ch 4 - intro to quantum mechanics

What do all waves have in common

Oscillatory variation of some property with time at a fixed location ; described by the same equations

Kinds of waves and oscillating quantities

Water : height of surface

Sound : density of air

Light : electric and magnetic fields

Chemical : concentrations of chemical species

Amplitude (A)

Max displacement from undistrubed (crest height or trough depth). Always positive

Wavelength (λ)

Distance between successive crests or troughs; measured peak-to-peak

Frequency (ν)

Number of cycles per passing second passing a point; units s^-1 or Hertz (Hz)

Period (T)

Time for one complete cycle : T= 1/ν

Relation ship of speed , ν, and λ

Wave speed v = λν . Speed = distance traveled / time elapsed

What is electromagnetic radiation

A wave of oscillating electric and magnetic fields carrying energy and momentum; does not require a medium

Maxwells contribution

1865 theory describing light as EM radiation

Orientation of E and B Fields

Perpendicular to each other and to direction of propagation

Fundamental equation for electric field

E( x, t) = Emax cos[ 2pie (x/ wavelength - frequency ( time)]

Speed of Light

C = νλ , c = 3.00 × 10^8

EM spectrum regions

radio → microwave → infrared → visible → ultraviolet → x-ray → gamma

Wavelength decreases, frequency and energy increase across this order

Interference

Overlap of light waves → constructive (bright) or destructive (dark) patterns

Diffraction

Spreading of waves through slits; double slit experiment shows interference pattern

Ionizing radiation

UV, X-ray, gamma (λ shorter than ~340 nm ) can ionize atoms, damage tissue

Why did rutherfords model fail

Accelerating electrons should emit EM radiation , lose energy, and collapse into the nucleus

What classical law predicts collapse

Maxwells electromagnetic theory of radiation from accelerating charges

What was the key Mystery

Atoms are stable despite predictions of collapse

What concept solved it

Energy quantization - electrons can occupy only discrete energy levels

First evidence of energy quantization

Black body radiation experiments required quantized energy to match observations

2nd evidence of energy quantitative

Gas phase atomic experiments showed discrete energy changes, explained with energy level diagrams

What is black body radiation

Thermal EM radiation emitted by all objects ; spectrum depends on temp

Classical prediction and issue

Predicts infinite `intensity at high v (Uv catastrophe)

Planks key hypothesis

Energy of oscillators = nav

Planck constant (h)

6.63 × 10^-34 Js

Physical meaning 0f quantizati0m

Energy exchanges occur in integer multiples of hv; cannot vary smoothly

Observational evidence

Peak wavelength shifts to shorter λ As T rises; intensity finite at high v

What is a spectrograph

Device that passes light through a narrow slit and prism , dispersing it into component wavelengths and recording the spectrum

Difference between emission and absorption spectra

Emission : bright discrete lines on dark background

Absorption : dark lines on continuous spectrum

Evidence for discrete energy levels

Each element shows a unique patterns of lines

Bohrs contribution

Electrons occupy quantized energy states ; light is emitted / absorbed during transitions with delta E = hv

Absorption vs emission sign of delta e

Absorption : delta e > 0

Emission : delta e< 0

Practical use of spectra ?

Identify elements and determine energy levels

Purpose of frank hertz experiment

Demonstrate that atomic energy levels are quantized by measuring electron energy loss during collisions

Apparatus and method

Heated collisions emits electrons → accelerated toward anode → pass through to collector. Measure current vs accelerating voltage in low pressure mercury gas

Observation at threshold voltage

Sharp current drop → electrons lose specific energy exciting mercury atoms

Formula linking threshold to photon frequency

V = eVthr/ h

Multiple thresholds meaning

Multi-le excited states in atom; each threshold corresponds to a higher energy level

Problem Bohr solved

Classical electrons should radiate and collapse; need stable atom explanation

Bohrs quantization

Angular momentum L= nh/ 2pie

Condition for stationary orbit ?

Coulomb force provides centripetal acceleration

Orbits radius formula

Rn = n²/Z (a0) where a0 = 5.29× 10^-11 m

Electron speed?

Vn = Ze²/2Eo nh

Energy levels

En = -2.18×10^-18 Z²/n² J

Emission / absorption rule ?

