}\) table to show that reflections do not commute with rotations in most point groups.
Worksheet Answer Key & Guidance
Part 1: Element Identification
Square (\(D_{4h}\)): Elements include \(E, C_4, C_2 (z), 2C_2', 2C_2'', i, S_4, \sigma_h, 2\sigma_v, 2\sigma_d\). For this intro level, focus on \(C_4, C_2, \sigma_h, \sigma_v\).Tetrahedron (\(T_d\)): Elements include \(E, 8C_3, 3C_2, 6S_4, 6\sigma_d\). High spatial reasoning is required here; use physical models if available.
Part 2: \(C_{3v}\) Table Keys
\(\sigma_a \cdot C_3 = \sigma_b\)
Notice that \(\sigma_a \cdot C_3 \neq C_3 \cdot \sigma_a\). This demonstrates the non-Abelian nature of \(C_{3v}\).
Point Group Slides Molecular Point Groups
Schoenflies Notation & Classification
Geometry vs. Properties
Ammonia (\(NH_3\)) and Water (\(H_2O\)) both contain Hydrogen. Both have lone pairs.
Why does Water belong to point group \(C_{2v}\) while Ammonia belongs to \(C_{3v}\)?
How does this difference dictate their vibration and absorption spectra?
H2O Bent Shape
NH3 Pyramidal
The Schoenflies Hierarchy
Low Symmetry
C1 Identity only
Cs Plane only
Ci Inversion only
High Symmetry
Td Tetrahedral
Oh Octahedral
Ih Icosahedral
C∞v Linear (no i)
Rotational
Cnv n-fold + \(\sigma_v\)
Cnh n-fold + \(\sigma_h\)
Dnh n-fold + nC2 + \(\sigma_h\)
The Classification Algorithm
Step 1
Identify the principal (highest order) axis \(C_n\).
Step 2
Are there \(n\) \(C_2\) axes perpendicular to the principal axis?
Step 3
Look for reflection planes (\(\sigma_h, \sigma_v, \sigma_d\)) in hierarchical order.
Case Study: Benzene
D6h
Benzene is a highly symmetric planar molecule. Why \(D_{6h}\)?
Principal Axis: \(C_6\)
6 \(C_2\) axes perpendicular to \(C_6\)
Horizontal Plane: \(\sigma_h\) (Molecular plane)
6 Vertical Planes (\(3\sigma_v + 3\sigma_d\))
One of the most symmetric non-cubic molecules.
Linear Molecules
C∞v
Linear, no center of inversion. Has an infinite-fold rotation axis along the bond.
HCl Heteronuclear
D∞h
Linear with a center of inversion. Contains \(C_\infty\) plus perpendicular \(C_2\)'s and \(\sigma_h\).
N2, CO2 Homonuclear/Centrosymmetric
Schoenflies Classification Guide Schoenflies Notation Guide
Systematic Molecular Classification Reference
PHYSICS REF: 02
The Point Group Decision Tree
Is the molecule linear?
YES
Center of inversion?
D∞h C∞v
NO
High symmetry (\(T_d, O_h, I_h\))?
Find principal axis \(C_n\) (highest \(n\)).
Are there \(n\) \(C_2\) axes perpendicular to \(C_n\)?
YES (D Groups)
\(\sigma_h\)? Dnh
\(\sigma_d\)? Dnd
None? Dn
NO (C Groups)
\(\sigma_h\)? Cnh
\(\sigma_v\)? Cnv
\(S_{2n}\)? S_{2n}
None? Cn
Notation Key Elements Example C1 Only \(E\) CHBrClF Cs One \(\sigma\) CH2BrCl Ci Center of inversion \(i\) Staggered C2H2Cl2Br2 C2v \(C_2, 2\sigma_v\) H2O, SO2 C3v \(C_3, 3\sigma_v\) NH3, PCl3
Notation Key Elements Example D2h \(3C_2, 3\sigma, i\) Ethylene (C2H4) D3h \(C_3, 3C_2, \sigma_h\) BF3 D6h \(C_6, 6C_2, \sigma_h, i\) Benzene Td Tetrahedral CH4, CCl4 Oh Octahedral SF6
Classification Pro-Tips
• Principal Axis: Always align the principal axis with the \(z\)-axis for matrix operations later.
