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PHY206: Atomic PHY206: Atomic SpectraSpectra
Lecturer: Dr Stathes PaganisLecturer: Dr Stathes Paganis Office: D29, Hicks BuildingOffice: D29, Hicks Building Phone: 222 4352Phone: 222 4352 Email: Email: paganis@NOSPAMmail.cern.chpaganis@NOSPAMmail.cern.ch Text: A. C. Phillips, ‘Introduction to QM’Text: A. C. Phillips, ‘Introduction to QM’ http://www.shef.ac.uk/physics/teaching/phy20http://www.shef.ac.uk/physics/teaching/phy20
66 Marks: Final 70%, Homework 2x10%, Marks: Final 70%, Homework 2x10%,
Problems Class 10%Problems Class 10%
PHY206: Spring Semester Atomic Spectra 2
Course Outline (1)Course Outline (1) Lecture 1 : Bohr TheoryLecture 1 : Bohr Theory
Introduction Bohr Theory (the first QM picture of the atom) Quantum Mechanics
Lecture 2 : Angular Momentum (1)Lecture 2 : Angular Momentum (1) Orbital Angular Momentum (1) Magnetic Moments
Lecture 3 : Angular Momentum (2)Lecture 3 : Angular Momentum (2) Stern-Gerlach experiment: the Spin
Examples Orbital Angular Momentum (2)
Operators of orbital angular momentum
Lecture 4 : Angular Momentum (3)Lecture 4 : Angular Momentum (3) Orbital Angular Momentum (3)
Angular Shapes of particle Wavefunctions Spherical Harmonics
Examples
PHY206: Spring Semester Atomic Spectra 3
Course Outline (2)Course Outline (2) Lecture 5 : The Hydrogen Atom (1)Lecture 5 : The Hydrogen Atom (1)
Central Potentials Classical and QM central potentials
QM of the Hydrogen Atom (1) The Schrodinger Equation for the Coulomb Potential
Lecture 6 : The Hydrogen Atom (2)Lecture 6 : The Hydrogen Atom (2) QM of the Hydrogen Atom (2)
Energy levels and Eigenfunctions Sizes and Shapes of the H-atom Quantum States
Lecture 7 : The Hydrogen Atom (3)Lecture 7 : The Hydrogen Atom (3) The Reduced Mass Effect Relativistic Effects
PHY206: Spring Semester Atomic Spectra 4
Course Outline (3)Course Outline (3)
Lecture 8 : Identical Particles (1)Lecture 8 : Identical Particles (1) Particle Exchange Symmetry and its Physical
Consequences
Lecture 9 : Identical Particles (2)Lecture 9 : Identical Particles (2) Exchange Symmetry with Spin Bosons and Fermions
Lecture 10 : Atomic Spectra (1)Lecture 10 : Atomic Spectra (1) Atomic Quantum States
Central Field Approximation and Corrections
Lecture 11 : Atomic Spectra (2)Lecture 11 : Atomic Spectra (2) The Periodic Table
Lecture 12 : Review LectureLecture 12 : Review Lecture
PHY206: Spring Semester Atomic Spectra 5
AtomsAtoms, , ProtonsProtons, , QuarksQuarks and and GluonsGluons
Atomic Nucleus
Proton
gluons
Atom
Proton
PHY206: Spring Semester Atomic Spectra 7
Early Models of the AtomEarly Models of the Atom
Rutherford’s model Rutherford’s model
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
PHY206: Spring Semester Atomic Spectra 8
atoms should atoms should collapsecollapse
Classical Electrodynamics: charged particles radiate EM energy (photons) when their velocity vector changes (e.g. they accelerate).
This means an electron should fall into the nucleus.
Classical Physics:Classical Physics:
PHY206: Spring Semester Atomic Spectra 9
Light: the big puzzle in the 1800s 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
PHY206: Spring Semester Atomic Spectra 11
Emission spectrum of HydrogenEmission spectrum of Hydrogen “ “Continuous” spectrumContinuous” spectrum “ “Quantized” spectrumQuantized” spectrum
Any E ispossible
Only certain E areallowed
E E
Relaxation from one energy level to another by Relaxation from one energy level to another by emitting a photon, withemitting a photon, withE = hc/E = hc/
If If = 440 nm, = 440 nm, = 4.5 x 10= 4.5 x 10-19-19 J J
PHY206: Spring Semester Atomic Spectra 12
The goal: use the emission spectrum to determine the energy levels for the hydrogen atom (H-atomic spectrum)
Emission spectrum of HydrogenEmission spectrum of Hydrogen
PHY206: Spring Semester Atomic Spectra 13
Joseph Balmer (1885) first noticed that Joseph Balmer (1885) first noticed that the frequency of visible lines in the H the frequency of visible lines in the H atom spectrum could be reproduced by:atom spectrum could be reproduced by:
1
22
1
n2n = 3, 4, 5, …..
The above equation predicts that as n The above equation predicts that as n increases, the frequencies become more increases, the frequencies become more closely spaced.closely spaced.
