PHY206: Atomic Spectra

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: [email protected]@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%

mailto:[email protected]

http://www.shef.ac.uk/physics/teaching/phy206

http://www.shef.ac.uk/physics/teaching/phy206

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


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


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


AtomsAtoms, , ProtonsProtons, , QuarksQuarks and and GluonsGluons

Atomic Nucleus

Proton

gluons

Atom

Proton


Atomic StructureAtomic Structure


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


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:


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


Emission lines of some elements(all quantized!)


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


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


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)


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


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


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!


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)


• 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


• 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)


• 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



• 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


• 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



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.


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

,


QM: we can only talk about the QM: we can only talk about the probability to find the electron probability to find the electron around the atom – there is no orbit!around the atom – there is no orbit!

PHY206: Atomic Spectra

Documents

h atom spectrum

atomic spectralecturer

atomic spectralight

hydrogen gas

hatom quantum stateslecture

energy levels

frequency of visible

identical particles