university of san francisco modern physics for frommies ii the universe of schr ödinger’s cat

27 January 2010 Modern Physics II Lecture 3 1

University of San Francisco

Modern Physics for Frommies II

The Universe of Schrödinger’s Cat

Lecture 3


Agenda (1)• Administrative matters• More on Wave-Particle Duality

– Young’s Double-Slit Experiment• Wave Nature of Matter

– De Broglie’s Hypothesis• Early Models of the Atom

– Plum Puddings and Blackberry Jam– Rutherford scattering and the nucleus

• Another UV Catastrophe• Troubles with Atomic Spectra


Agenda (2)

– The Atom of Niels Bohr• de Broglie’s Hypothesis and the Bohr Atom

– The Atom of Bohr Kneels1

1 “The Atom of Niels Bohr” and “The Atom of Bohr Kneels” are chapter tiles taken from a book that I read as a child and whose title I cannot remember. I think the author was George Gamow to whom I am indebted.


Correction to previous course wiki URL:

http://modphysifrommi1.wiki.usfca.edu/

Thanks to Bud Bronstein, one of us, and to Ginny Wallace of USF’s ITS

Administrative Matters

Correction to CLEA URL:

http://www.gettysburg.edu/~marschal/clea/CLEAhome.html

Thanks to Jonathan Marsh

http://modphysifrommi1.wiki.usfca.edu/

http://ww3.gettysburg.edu/%7Emarschal/clea/CLEAhome.html


First Physics and Astronomy Colloquium:

Wednesday, 10 February 2010 at 4 PMProfessor Richard Muller, Department of Physics, UC BerkeleyTopic TBARefreshments at 3:30 PMHarney Science Center Room 127


Waves and Interference:

Superposition principle: Linear combinations of solutions are also solutions.

Because of phase, addition is not simply the addition of intensities. It is almost “vectorial” in nature.

Simplest cases:

Constructive interference: waves are in phase or out of phase by n • (360º or 2 rad.) where n = 0, 1, 2, 3,…An integral number of wavelengths or cycles.

Destructive interference: waves are out of phase by(n +1/2) • (360º or 2 rad.).An odd ½ integral number of wavelengths


Constructive

Waves add

y = y1 + y2

at every point in space and/or in time.

2 = 1


Destructive

Waves subtract

y = y1 - y2

at every point in space and/or in time.2 = 1 -180º


Young’s Double Slit Experiment:

Thomas Young (1773-1829), a British Physician, performed this experiment in 1804.


3 5Now look at 2 ,3 , etc. and at , , etc.

2 2

1Extra dist

Extra dista

ance 0,1,2... destructive interfere

nce 0,1, 2... constructive interferenc

ce2

e

nm

m

m

m

sin 0,1,2... constructive interfere

1sin 0,1,2... destructive interference

2

nced m

m

m

d m


Effect of changing or d :

Increase , pattern spreads out

Decrease d, pattern spreads

Difference between sound and light diffracting around doorway

Use of diffraction pattern to analyze wavelengths


Now let’s turn down the power so only one photon passes through the apparatus at a time.

HeNe laser: = 632.991 nm,8

149

3.00 10 m/sec4.73 10 Hz

632.991 10 m

cf

34 14 196.63 10 J sec 4.73 10 Hz 3.13 10 JE hf

10,000 photons/sec = 3.13 fW of power

Replace screen with photographic film or a moveable photomultiplier tube.


Count photons for a fixed time at various positions across the slits and for one slit blocked and both slits open.

What is each photon interfering with if it is alone in the apparatus?

In a sense, each photon passes through both slits and interferes with itself!!


de Broglie’s Hypothesis

Major symmetry fan:

Waves sometimes act like particles (given)

Particles sometimes act like waves (hypothesis)

h

p Sometimes called the de Broglie

wavelength of a particle

Sounds nuts, but remember h is very small ( 10-34 J·sec)

A couple of examples may restore your gullibility.

Louis de Broglie (1892-1987)


Wavelength of a ball:

0.20 kg moving with speed of 15 m/sec

34

346.6 10 J sec2.2 10 m

0.20 kg 15 m/sec

h h xx

p mv

Very small, something like 20 Planck lengths

of any ordinary object is much to small to be detected.

Interference and diffraction are significant only when the sizes of objects or slits are not much larger than


Wavelength of an electron:

Accelerated thru a potential difference of 100 V so we do not need to use relativistic mechanics.

2

19

631

1

2

2 1.6 10 C 100 V25.9 10 m/sec

9.1 10 kg

K mv eV

xeVv x

m x

34

10

-31 6

6.6 10 J sec1.2 10 m

9.1 10 kg 5.9 10 m/sec

h h x

p mv

Scale of atomic size


100 eV e

-

d

d sin

Crystal lattice, atomic separation d

Path length difference d sin

Suppose =24º is smallest angle for diffraction maximum, what is d?

Electron Diffraction and Interference


For diffraction maximum, path difference is integral multiple of the de Broglie wavelength. For smallest

d sin =

2 2

2

34

31 19

2 2

2

6.63 10 J sec0.123 nm

2 9.11 10 kg 100 eV 1.6 10 J/eV

e e

e

p hK

m m

h

m K

0.123 nm0.30 nm

sin sin 24d


Neutrons with v = 2 km/sec

34

4

-27 3

6.63 10 J sec8 10 m

1.68 10 kg 2 10 m/sec

h

p

This wavelength is that of near infrared radiation

Easy to make double slits that will give interference if neutrons act like waves.

