Ch 6 Lecture 1- Quantum Mechanics
${{}}\frac{\differentialD f}{\differentialD x}{}\frac{df}{\differentialD t } Overview of Quantum Mechanics
Introduction to basic concepts needed for quantum mechanics in a general chemistry context.
Key Properties of Light
Definition of Light:
Light is described as a wave, specifically an oscillation that travels through space.
The light wave's propagation can be imagined from left to right.
Wave Equation:
Described by the equation: C = oldsymbol{λν}
Where:
C is the speed of light (assumed to be 3imes10^8{ m/s} ).
λ (lambda) is the wavelength: the distance between one part of the wave and the next equivalent part (peak, crossing point, or trough).
ν (nu) is frequency: the number of oscillations per unit time.
Intensity:
Defined as the height of the wave, related to the brightness of light.
If the brightness doubles, the intensity is also doubled.
Energy Relationships:
The energy of a light wave is proportional to one over its wavelength: E imes ext{wavelength} .
Energy is also believed to be proportional to intensity prior to the advent of quantum mechanics, which modifies this assumption.
The Nature of Light
Vector Fields:
Light is represented by two 3D vector fields:
Electric Field (E)
Magnetic Field (B)
These fields are depicted on a Cartesian plane, with B showing fluctuations in one direction and E in another, collectively forming an electromagnetic wave.
Wave Equation for Light:
Reference point for later discussions, contrasting with the Schrödinger equation for matter particles.
Assumption: Light wave velocity is 1 for simplicity.
Definition of terms:
f as a function representing the wave.
\frac{{df}}{\differentialD x}\frac{df}{\differentialD t} signifies the partial derivatives of the function with respect to space and time.
This leads to a breakdown in mathematical treatments of light waves.
Linear Wave Equations
Properties of Linear Equations:
The light wave equation is linear, implying that the sum of the solutions yields another solution.
Interference Patterns:
Constructive Interference:
Occurs when two waves meet in phase, amplifying their amplitude (e.g., +1 + +1 = +2).
Destructive Interference:
Happens when waves are out of phase, reducing their amplitude (e.g., +1 + -1 = 0).
Conservation of Matter Implications:
Energy conservation principles preclude matter from displaying similar interference patterns observed in light waves.
Introduction to Energy Quantization
Concept of Quantized Energy:
Energy consists of infinitesimal components denoted by ε_i .
Total energy is the sum of all individual energy components: E=+n\varepsilon_{i} .
Partition Function:
Introduced as a useful function for computing properties in statistical mechanics:
Q = ext{e}^{-ε/kT} (unitless).
Manipulation of Equations:
Methodical restructuring of energy equations through multiplication and subtraction, introducing the variable β = rac{1}{kT}.
Energy Derivative Relations:
Relationship between energy and its representation in terms of derivatives.
Planck's Contribution to Black Body Radiation
Black Body Definition:
A hypothetical object that absorbs and emits all frequencies of radiation uniformly.
Experimental Problem:
Classic literature outlines the discrepancy between experimental data and predictions made by the Rayleigh-Jeans law.
Planck’s Fix:
Introduces the concept of quantized energy packets, or quanta.
Energy emanating from the black body becomes:
E = hν/(e^{hν/kT} - 1)
Where h is Planck's constant.
Understanding the Original Error:
Prior theories led to a runaway energy prediction in the ultraviolet spectrum, necessitating Planck’s insights to rectify the issue.
Key Relationships in Quantum Mechanics
Formula Recap:
For one quantum of energy: E = hν.
Light speed equation: c = λν.
Rearranging allows one to express frequency as: ν = rac{c}{λ} leading to the energy expression E = rac{hc}{λ}.
Energy Proportionality:
Energy is proportional to both frequency and rac{1}{λ} (inverse wavelength).
Momentum Connections:
The total energy for a free particle is E = rac{mv²}{2}.
Expressed alternatively: E = rac{P²}{2m}, establishing connections between energy and momentum in a wave-particle duality framework.
Final Assessment:
Consolidating previously discussed principles to encapsulate foundational quantum mechanics.
Assignment Recommendations:
Encourages students to review concepts, practice the equations, and prepare for the next discussion on quantum mechanics.
Practice Problems:
Practice Problems:
Calculate the frequency of light with a wavelength of 500 nm500 nm. (C=3×108 m/sC=3×108 m/s)
What is the energy of a photon with a frequency of 6×1014 Hz6×1014 Hz? (h = 6.626\times10^{-34}\text{ J·s})
An electron (m=9.109×10−31 kgm=9.109×10−31 kg) is moving at a velocity of 2.2×106 m/s2.2×106 m/s. Calculate its kinetic energy.
Using Planck's Black Body Radiation equation, describe what happens to the energy emitted as frequency (νν) approaches zero. Assume ehν/kT≈1+hν/kTehν/kT≈1+hν/kT for small νν.
If the energy of a light wave is doubled, how does this affect its wavelength, assuming all other factors remain constant?