Organic Chemistry I Lecture 7 - Quantum Mechanical Kinetic Energy

Name: Organic Chemistry I Lecture 7 - Quantum Mechanical Kinetic Energy
Uploaded: 2018-09-01T01:13:54.000Z
Duration: 1 h 38 min 18 s

Crystal Structure and Atomic Vibrations

Precision in Molecular Measurements

The crystal structure of a complex molecule is presented, showcasing remarkable precision with bond distances reported to ±0.002 angstroms.

Typical atomic vibrations are about 0.05 angstroms, which is significantly larger than the measurement precision, indicating that atoms are constantly in motion.

No molecule appears static; instead, atoms are always slightly displaced due to their vibrational movements.

Purpose of Precise Measurements

The primary reason for such precise measurements is to accurately determine average atomic positions for further analysis rather than merely knowing bond distances.

Accurate positioning is crucial for understanding bonding interactions; incorrect atom positions can lead to misleading conclusions about molecular behavior.

Analyzing Electron Density

Difference Electron Density Maps

A difference electron density map reveals how electron density shifts when molecules form from individual atoms, showing expected patterns between carbon atoms and CH bonds.

The first slice through the molecule confirms anticipated results but also highlights the beauty of the data produced by meticulous work.

Unexpected Findings in Bonding

A second slice shows bonding structures clearly but reveals an unexpected absence of electron density in one bond associated with three-membered rings.

This missing bond indicates "pathological bonding," challenging traditional Lewis structures and expectations regarding shared electrons.

Pathological Bonding Observations

Anomalies in Bond Structures

Further examination through another plane slice again shows the absence of expected electron density for certain bonds within the structure.

Observations indicate that some bonds appear denser yet do not align with connecting lines between nuclei, suggesting they may be bent or distorted.

Limitations of Traditional Models

While Lewis pairs and octets serve as useful tools for tracking valence electrons, they fall short in accurately describing real-world electron distributions observed experimentally.

Is There a Better Bond Theory than Lewis' Theory?

Electron Sharing and Bonding Insights

The discussion begins with the concept of electron sharing, highlighting that the distortion of electron density spheres around atoms is only about 5% of what Lewis predicted regarding electron pairs.

Unshared pairs exist as Lewis suggested, but their presence is also less than 5% of his predictions, prompting questions about the adequacy of Lewis' bonding theory.

Introduction to Chemical Quantum Mechanics

The speaker introduces chemical quantum mechanics as a superior theory for understanding bonding, emphasizing its quantitative nature compared to Lewis' qualitative approach.

This course will focus on applying quantum mechanics specifically to bonding rather than solving mathematical problems typically emphasized in physics or physical chemistry.

Schrodinger's Wave Equation

The course will explore insights into bond formation through Schrodinger's wave equation, which was invented in 1926.

A brief background on Erwin Schrödinger is provided, noting his initial obscurity before making significant contributions to physics.

Historical Context and Development

An anecdote recounts Schrödinger’s early career and interactions with notable physicists like Felix Bloch during colloquia at the University of Zurich.

During these discussions, Schrödinger presented ideas inspired by Louis de Broglie's association between waves and particles.

Understanding the Wave Function (Psi)

The speaker explains Schrödinger's equation (Hψ = Eψ), introducing "psi" (Ψ), which represents a wave function—a concept that puzzled even leading physicists at the time.

A humorous reference highlights how young physicists like Felix Bloch were curious about the meaning behind psi while acknowledging its utility in calculations.

What Is Matter? Insights from Schrödinger's Lecture

Understanding Schrödinger's Confusion with Psi

The question of what "psi" represents has puzzled many, including Schrödinger himself, who expressed discomfort with its meaning even decades after his initial inquiries.

The lecture emphasizes that confusion about psi is common and encourages learners not to feel disheartened by the complexity of its interpretation.

Defining Psi as a Wave Function

Psi is introduced as a wave function, which requires understanding what it is a function of—specifically, the position of particles in space.

Different functions can share the same variable; for example, sine and cosine are both functions of an angle but represent different mathematical relationships.

Characteristics and Variables of Psi

Wave functions are named using quantum numbers (n, l, m), but these names do not define what the function depends on; they merely label them.

Psi is fundamentally a function of particle positions (x, y, z), time (though less emphasized in this course), and potentially spin for certain particles.

Complexity in Multi-Particle Systems

For n particles in three-dimensional space, psi requires 3n arguments to specify their positions accurately.

The discussion progresses from one-particle systems to more complex scenarios involving multiple electrons and atoms before addressing molecular structures.

Introduction to Quantum Mechanics through the Schrödinger Equation

The focus shifts to understanding quantum mechanics via the time-independent Schrödinger equation: Hψ = Eψ. This equation describes stationary states where atomic clouds are analyzed without considering time-dependent changes.

In this context, E represents the system's energy while H acts as an operator that modifies psi rather than being directly canceled out.

Energy Components within Quantum Systems

The left side of the equation involves kinetic and potential energy components derived from how H interacts with psi.

Potential energy can be calculated using Coulomb’s law if particle charges and positions are known. This foundational concept links classical physics with quantum mechanics.