Such patterns are observed

In fact , this result has been extended to include whole atoms, molecules and exotic concoctions like buckyballs.


Electron Microscopy

Rayleigh criterion

1.22

D

Decrease improved resolution

Optics becomes more difficult as gets smaller

Glass cutoff 230 nm

Quartz cutoff 180 nm

NaF cutoff 130 nm


Use electrons and focus with magnetic “lenses”.

E- accelerated through 100kV has 0.004 nm

In practice, aberrations in magnetic lenses currently limit this resolution to about 0.1 to 0.5 nm at best.

1000 times better than a light microscope

This is M = 106. Hard to do, common Ms are 104 - 105


Early Atomic Models

Early atomic modelsThomson and the electron, the Lorentz modelRutherford scattering and the nuclear atom

Lines rather than continuaUV catastrophe II type instabilityLines rather than continua

The Atom of Niels BohrThe hydrogen spectrum and the Rydberg formulaQuantization of angular momentum and the Bohr

modelElectron waves and the Bohr model


ca. 1890

electrons

Rutherford scattering (1911)

(+) charge confined to nucleus with r = 10-15-10-14 m


Rutherford Scattering and the Nuclear Atom

1911 Ernest Rutherford (1871-1937) et al. concluded a study of particle scattering from metal foils.

An excess of events at large scattering angles lead him to the conclusion that the positve charge in atoms is concentrated in a small, massive charged core, the nucleus.

Atomic size ~ 10-10 m = 1 Å

Nuclear size ~ 10-15 – 10-14 m = 1 fm


Atomic Instability - Another UV CatastropheClassical Rutherford atom (1911)

e- are accelerated and should radiate.

e- lose energy and spiral into nucleus

Atoms should be unstable and the universe should only have lasted a small fraction (10-9) of a second

Oh, and there are other problems.


Line Spectra

Emission from rarified gasses


Absorption in rarified gasses


Line spectra cannot be explained by a classical model


The Hydrogen Spectrum


!885 J. J. Balmer (1825-1898) showed that the 4 visible lines ( 410, 434, 486 and 656 nm) fit the formula below.

2 2

7 -1

1 1 1, 3, 4,

2

1.0974 10 Rydberg constan m t

R nn

R

Later measurements extended this Balmer series of lines into the UV. The lines become closer together with decreasing l and become indistinguishable near 365 nm.

365 nm corresponds to n =


Later experiments found similar series in the UV (Lyman series) and the IR (Paschen series).

Balmer’s formula can be generalized as the Rydberg formula.

2 2

1 1 1

1 Lymann series

2 Balmer series 1, 2, ,

3 Paschen series

Rn k

n

n k n n

n

These are the experimental data which Niels Bohr attempted to explain with his modification of the Rutherford model


Neils Bohr (1885-1962)

Bohr’s Postulates

Electrons in atoms cannot lose energy continuously, but must do so in quantum “jumps”.

Electrons move about the nucleus in circular orbits, but only certain orbits are allowed.


An electron in an orbit has a definite energy and moves in the orbit without radiating energy. The allowed orbits are referred to as stationary states.

Emission and absorption of radiation can only occur in conjunction with a transition between 2 stationary states. This results in an emitted or absorbed photon of frequency such that

hf = E1 - E2

What makes an orbit allowed?

Maybe energy is not the only quantized quantity.

No

tice

d that

can be

shown to have un

Requir

its of angula

e that

r moment

where 1,2

2,

um

3,L n n

h


This quantization of L had no firm theoretical foundation, Bohr tried various “quantum conditions”. This one worked.

Imposing quantization of angular momentum and his other postulates on classical electrodynamics and classical orbital mechanics allowed Bohr to derive the Rydberg forrnula.

Stability of atoms is insured. Ground state is the lowest state. There is no lower state it can attain by the emission of more energy


Binding energy is the energy one must supply to an electron in a state, n, to remove that electron from the atom.


de Broglie Waves and Bohr’s Quantization

Bohr’s model was largely ad hoc. Assumptions were made so that theory would agree with experiment.

No reason why orbits should be quantized.

No reason why ground state should be stable

De Broglie proposed that an electron in a stable orbit is actually a circular standing matter wave.

h

mv


The circumference of the wave must contain an integral number of wavelengths

2 1, 2,3,

Bohr's condit

or

2

2

ion

n

n

n

r n n

nhr

mvnh

mvr n

Bohr published his model in 1913. de Broglie did not propose matter waves until 1923. Bohr tried many quantization conditions in attempting to explain the experimental data. Quantization of angular momentum worked.


The Atom of Bohr KneelsSuccesses:

Predicts the correct Rydberg constant for alkali atoms and things like He+ and Li++. Replace e with Zeffe in the Coulomb force

Failures:

Is successful only for single electron atoms

Fails to explain the “fine structure” of spectral lines, even for alkali atoms

The ground state of Hydrogen has L=0 not 1

Cannot explain the bonding of atoms into molecules.


The Bohr model is an ad hoc theory which fits the hydrogen spectrum. It is a semiclassical theory.

We now know that it does not correctly describe atoms. This description requires a true quantum mechanical theory.

Important 1st step from a purely classical theory to a quantum mechanical one.

“appeared to me like a miracle and appears as a miracle even today.” - A. Einstein, ca. 1940


Erwin Schrödinger

1887 - 1961

Werner Heisenberg

1901 - 1967

university of san francisco modern physics for frommies ii the universe of schr ödinger’s cat

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