Delta E =hv

Ionization energy for H

2.18 × 10^-18 J

Spectral prediction

Accurately reproduces hydrogen emission series

Bohr model shortcoming

Gave energy levels but not the reason for quantization

What is wave particle duality

Phenomenon where matter and radiation exhibit both wave and particle properties depending on the experiment

Einstein contribution

Explained photoelectric effect by treating light as particles (photons ) with energy E= hv

De broglie hypothesis

Particles have wavelength λ= h/p ; predicted electron diffraction and standing waves

Why important ?

Provided conceptual basis for modern quantum mechanics

What is the photo electric affect

Ejection of electrons from a metal when illuminated by light or sufficient frequency

Key experimental findings of photo electric

1) no emission below threshold frequency

2) immediate emission above threshold

3) kinetic energy depends on frequency, not intensity

Einsteins equation

Emax = hv - work function of metal

What is wave particle duality

Light and matter shows both wave and particle properties depending on the experiment

What is the photoelectric effect

Emissions of electrons from a metal surface when light of succidenr frequency shines on it, producing an electric current

What is the metal surface called in the photoelectric effect

Photocathode

What is the emitted electric current called

Photocurrent

Does increasing light intensity alone cause photoemission

No, no electrons are emitted until light frequency exceeds a threshold v0 specific to the metal

Once the threshold frequency is exceeded how does the photo current depend on intensity

It is directly proportional to light intensity

What happens with low frequency, high intensity red light vs low intensity blue light

Low frequency red light produces no emission ; dim high frequency blue light can eject electrons

Why do emitted electrons have a range of kinetic energies

Some originate deeper in the metal and lose energy through collisions before escaping

How is the maximum kinetic energy Emax of photo electrons measured

By applying a stopping potential vmax : Emax = e Vmax

What is a photon

A quantum bundle of light energy with energy ephoton = hv

Einisteins energy equation for the photoelectric effect

Emax = hv - work function

What does the work function represent

The minimum energy needed to free an electron from the metals surface

How does Emax depend on light frequency

It’s linear with slope h and y-intercept work function

How did the photo elctric effect challenge classical wave theory

Classical theory predicted energy depends on light intensity not frequency . Experiments showed that frequency is key

What concept emerged from these findings

Wave particle duality : light exhibits both wave like and particle like behavior

How does quantum mechanics address lights nature

Light behaves as a wave or particle depending on the experimental conditions

What new idea did Louis de broglie propose

Particles such as electrons have wavelike properties

What is a traveling wave

A wave that moves through space , like em radiation

What is a standing wave

A wave confined between fixed ends , with nodes and anti nodes

Conditions for standing waves in a string of length L

Nλ/2= L where n = 1,2,3….

What phenomenon reveals the wavelike nature of particles

Interference and diffraction . When waves overlap, constructive interference , bright spots and destructive interference , dark spots occur producing a diffraction pattern

How is electron diffraction analogous to light diffraction

Electrons striking a thin crystal foil producing alternating bright/dark patterns, just as light produces fringes when passing through a double slit

Formula for an electron does broglie wavelength in terms of kinetic energy

λ= h/ square root (2meEk)

Diffraction condition for electrons from a crystal surface

Naλ= asintheta and nbλ=b sin theta

What does angle theta represent

The scattering angle relative to the surface normal where constructive interference occurs

What does the diffraction pattern reveal

The atomic arrangement and spacing of surface atoms

How does de broglie standing waves description of electrons lead to interdminacy

Bc the electron wave is spread uniformly around an orbit the angular position cannot be specified , the momentum is definite (p=h/λ) but the position is interdeminate

What does the term indeterminacy better describe the principal than uncertainty

Interdiminate means the property truly has no definitive value where uncertainty implies we could reduce error with better experiments

Core postulate of Boris model

Electrons orbit in discrete stable orbits without radiating energy

Angular momentum quantization formula

L= n (h/2pi)

De broglie wavelength formula

λ = h/p

What does a particle in a box model represent

A particle confined to move only between two infinity high potential walls, free inside but unable to escape

Potential energy inside and outside the box.

Inside : V(x) =0, outside : V(x) =infinity

Boundary conditions for the wave function

Ψ(0)=0 and Ψ(L)=0

General solution satisfying boundary conditions

ΨN (x)= square root (2/L) sin (npix/ L), n = 1,2,3…

Why n=0 is not allowed

Would give Ψ=0 everywhere → no particle

Number of notes for state n

N-1 nodes

Formula for quantized energy levels En

En= n² h² / 8mL²

Relation between energy and quantum number

En is directly proportional to quantum number

What is zero point energy

The finite ground state energy E1;

How does. N affect wavelengths and nodes

Higher n → shorter wavelength, more nodes, more energy