Transcript Atomic Spectra

PHY206: Atomic Spectra
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Lecturer: Dr Stathes Paganis
Office: D29, Hicks Building
Phone: 222 4352
Email: [email protected]
Text: A. C. Phillips, ‘Introduction to QM’
http://www.shef.ac.uk/physics/teaching/phy206
Marks: Final 70%, Homework 2x10%, Problems Class 10%
Course Outline (1)
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Lecture 1 : Bohr Theory
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Lecture 2 : Angular Momentum (1)
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Introduction
Bohr Theory (the first QM picture of the atom)
Quantum Mechanics
Orbital Angular Momentum (1)
Magnetic Moments
Lecture 3 : Angular Momentum (2)
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Stern-Gerlach experiment: the Spin
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Orbital Angular Momentum (2)
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Examples
Operators of orbital angular momentum
Lecture 4 : Angular Momentum (3)
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Orbital Angular Momentum (3)
Angular Shapes of particle Wavefunctions
 Spherical Harmonics
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Examples
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Course Outline (2)
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Lecture 5 : The Hydrogen Atom (1)
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Central Potentials
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QM of the Hydrogen Atom (1)
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The Schrodinger Equation for the Coulomb Potential
Lecture 6 : The Hydrogen Atom (2)
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QM of the Hydrogen Atom (2)
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Classical and QM central potentials
Energy levels and Eigenfunctions
Sizes and Shapes of the H-atom Quantum States
Lecture 7 : The Hydrogen Atom (3)
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The Reduced Mass Effect
Relativistic Effects
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Course Outline (3)
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Lecture 8 : Identical Particles (1)
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Lecture 9 : Identical Particles (2)
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Particle Exchange Symmetry and its Physical Consequences
Exchange Symmetry with Spin
Bosons and Fermions
Lecture 10 : Atomic Spectra (1)
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Atomic Quantum States
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Lecture 11 : Atomic Spectra (2)
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Central Field Approximation and Corrections
The Periodic Table
Lecture 12 : Review Lecture
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Atoms, Protons, Quarks and Gluons
Atomic Nucleus
Atom
Proton
Proton
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gluons
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Atomic Structure
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Early Models of the Atom
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Rutherford’s model
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Planetary model
Based on results of thin foil
experiments (1907)
Positive charge is
concentrated in the center of
the atom, called the nucleus
Electrons orbit the nucleus
like planets orbit the sun
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Classical Physics: atoms should collapse
Classical Electrodynamics: charged particles This means an electron should fall
radiate EM energy (photons) when their
into the nucleus.
velocity vector changes (e.g. they accelerate).
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Light: the big puzzle in the 1800s
Light from the sun or a light bulb has
a continuous frequency spectrum
Light from Hydrogen gas has a
discrete frequency spectrum
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Emission lines of
some elements
(all quantized!)
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Emission spectrum of Hydrogen
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“Quantized” spectrum
DE
DE
“Continuous” spectrum
Any DE is
possible
Only certain
DE are
allowed
Relaxation from one energy level to another by emitting a photon,
with DE = hc/l
If l = 440 nm, DE = 4.5 x 10-19 J
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Emission spectrum of Hydrogen
The goal: use the emission spectrum to determine the
energy levels for the hydrogen atom (H-atomic spectrum)
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Balmer model (1885)
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Joseph Balmer (1885) first noticed that the
frequency of visible lines in the H atom spectrum
could be reproduced by:
1 1
 2 2
2 n
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n = 3, 4, 5, …..
The above equation predicts that as n increases,
the frequencies become more closely spaced.
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Rydberg Model
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Johann Rydberg extended the Balmer model by finding more
emission lines outside the visible region of the spectrum:
1
1
  Ry  2  2 
 n1 n 2 
n1 = 1, 2, 3, …..
n2 = n1+1, n1+2, …
Ry = 3.29 x 1015 1/s
this model the energy levels of the H atom are proportional
 In
to 1/n2
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The Bohr Model (1)
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Bohr’s Postulates (1913)
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Bohr set down postulates to account for (1) the stability of the
hydrogen atom and (2) the line spectrum of the atom.