• Horizontal Plane (\(\sigma_h\)): This plane MUST be perpendicular to the principal axis.
• D vs C: If you find any \(C_2\) perpendicular to your principal \(C_n\), it's a D group. If not, it's C or S .
Molecular Sorting Activity Molecular Sorting
Symmetry Group Classification Lab
Student Identifiers
Your Objective
Classify the following molecules into their correct Schoenflies point groups. For each, you must identify the principal axis and at least two supporting symmetry elements .
1. Formaldehyde (\(CH_2O\))
PLANAR
[Molecule Sketch Area]
Principal Axis
Point Group
Evidence Elements
2. Allene (\(C_3H_4\))
NON-PLANAR
[Molecule Sketch Area]
Principal Axis
Point Group
Evidence Elements
3. Boron Trifluoride (\(BF_3\))
TRIGONAL PLANAR
[Molecule Sketch Area]
Principal Axis
Point Group
Evidence Elements
4. Sulfur Hexafluoride (\(SF_6\))
OCTAHEDRAL
[Molecule Sketch Area]
Principal Axis
Point Group
Evidence Elements
Deep Dive Analysis
Compare Molecule 1 (Formaldehyde) and Molecule 3 (Boron Trifluoride) . Both are planar and have vertical mirror planes. Why is one a C group and the other a D group? Be specific about the symmetry elements that differentiate them.
Identify a molecule (or macroscopic object) that belongs to the \(C_1\) point group. What does this imply about the "symmetry" of the object?
Matrix Representation Slides Matrix Representations
Symmetry in Coordinate Space
From Geometry to Algebra
"How do we tell a computer that a molecule has been rotated 120°?"
Visualizing symmetry is for humans. Matrices allow us to compute physical properties through linear algebra.
\[ \vec{r}' = \mathbf{M} \vec{r} \]
Where:
Reflection Matrices
Reflection across the \(xy\)-plane (\(\sigma_{xy}\)):
Points \( (x, y, z) \) map to \( (x, y, -z) \).
\[ \sigma_{xy} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix} \]
Reflection across the \(yz\)-plane (\(\sigma_{yz}\)):
Points \( (x, y, z) \) map to \( (-x, y, z) \).
\[ \sigma_{yz} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
Rotation Matrices
The general matrix for a rotation \(\theta\) about the \(z\)-axis:
\[ C_n(z) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
\(\theta = \frac{2\pi}{n}\)
Special Case: C2
\(\theta = 180^\circ \implies \cos\theta = -1, \sin\theta = 0\)
diag(-1, -1, 1)
Special Case: C3
\(\theta = 120^\circ \implies \cos\theta = -0.5, \sin\theta = 0.866\)
The Power of the Trace (\(\chi\))
The sum of the diagonal elements of a transformation matrix is called the Character (\(\chi\)).
\[ \chi(R) = \sum_{i} \mathbf{M}_{ii} \]
The character is invariant under basis transformations! This is the fundamental link to Character Tables in Lesson 5.
Examples of \(\chi\) in 3D:
Identity (\(E\)) 3
Inversion (\(i\)) -3
Reflection (\(\sigma\)) 1
Rotation (\(C_n\)) 2\cos\theta + 1
Transformation Math Problem Set Transformation Math
Problem Set: Matrix Representations of Symmetry
STUDENT:
PROBLEM SET #3 • PHYSICS
Instructions: Use linear algebra to represent geometric symmetry operations. For all rotation operations, assume the axis of rotation is the \(z\)-axis unless specified otherwise. Show your matrix construction and calculate the trace (character) for each.
