Dr. Bill Pezzaglia

QM Part 2

Updated: 2012Aug28

Quantum Mechanics:Wave Theory of Particles

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Quantum Mechanics

A. Bohr Model of Atom

B. Wave Nature of Particles

C. Schrodinger Wave Equation

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A. Bohr Model of Atom

1. Bohr’s First Postulate• Electron orbits are quantized by

angular momentum• Orbits are stable, and contrary to

classical physics, do not continuously radiate

• Principle Quantum number “n” (an integer whose lowest value is n=1)

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Niels Bohr1885-19621922 Nobel Prize

1. Bohr’s First Postulate

(a) Quantized Angular Momentum• 1912 first ideas by J.W. Nicholson• Postulates angular momentum of electron in

atom must be a multiple of

4

nmvrL

2h

1. Bohr’s First Postulate

(b) Stationary Orbits• Classical physics says accelerating charges

(i.e. electrons in circular orbits) should radiate energy away, hence orbits decay.

• Bohr says orbits are stable and do not radiate

• Principle quantum number “n” has a lowest value of n=1 (lowest angular momentum of one h-bar).

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(c) The Bohr Radius

With only a little algebra can solve for radius of electron’s orbit in the atom. Details (can ignore!):•Classical equation of motion

•Substitute:

•Solve for radius:

•Bohr Radius:

6

20

2

4

)(

r

eZe

r

vm

mr

n

mr

Lv

Z

anrn

02

nmme

ha 053.0

20

2

0

2. Bohr’s Second Postulate

(a) The sudden transition of the electron between two stationary states will produce an emission (or absorption) of radiation (photon) of frequency given by the Einstein/Planck formula

7

fi EEhf

(b) Energy of nth orbit

DETAILS (ignore)

•Viral Theorem: For inverse square law force:

•Hence total energy:

•Use Electrostatic energy formula, we get:

8

PEKE 21

PEPEKEE 21

r

ZeE

0

2

8

(b) Energy of nth orbit

DETAILS (ignore)•Substitute Bohr’s radius formula for n-th orbit gives energy of nth orbit:

•Importance is Bohr was able to calculate Rydberg’s constant from scratch!

9

2

2

2

2

)6.13(n

Zev

n

hcRZEn

ch

me

hca

eR 2

03

4

00

2

88

(c) Bohr Derives Balmer’s Formula

• From Einstein-Planck Formula:

• Substituting his energy formula (and divide out factor of hc), he derives Balmer’s formula!

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fi EEhc

hf

22

2 111

fi nnRZ

3. Bohr’s Correspondence Principle

• 1923: Classical mechanics “corresponds” to quantum system for BIG quantum numbers.

• When “n” is big, it behaves classically

• When “n” is small, it behaves “quantumly” (is that a word?)

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B. Wave Nature of Particles

1. deBroglie Waves

2. Particle in a Box

3. Heisenberg Uncertainty

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1. deBroglie Waves (1924)a) Suggest particles have wavelike

properties following same rules as photon.

• Proof: 1927 Electron diffraction experiment of Davisson & Germer (Nobel Prize 1937)

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fhE

h

P

(b) deBroglie’s Bohr ModelDETAILS (ignore)•Bohr’s model had an ad-hoc assumption that orbits had quantized angular momentum (multiples of h-bar)

•deBroglie postulates that only “standing waves” can yield stationary orbits, i.e. circumference must be multiple of the wavelength

•Hence allowed momentums are:

•Or angular momentums mustbe quantized:

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rn 2

r

nhhp

2

2h

nrpL

2. Particle in a Boxa) Standing wave patterns• Analogous to waves on

a string with fixed ends.

• Momentum hence is quantized to values:

15

L

nhhpn 2

n

Ln

2

2. Particle in a Box(b) Energy is hence quantized to

values:

• The particle can never have zero energy! The lowest is n=1

• The smaller the box, the bigger the energy. If wall is height “z”, for small enough “L”, the particle will jump and escape!