1. Energy level postulate An electron can have only specific
energy levels in an atom.
– Electrons move in orbits restricted by the requirement that the
angular momentum be an integral multiple of h/2p, which means that
for circular orbits of radius r the z component of the angular
momentum L is quantized:
L  mvr  n
2. Transitions between energy levels An electron in an atom
can change energy levels by undergoing a “transition” from
one energy level to another.
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The Bohr Model (2)
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Bohr derived the following formula for the energy
levels of the electron in the hydrogen atom.
Bohr model for the H atom is capable of reproducing
the energy levels given by the empirical formulas of
Balmer and Rydberg.
2
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18 Z 
E  2.178 x10  2 
n 
• Ry x h = -2.178 x 10-18 J
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Energy in Joules
Z = atomic number (1 for H)
n is an integer (1, 2, ….)
The Bohr constant is the same
as the Rydberg multiplied by
Planck’s constant!
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The Bohr Model (3)
2
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Z
18
E  2.178 x10  2 
n 
• Energy levels get closer together
as n increases
• at n = infinity, E = 0
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Prediction of energy spectra
• We can use the Bohr model to predict what DE is
for any two energy levels
DE  E final  E initial
 1 
 1 
18
18
DE  2.178x10 J 2   (2.178x10 J) 2 
 ninitial 
 n final 
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 1
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1
DE  2.178x1018 J 2  2 
 n final ninitial 
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Example calculation (1)
• Example: At what wavelength will an emission from
n = 4 to n = 1 for the H atom be observed?
 1
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1
DE  2.178x1018 J 2  2 
 n final ninitial 
1
4
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1
DE  2.178x10 J1   2.04x1018 J
 16 
18
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18
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DE  2.04 x10 J 
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hc
l
l  9.74 x108 m  97.4nm
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Example calculation (2)
• Example: What is the longest wavelength of light
that will result in removal of the e- from H?
18
DE  2.178x10
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 1
1 
J 2  2 
 n final ninitial 
1
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DE  2.178x1018 J0 1  2.178x1018 J
18
DE  2.178x10 J 
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hc
l
l  9.13x108 m  91.3nm
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Bohr model extedned to higher Z
• The Bohr model can be extended to any single
electron system….must keep track of Z
(atomic number).
2
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Z
18
E  2.178 x10  2 
n 
Z = atomic number
n = integer (1, 2, ….)
• Examples: He+ (Z = 2), Li+2 (Z = 3), etc.
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Example calculation (3)
• Example: At what wavelength will emission from
n = 4 to n = 1 for the He+ atom be observed?
 1
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1
DE  2.178x1018 J Z 2  2  2 
 n final n initial 
2
1
4
 1
DE  2.178x10 J41   8.16x1018 J
 16 
hc
18
l  2.43x108 m  24.3nm
DE  8.16x10 J 
l
l H  l He
18
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Problems with the Bohr model
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Why electrons do not collapse to the nucleus?
How is it possible to have only certain fixed orbits
available for the electrons?
Where is the wave-like nature of the electrons?
First clue towards the correct theory: De Broglie relation (1923)
E  h  hc / l
wher e c  l
E  mc 2
Einstein
h
h
l

mc p
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De Broglie relation: particles with certain
momentum, oscillate with frequency hv.
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Quantum Mechanics
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Particles in quantum mechanics are expressed by
wavefunctions
Wavefunctions are defined in spacetime (x,t)
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They could extend to infinity (electrons)
They could occupy a region in space (quarks/gluons inside proton)
In QM we are talking about the probability to find a
particle inside a volume at (x,t)
 2 3
 r , t  d r
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So the wavefunction modulus is a Probability
Density (probablity per unit volume)
In QM, quantities (like Energy) become eigenvalues
of operators acting on the wavefunctions
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QM: we can only talk about the probability to find
the electron around the atom – there is no orbit!
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