Lecture 11

> Uncertainty Principle

> Atomic Structure, Spectra

*Beiser, Mahajan & Choudhury, Concepts of Modern Physics 7/eFrench, Special Relativity*Nolan, Fundamentals of Modern Physics 1/eSerway, Moses & Moyer, Modern Physics 3/eTaylor & Wheeler, Spacetime Physics 2/eTipler & Llewellyn, Modern Physics 5/e

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Uncertainty Principle

2

> Measuring position and wave number + Probable location of a particle: |Ψ|2

→ →+ Narrow wave groups Less definite λ Imprecise p → →+ Definite λ Wider location Imprecise x

> Heisenberg's Uncertainty Principle 1927 + “It is impossible to know both the exact position and exact momentum of an object at the same time.” + Wave group: particle motion, many single waves + Fourier transform: multiple k's, position description

Villacorta--DLSUM-PHYFUN4-L11-1718Term02 Beiser, ch02

Uncertainty Principle contd

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> Minimum value for standard deviations ∆x & ∆k + Gaussian: ½ + Others: greater than ½ + From de Broglie's formula:

> Final form:

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Gaussian

Uncertainty Principle contd

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> Interpretation + Limits to knowing position and momentum at the same time + Future predictions difficult to know because present info is limited + Probabilities of finding a particle at this position or at w/ this momentum

> Use h-bar to simplify the equation

> Leading to

> Alternative example: measuring an electron →+ Photon λ: long λ small ∆p →short λ small ∆x + Error trade-off

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Uncertainty Principle contd

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> Lower limits are imposed by the uncertainty principle: it can relate not just position and momentum but also energy, frequency, etc.

> Energy and time also have uncertainties + Measuring the energy/frequency requires a longer observation time + From the equations, we get

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leading toor

Atomic Model

6

> Model of the atom + Continuous matter, discontinuous atoms + Protons, neutrons, electrons + Niels Bohr, hydrogen atom

> Rutherford's experiment + Early model: Thompson's plum pudding + Testing the model: Geiger-Marsden experiment ~ α-particles (He atoms – 2 electrons) hitting gold foil ~ Zinc screen ~ α-particles would pass through w/o deflection due to even dist of charges ~ Some large deflections were observed ~ α mass: 8k x electron mass; α speed: 2 x 107 m/s + Explanation: positive charges are concentrated at the center

Villacorta--DLSUM-PHYFUN4-L11-1718Term02 Beiser, ch04

Atomic Model contd

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> Further observations ↔+ Nuclear charge deflection angle + Nuclear charge: e–, multiples + Positive charge e+: atomic number Z + Atom, matter: empty space

> Rutherford scattering

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Inverse 4th power leads tovery few deflections at large θ

Atomic Model contd

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> Dimensional considerations + Rutherford: nuclear radius << deflection radius R

+ KE of approaching α-particle

+ Deflection radius + For the experiment: ~ Z: 79 for gold ~ KE: 7.7 MeV ~ 1.2 x 10–12 J

+ Deflection radius R: 3.0 x 10–14 m

> Inaccurate at larger energies

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Very few N for large θ

Electron Orbit

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> Consider the energy of the electron as it goes around the nucleus: E = K + P

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Electric PE

Electron velocity

E < 0 for closed orbitsE of the nucleus-electron system

> Orbiting e + Circular motion + Acceleration + Radiating E accd to EM theory + Decreasing radius

> Rutherford's analysis coincides w/ quantum theory coincidentally

> A quantum explanation is needed

Atomic Spectra

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> Spectral lines + Planck energy, wavelength + For rarefied gases, interactions are few and lead to emissions when molecules interact + The emitted radiation is unique to the gas

> Spectroscopy + Emissions from excited gas molecules + Emission spectra + Absorption spectra

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Atomic Spectra contd

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> Different equations describe the emitted light

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Balmer

Lyman

Summary

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> The Uncertainty Principle describes the limit of accurate measurement

> The structure of the atom suggests the light that it emits.

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Sample Problems

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1. The position and momentum of a 1.00-keV electron are simultaneously determined. If its position is located to within 0.100 nm, what is the percentage of uncertainty in its momentum? (Beiser)

2. An unstable elementary particle called the eta meson has a rest mass of 549 MeV/c2 and a mean lifetime of 7.00 x 10–19 s. What is the uncertainty in its rest mass? (Beiser)

3. Find the frequency of revolution of the electron in the classical model of the hydrogen atom. In what region of the spectrum are electromagnetic waves of this frequency? (Beiser)

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