Understanding Kinetic and Potential Energy in Quantum Mechanics

Calculating Potential Energy

The potential energy can be calculated using a systematic approach once the system is defined, involving distances and charges.

The formula for potential energy is straightforward when you know the wave function (Hψ/ψ).

Kinetic Energy: Historical Context

A discussion on kinetic energy reveals past misconceptions, referencing Thompson's incorrect proposal from 1923.

Coulomb's Law has been validated through extensive spectroscopic evidence related to atomic spectra.

Classical vs. Quantum Kinetic Energy

In classical mechanics, kinetic energy is simply the sum of individual particle energies (1/2 mv²).

This classical formula is insufficient for quantum systems; thus, a new formulation is necessary.

Quantum Kinetic Energy Formulation

The correct expression for kinetic energy in quantum mechanics involves constants and second derivatives of the wave function.

It incorporates terms like H²/(8π²), emphasizing that it depends on curvature rather than just velocity.

Simplifying Kinetic Energy Concepts

Focusing on one particle simplifies understanding; kinetic energy becomes a negative constant times the second derivative of the wave function divided by ψ.

The second derivative indicates how curved the wave function is, which directly influences kinetic energy calculations.

Curvature and Scaling Effects

An explanation of how scaling affects slope but not curvature illustrates that kinetic energy relies more on shape than magnitude.

Doubling psi does not quadruple curvature; hence, kinetic energy remains unchanged under such transformations.

Solving Quantum Problems

In quantum mechanics problems, given parameters include mass and charge of particles along with potential laws like Coulomb’s law.

Finding the wave function (ψ), which represents particle positions over time or spin, is crucial to solving these problems.

Understanding the Role of Psi in Quantum Mechanics

The Nature of Psi

Psi must be defined such that changes in kinetic energy compensate for changes in potential energy, ensuring a correct representation.

It is crucial for psi to remain finite, single-valued, continuous, and integrable to maintain mathematical validity.

Finding the Area Under Psi Squared

Determining the area under psi squared is essential; it relates to understanding how kinetic and potential energies interact.

The lecture will progress from one particle in one dimension to more complex systems like multi-electron atoms and molecules.

Approach to Problem Solving

Initially, problems are approached by starting with known answers (psi values), then deducing questions related to kinetic energy.

For example, if psi is represented as sin(x), students can derive kinetic energy using its curvature.

Kinetic Energy Calculations

The relationship between kinetic and potential energy is established through their variations with position; they must sum up to a constant value.

A sine function indicates a particle not influenced by external forces, resulting in constant potential energy across positions.

Exploring Variations of Sine Functions

Modifying the sine function (e.g., sin(2x)) results in shorter wavelengths and higher kinetic energies due to increased frequency.

Kinetic energy becomes proportional to 1 over wavelength squared; thus, shorter wavelengths yield greater kinetic energies.

Examining Exponential Functions

Understanding Exponential Behavior

Analyzing e^x reveals that its second derivative remains e^x; this leads to negative kinetic energy scenarios.

Negative kinetic energy implies total energy is lower than potential energy—a concept not typically encountered in classical mechanics.

Implications for Atomic Structure

This phenomenon isn't merely theoretical; it occurs within every atom where electrons may exhibit negative kinetic energies at certain distances from nuclei.

As distance increases from atomic nuclei, the influence of Coulomb's law diminishes significantly, leading to these unique energetic states.

Determining Potential Energy from Psi

Mapping Potential Energy Relationships

By analyzing the wave function (psi), which varies between positive and negative values across positions, we can infer corresponding potential energies based on calculated kinetic energies.

Understanding Kinetic Energy and Curvature in Wave Functions

The Jeopardy Approach to Kinetic Energy

The discussion begins with the "jeopardy approach," focusing on how curvature (negative constant) relates to amplitude and kinetic energy, which can be positive, zero, or negative.

Analyzing Graph Positions for Kinetic Energy

A question is posed about determining kinetic energy at a specific graph position based on curvature divided by amplitude. Students are engaged to assess the curvature at various points.

Curvature Analysis

At one point, it is identified that there is zero curvature; thus, the kinetic energy at this point is also zero. This leads to a color coding system where gray represents zero kinetic energy.

Another straight point confirms zero curvature again, reinforcing that without curvature information, certainty about kinetic energy remains elusive.

Questionable Points of Kinetic Energy

Further analysis reveals another straight section with zero amplitude leading to uncertainty regarding its kinetic energy value.

As students identify curving directions (up or down), they learn that negative curvature indicates negative kinetic energy while positive curvature suggests positive values.

Finalizing Kinetic Energy Values

The instructor clarifies that both upward and downward curvatures can yield positive energies depending on their relation to the baseline (psi equals zero).

It’s concluded that if the curve moves away from psi equals zero, it results in negative kinetic energy.

Summary of Potential and Total Energies

The relationship between potential and total energies is discussed: when total energy exceeds potential energy, kinetic energy is positive; conversely, when potential exceeds total, kinetic becomes negative.

This structured breakdown provides clarity on how wave function characteristics influence physical properties like kinetic and potential energies through graphical analysis.