Dr. Bill Pezzaglia

QM Part 2

Updated: 2010May11

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

• 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

• Viral Theorem: For inverse square law force:

• Hence total energy:

• Use Electrostatic energy formula, we get:

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PEKE 21

PEPEKEE 21

r

ZeE

0

2

8

(b) Energy of nth orbit

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

• Where he can 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 Model• 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

1c. Phase Velocity• Velocity of waves are FASTER than light

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p

h

h

Efv

v

c

mv

mc

p

E 22

Where “v” is the classical speed of the particle (aka “group velocity”)

(d) Interpretation• deBroglie thought that the “wave” of a

particle had two aspects.

• The “group velocity” described the localized “particle” nature of the classical particle

• The “phase velocity” was associated with the “pilot wave” which traveled ahead and behind the particle (faster than light), sensing the environment.

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

a string with fixed ends.

• Momentum hence is quantized to values:

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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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3a. Heisenberg QM• 1925 First formulation of “quantum

mechanics” which correctly describes energy levels and quantum jumps.

• It’s a mathematical theory, which assumes that position and momentum do not commute:

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2ih

pxxp

3b. 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. Bohr Sommerfeld Model

1916 use elliptical orbits to different energies (new quantum number “l”).

Also, quantumnumber “m” todescribe orientation,where if l=2, mcould be{-2,-1,0,1,2}

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1c. Bohr’s Periodic Table1921 uses quantum numbers to explain periodic table (Pauli’s contribution is that each state has 2 electrons in it, another quantum number)

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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

• 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 Electron Orbits• S orbits hold 2 electrons

• P orbits hold 6 electrons

• D orbits hold 10 electrons

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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).

• Hund’s Rules (Bohr’s assistant)

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Madelung Rule 32

Hund’s Rules

1. Rule of Maximum Multiplicity: maximize the spin (e.g. put one electron into each of the three p orbits with spins parallel, i.e. maximize unpaired electrons).

2. For a given multiplicity, the term with the largest value of L (orbital angular momentum), has the lowest energy

3. The level with lowest energy (where J=L+S)1. Outer shell Less than half filled: minimum J

2. Outer shell more than half filled: maximum J

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3c. Max Born• 1924 coins the term “Quantum Mechanics”• 1925 helps with Heisenberg’s matrix form

of quantum mechanics

• 1928 The square of the quantum wave is proportional to the probability of finding the particle at that position.

• Hence you can think of the quantum wave as having a “classical” probability density , and an “imaginary” quantum phase part.

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References/Notes35

• 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