1
The Inversion Operator
Construct the 3D matrix for the inversion operation \(i\). Prove that applying \(i\) twice returns the system to identity (\(i^2 = E\)).
\(\chi(i) = \) ____
2
Point Group \(C_{2v}\) (H2O)
The water molecule lies in the \(xz\)-plane. Identify and construct the four 3x3 matrices for the operations of the \(C_{2v}\) group: \(E\), \(C_2(z)\), \(\sigma_{xz}\), and \(\sigma_{yz}\).
Matrix: \(C_2(z)\)
[ 3 x 3 Matrix ]
Matrix: \(\sigma_{xz}\)
[ 3 x 3 Matrix ]
Matrix: \(\sigma_{yz}\)
[ 3 x 3 Matrix ]
Verification Task:
Multiply \(\sigma_{xz}\) and \(\sigma_{yz}\). Which group operation is the result equivalent to?
3
Improper Rotation \(S_4\)
The operation \(S_4\) is defined as a 90° rotation about the \(z\)-axis followed by reflection through the \(xy\)-plane. Derive the resulting 3x3 matrix by multiplying the two constituent operation matrices.
×
=
[ Resulting S4 Matrix ]
Challenge: Determinants
Calculate the determinant for each of your matrices in Problem 2 and 3. What do you notice about the determinants of proper rotations vs. improper operations (reflections and improper rotations)?
Transformation Math Answer Key Answer Key
Transformation Math • Lesson 3
Confidential
1
The Inversion Operator
Matrix Construction:
\[ i = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix} \]
Proof: \( i \cdot i = (-1)^2 \mathbf{I} = \mathbf{I} = E \). Any point \((x,y,z)\) mapped twice through the origin returns to its original position.
Character Value:
\(\chi(i) = -3\)
2
Point Group \(C_{2v}\) (H2O)
\(C_2(z)\)
\[ \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
\(\sigma_{xz}\)
\[ \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
\(\sigma_{yz}\)
\[ \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
Verification Key:
\(\sigma_{xz} \cdot \sigma_{yz} = C_2(z)\)
3
Improper Rotation \(S_4\)
\(\sigma_{xy}\)
×
\(C_4(z)\)
=
\(S_4(z)\)
Challenge Determinant Key:
• Proper Rotations (\(E, C_2, C_n\)): \(\det = +1\) (Orientation-preserving)
• Improper Operations (\(i, \sigma, S_n\)): \(\det = -1\) (Orientation-reversing / Chiral-switching)
Noether Theorem Slides Noether’s Theorem
Symmetry & Conservation in Physics
The Geometry of Law
"If the universe were shaped differently, would energy still be conserved?"
Conservation laws aren't just arbitrary rules—they are the inevitable consequences of the symmetries of space and time.
Continuous Symmetry
The Core Principle
Every differentiable symmetry of the action of a physical system has a corresponding conservation law.
— Emmy Noether (1915)
Symmetry
Invariance of the Lagrangian
Conservation
Noether Current / Charge
Spatial Translation
The Logic:
If the laws of physics are the same at \(x\) as they are at \(x + \Delta x\), then the system's Linear Momentum is conserved.
"The experiment works the same in London as it does in Paris."
Lagrangian Formalism
\[ \frac{\partial \mathcal{L}}{\partial x_i} = 0 \implies \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}_i} \right) = 0 \]
\[ p_i = \text{const} \]
The Fundamental Links
Time Translation
\( t \to t + \Delta t \)
Energy
Spatial Translation
\( \vec{r} \to \vec{r} + \Delta \vec{r} \)
Momentum
Rotation
\( \theta \to \theta + \Delta \theta \)
Angular Momentum
Continuous vs. Discrete Symmetry
Discrete (Crystal/Shape)
Not Noether-type
Leads to Selection Rules
Lattice Momentum
Continuous (Space/Time)
Noether's Theorem applies
Global Conservation Laws
Essential for Particle Physics
Lagrangian Symmetry Handout Lagrangian Symmetry
Noether’s Theorem: From Invariance to Conservation
DERIVATION REF: 04
"If the Lagrangian \(\mathcal{L}(q, \dot{q}, t)\) is invariant under a continuous transformation \(q \to q + \delta q\), then there exists a quantity \(J\) such that \(\frac{dJ}{dt} = 0\)."