Balmer model (1885)Balmer model (1885)
PHY206: Spring Semester Atomic Spectra 14
Rydberg ModelRydberg Model
Johann Rydberg extended the Balmer model by Johann Rydberg extended the Balmer model by finding more emission lines outside the visible finding more emission lines outside the visible region of the spectrum:region of the spectrum:
Ry1
n12
1
n22
n1 = 1, 2, 3, …..
In this model the energy levels of the H atom In this model the energy levels of the H atom are proportional to 1/nare proportional to 1/n22
n2 = n1+1, n1+2, …
Ry = 3.29 x 1015 1/s
PHY206: Spring Semester Atomic Spectra 15
The Bohr Model (1)The Bohr Model (1)
Bohr’s Postulates (1913)Bohr’s Postulates (1913) 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/2, which means that for circular orbits of radius r the z component of the angular momentum L is quantized:
2. Transitions between energy levels An electron in an atom can change energy levels by undergoing a “transition” from one energy level to another.
nmvrL
PHY206: Spring Semester Atomic Spectra 16
The Bohr Model (2)The Bohr Model (2) Bohr derived the following formula for the energy levels of Bohr derived the following formula for the energy levels of
the electron in the hydrogen atom.the electron in the hydrogen atom. Bohr model for the H atom is capable of reproducing the Bohr model for the H atom is capable of reproducing the
energy levels given by the empirical formulas of Balmer energy levels given by the empirical formulas of Balmer and Rydberg.and Rydberg.
2
21810178.2n
ZxE
Energy in JoulesZ = atomic number (1 for H)n is an integer (1, 2, ….)
• Ry x h = -2.178 x 10-18 J The Bohr constant is the same as the Rydberg multiplied by Planck’s constant!
PHY206: Spring Semester Atomic Spectra 17
2
21810178.2n
ZxE
• Energy levels get closer together as n increases
• at n = infinity, E = 0
The Bohr Model (3)The Bohr Model (3)
PHY206: Spring Semester Atomic Spectra 18
• We can use the Bohr model to predict what E is for any two energy levels
E E final E initial
E 2.178x10 18J1
n final2
( 2.178x10 18J)
1
ninitial2
E 2.178x10 18J1
n final2
1
ninitial2
Prediction of energy spectraPrediction of energy spectra
PHY206: Spring Semester Atomic Spectra 19
• Example: At what wavelength will an emission from n = 4 to n = 1 for the H atom be observed?
E 2.178x10 18J1
n final2
1
ninitial2
1 4
E 2.178x10 18J 11
16
2.04x10 18J
E 2.04x10 18J hc
9.74x10 8m97.4nm
Example calculation (1)Example calculation (1)
PHY206: Spring Semester Atomic Spectra 20
• Example: What is the longest wavelength of light that will result in removal of the e- from H?
E 2.178x10 18J1
n final2
1
ninitial2
1
E 2.178x10 18J 0 1 2.178x10 18J
E 2.178x10 18J hc
9.13x10 8m91.3nm
Example calculation (2)Example calculation (2)
PHY206: Spring Semester Atomic Spectra 21
• The Bohr model can be extended to any single electron system….must keep track of Z (atomic number).
• Examples: He+ (Z = 2), Li+2 (Z = 3), etc.
2
21810178.2n
ZxE
Z = atomic number
n = integer (1, 2, ….)
Bohr model extedned to higher ZBohr model extedned to higher Z
PHY206: Spring Semester Atomic Spectra 22
• Example: At what wavelength will emission from n = 4 to n = 1 for the He+ atom be observed?
E 2.178x10 18J Z 2 1
n final2
1
ninitial2
2 1 4
E 2.178x10 18J 4 11
16
8.16x10 18J
E 8.16x10 18J hc
2.43x10 8m24.3nm
H He
Example calculation (3)Example calculation (3)
PHY206: Spring Semester Atomic Spectra 23
Problems with the Bohr modelProblems with the Bohr model
Why electrons do not collapse to the nucleus? Why electrons do not collapse to the nucleus? How is it possible to have only certain fixed How is it possible to have only certain fixed
orbits available for the electrons?orbits available for the electrons? Where is the wave-like nature of the electrons?Where is the wave-like nature of the electrons?
First clue towards the correct theory: De Broglie relation (1923)
2
e wher/
mcE
chchE
Einstein
p
h
mc
h De Broglie relation: particles with certain
momentum, oscillate with frequency hv.
PHY206: Spring Semester Atomic Spectra 24
Quantum MechanicsQuantum Mechanics Particles in quantum mechanics are Particles in quantum mechanics are
expressed by wavefunctionsexpressed by wavefunctions Wavefunctions are defined in spacetime (x,t)Wavefunctions are defined in spacetime (x,t)
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 In QM we are talking about the probability to find a particle inside a volume at (x,t)find a particle inside a volume at (x,t)
So the wavefunction modulus is a So the wavefunction modulus is a Probability Probability DensityDensity (probablity per unit volume) (probablity per unit volume)
In QM, quantities (like Energy) become In QM, quantities (like Energy) become eigenvalues of operators acting on the eigenvalues of operators acting on the wavefunctionswavefunctions
rdtr 32
,