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mL

hn

m

pEn 2

222

82

mgzmL

h

2

2

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2c. Wavepackets & Localization

• A wave is infinite in extent, so the “electron” is not localized.

• The superposition of waves of slightly different wavelengths will create a “localized” wavepacket, which roughly corresponds to classical particle

• But now it does not have a single momentum (wavelength); it has a spread of momenta, and the packet will tend to spread out with time.

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3. Heisenberg Uncertainty• “principle of indeterminacy” • “The more precisely the

position is determined, the less precisely the momentum is known in this instant, and vice versa.”

• 1927 Uncertainty Principle (which can be derived from [x,p]=ih …)

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4h

px

C. Wave Mechanics

1. More Quantum Numbers

2. Pauli Exclusion Principle

3. Schrodinger Wave Mechanics

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1. Zeeman Effect (1894)(a) Zeeman effect: splitting of spectral

lines due to magnetic fields, shows us sunspots have BIG magnetic fields

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1b. Angular Momentum Quantum Number

Zeeman effect implies “suborbits” which are affected differently by the magnetic field.

•Principle (Bohr) quantum number n=1, 2, 3, 4 …•2nd quantum number “l” where l <n

n=1 l =0 (“s” orbit)

n=2 l =0 or l=1 (“p” orbit)

n=3 l =0, 1, or 2 (“d” orbit)

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Multiplicity of states

3rd quantum number “m”

l =0 (“s” orbit) m=0

l =1 (“p” orbit) m=-1, 0, +1

l =2 (“d” orbit) m=-2, -1, 0, +1, +2

So the Zeeman effect is splitting the “p” orbits into three different lines (and “d” orbits into 5)

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2. Pauli Spin• 1924 proposes new quantum number to

explain “Anomalous Zeeman Effect” where “s” orbits split into 2 lines.

• 1925 Uhlenbeck & Goudsmit identify this as description of “spin” of electron, which creates a small magnetic moment

• 1927 Pauli introduces idea of “spinors” which describe spin half electrons

• Famous quote: when reviewing a very badly written paper he criticized it as “It is not even wrong”

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2b. Pauli Exclusion Principle (1925)

• Serious Question: Why don’t all the electrons fall down into the first (n=1) Bohr orbit?

• If they did, we would not have the periodic table of elements!

• Exclusion Principle: Each quantum state can only have one electron (e.g. 1s orbit can have two electrons, one with spin up, other with spin down)

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2c. Fermions & Bosons

Details•Fermions, which have spin ½ (angular momentum of h/4) obey the Pauli exclusion principle (e.g. electrons, neutrinos, protons, neutrons, quarks)

•Bosons, which have integer spin, do NOT obey the principle (e.g. photons, gravitons).

•This is why we can have “laser” light (a bunch of photons with their waves all in phase).

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3. Schrodinger 1926Bohr & Heisenberg’s quantum mechanics

used abstract mathematical operations (e.g. x and p don’t commute)

a) Schrodinger writes a generalized equation that deBroglie waves must obey when there is Potential Energy (such that the wavelength changes from point to point in space)

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ExV

xm

h

)(2 2

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3b Solution to Schrodinger Equation: Electron Orbits

• S orbits hold 2 electrons

• P orbits hold 6 electrons

• D orbits hold 10 electrons

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Madelung Rule: Filling electrons into an atom 28

Electron Configurations

• Bohr’s Aufbau (build up) Principle: Fill orbits of lowest energy first (e.g. the n=1 orbit before the n=2 orbit)

• Madelung Rule: for states (n,l), the states with lower sum “n+l” are filled first (because they have lower energy). For example, 4s (4,0) would be filled before 3d (3,2).

• Carbon: 1s2, 2s2, 2p2

• Titanium: 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3p2

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References/Notes30

• McEvoy & Zarate, “Introducing Quantum Theory” (Totem Books, 1996)

• http://www.aip.org/history/heisenberg/p08.htm (includes audio !)

• http://www.uky.edu/~holler/html/orbitals_2.html

• http://www.meta-synthesis.com/webbook/30_timeline/lewis_theory.php