1. The General Derivation
Consider a transformation \(q_i \to q_i + \epsilon \frac{\partial q_i}{\partial s}\). The change in the Lagrangian is:
\[ \delta \mathcal{L} = \sum_i \left( \frac{\partial \mathcal{L}}{\partial q_i} \delta q_i + \frac{\partial \mathcal{L}}{\partial \dot{q}_i} \delta \dot{q}_i \right) \]
Using Euler-Lagrange equations, we substitute \(\frac{\partial \mathcal{L}}{\partial q_i} = \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}_i} \right)\):
\[ \delta \mathcal{L} = \sum_i \left( \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}_i} \right) \delta q_i + \frac{\partial \mathcal{L}}{\partial \dot{q}_i} \frac{d}{dt}(\delta q_i) \right) \]
Applying the product rule in reverse:
\[ \delta \mathcal{L} = \frac{d}{dt} \sum_i \left( \frac{\partial \mathcal{L}}{\partial \dot{q}_i} \delta q_i \right) = 0 \]
The conserved current is: \( J = \sum_i \frac{\partial \mathcal{L}}{\partial \dot{q}_i} \frac{\delta q_i}{\epsilon} \)
A. Cyclic Coordinates
A coordinate \(q_k\) is cyclic if \(\frac{\partial \mathcal{L}}{\partial q_k} = 0\). This implies invariance under translation in \(q_k\).
Conserved: \( p_k = \frac{\partial \mathcal{L}}{\partial \dot{q}_k} \)
B. Angular Momentum
For a central potential \(V(r)\), the Lagrangian is invariant under rotation \(\theta \to \theta + \delta\theta\).
Conserved: \( L = m r^2 \dot{\theta} \)
Application Exercises
1. The Free Particle
Write the Lagrangian for a free particle in 1D. Identify the symmetry and use Noether's logic to find the conserved quantity.
2. Time Invariance
If the Lagrangian has no explicit time dependence (\(\frac{\partial \mathcal{L}}{\partial t} = 0\)), what quantity is conserved? Briefly explain the physical meaning.
NOTE: Noether's theorem applies to both Classical Mechanics and Quantum Field Theory. PHYSICS DEPT • ADVANCED DYNAMICS
Selection Rules Slides Selection Rules
Symmetry & Spectroscopic Transitions
Forbidden Beauty
"How can we know which colors a molecule absorbs without doing a single integral?"
In quantum mechanics, if the "shape" of the transition is asymmetric, the probability of it happening is zero.
\[ \int \psi_f^* \hat{\mu} \psi_i \, d\tau = 0 \]
The Vanishing Integral
The Transition Integral
\(\psi_i\)
Initial State
The wavefunction before excitation.
\(\hat{\mu}\)
Dipole Operator
Usually transforms as \(x, y,\) or \(z\).
\(\psi_f\)
Final State
The excited state wavefunction.
Rule: The product \(\psi_f \otimes \hat{\mu} \otimes \psi_i\) must contain the Totally Symmetric Representation (\(A_1\) or \(A_g\)) for the transition to be "allowed."
How to Predict Transitions
1
Identify the point group of the molecule (e.g., \(C_{2v}\)).
2
Find the symmetry labels (\(A_1, B_2,\) etc.) for your wavefunctions.
3
Use the Direct Product Table to multiply the representations.
\( \otimes \) \(A_1\) \(B_1\) \(A_1\) \(A_1\) \(B_1\) \(B_1\) \(B_1\) \(A_1\)
Example: Multiplication in C2v
The Laporte Rule (Centrosymmetric)
In molecules with a center of inversion (\(i\)), transitions between states of the same parity are forbidden .
Parity Rule:
\(g \to u\) Allowed
\(u \to g\) Allowed
\(g \to g\) Forbidden
i
Centrosymmetric molecules like Octahedral Complexes (Oh) follow this strictly.
Character Table Reference Character Data
Reference Tables for Symmetry & Spectroscopy
REF: PT-G-05
C2v
Point Group: \(C_{2v}\) (H2O, SO2)
\(C_{2v}\) \(E\) \(C_2\) \(\sigma_v(xz)\) \(\sigma_v'(yz)\) Linear Functions \(A_1\) 1 1 1 1 \(z\) \(A_2\) 1 1 -1 -1 \(R_z\) \(B_1\) 1 -1 1 -1 \(x, R_y\) \(B_2\) 1 -1 -1 1 \(y, R_x\)
C3v
Point Group: \(C_{3v}\) (NH3)
\(C_{3v}\) \(E\) \(2C_3\) \(3\sigma_v\) Linear Functions \(A_1\) 1 1 1 \(z\) \(A_2\) 1 1 -1 \(R_z\) \(E\) 2 -1 0 \((x,y), (R_x, R_y)\)
Direct Product Table Rules
General Rules
1 \(A \otimes A = A\), \(B \otimes B = A\), \(A \otimes B = B\)
2 \(1 \otimes 1 = 1\), \(2 \otimes 2 = 1\), \(1 \otimes 2 = 2\)
3 \(g \otimes g = g\), \(u \otimes u = g\), \(g \otimes u = u\)
4 \('\) \(\otimes\) \('\) = \('\), \('\) \(\otimes\) \(''\) = \(''\), \(''\) \(\otimes\) \(''\) = \('\)
Selection Logic
A transition between \(\psi_i\) and \(\psi_f\) is Electric-Dipole Allowed if the direct product \(\Gamma(\psi_f) \otimes \Gamma(\mu) \otimes \Gamma(\psi_i)\) contains the totally symmetric representation (\(A_1, A_{1g}, \Sigma^+\)).
Note: The Dipole Operator \(\mu\) transforms as the linear functions \(x, y,\) or \(z\) in the character table.
Symmetry Groups & Conservation Laws • Physics Reference
Spectroscopic Predictor Assessment Spectroscopic Predictor
Final Mastery Assessment
STUDENT:
SCORE:
Scenario: Transition Analysis in Ammonia (NH3)
Ammonia (\(NH_3\)) belongs to the \(C_{3v}\) point group. A researcher is studying a transition from the ground state orbital (symmetry \(A_1\) ) to an excited state orbital (symmetry \(E\) ).
Initial State \(\Gamma_i = A_1\)
Final State \(\Gamma_f = E\)
Point Group \(C_{3v}\)
1 Determine the transition dipole operator symmetry.
Identify the symmetry of the \(z\)-axis and the \((x,y)\) pair in the \(C_{3v}\) character table.
\(z\)-dipole (\(\mu_z\)) transforms as: _________
\(xy\)-dipole (\(\mu_{xy}\)) transforms as: _________
2 Analyze the \(z\)-polarized transition.
Calculate the direct product \(\Gamma_f \otimes \Gamma_{\mu_z} \otimes \Gamma_i\). Is this transition allowed?
3 Analyze the \(xy\)-polarized transition.
Calculate the direct product \(\Gamma_f \otimes \Gamma_{\mu_{xy}} \otimes \Gamma_i\). Is this transition allowed?
The Selection Synthesis
Based on your analysis above, if you shine unpolarized light on a sample of ammonia, will you see an absorption peak for this \(A_1 \to E\) transition? If so, which component of the light wave (the vertical electric field or the horizontal one) is doing the work?
PHYSICS SYMMETRY GROUP • END OF SEQUENCE ASSESSMENT
