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Early 1900s
Electrons are nice particles They obey laws of classical mechanics
Light behaves just like a wave should It reflects, refracts and diffracts
Then, things began to happen…
Electromagnetic Waves
EM waves include two oscillating components: electric field magnetic field
EM waves can travel in vacuum Mechanical waves (e.g. sound) need a medium (e.g. air)
EM Waves travel at the speed of light, c c = 299,792,458 m/s
frequency
wavelength
Electromagnetic Spectrum
Thermal Radiation
For human skin, T = 95oF, wavelength is 9.4 micrometer (infrared)
All hot bodies emit thermal radiation
Ultraviolet Catastrophe
Rayleigh-Jeans law described thermal radiation emitted by a black-body as
This implies that as the wavelength of an EM wave approaches zero (infinite frequency), its energy will become infinitely large! i.e. will get brighter and brighter
Experimentally, this was not observed… and this was referred to as the ultraviolet catastrophe
Rutherford Atom - Challenge
Belief: Attractive force between the positively charged nucleus and an electron orbiting around is equal to the centrifugal exerted on the electron. This balance determines the electron’s radius.
Challenge: A force is exerted on the electron, then, the electron should accelerate continuously according to F = ma. If this is the case, the electron should continuously lose its energy. According to classical physics, all accelerating bodies must lose energy. Then, the electrons must collapse with the nucleus.
Photoelectric Effect - Challenge
Light shining on a piece of metal results in
electron emission from the metal
There is always a threshold frequency of light below which no electron emission occurs from the metal.
Maximum kinetic energy of the electrons has nothing to do with the intensity of light. It is determined by the frequency of light.
Photoelectric EffectThe Experiment - 1
Light Sourceanode
cathode
Electrons emitted by the cathode are attracted to the positively charged anode.
A photocurrent begins to flow in the loop.
Photoelectric EffectThe Experiment - 2
Light Sourceanode
cathode
Electrons emitted by the cathode are repelled by the negatively charged anode.
The photocurrent decreases.
Photoelectric EffectThe Experiment - 3
Voltage
Current
IncreasingLight
Intensity
Regardless of the light intensity, the photocurrent becomes zero at V = - Vo
At this voltage, every emitted electron is repelledTherefore, qVo must be the maximum kinetic energy of the electron
This energy is independent of the light intensity
Vo
Photoelectric EffectThe Experiment - 4
fo
The maximum kinetic energy of electrons is determined by the frequency of light
The slope of this line is Planck’s constantIncreasing the light intensity only increases the number of photons
hitting the cathode
Frequency
Video: A Brief History of Quantum Mechanics http://www.youtube.com/watchv
=B7pACq_xWyw&list=PLXD9X52rMwwNJ85QDzv3G6cqbQ2ZVlsR8&index=2
Video:Max Planck & Quantum Physics https://www.youtube.com/watch?
v=2UkO_3NC3F4
Hydrogen Atom
Hydrogen Atom
Orbitalthree dimensional space around the nucleus of all the places we are likely to find an electron.
Orbitals & Quantum Numbers Atoms have infinitely many orbitals Each orbital can have at most two
electrons Each orbital represents a specific
Energy level Angular momentum Magnetic moment
Sub-levels
Quantum Numbers
Principal Quantum Number, n = 1, 2, 3, … Determines the electron energy
Azimuthal Quantum Number, l = 0, 1, 2, … Determines the electron’s angular
momentum Magnetic Quantum Number, m = 0, ± 1, ± 2,
… Determines the electron’s magnetic moment
Spin Quantum Number, s= ± 1/2 Determines the electron spin (up or down)
The energy, angular momentum and magnetic moment of an orbital are quantized
i.e. only discrete levels are allowed
Principal Quantum Number
Always a positive integer, n = 1, 2, 3, … Determines the energy of the electron in
each orbital. Sub-levels with the the same principal
quantum number have the same energy
Only certain (discrete) energy levels are allowed!
Azimuthal Quantum Number
Each n yields n – 1 sub levels
l = 0 l = 1 l = 2 l = 3
n = 1 a
n = 2 a a
n = 3 a a a
n = 4 a a a a
Magnetic Quantum Number
l = 0 l = 1 l = 2 l = 3
n = 1 m=0
n = 2 0 -1, 0,+1
n = 3 0 -1, 0,+1 -2, -1, 0,+1,+2
n = 4 0 -1, 0,+1 -2, -1, 0,+1,+2 -3,-2, -1, 0,+1,+2,+3
s p d f
1 orbital 3 orbitals 5 orbitals 7 orbitals
Spin Quantum Number
l = 0 l = 1 l = 2 l = 3
n = 1 m=0
n = 2 0 -1, 0,+1
n = 3 0 -1, 0,+1 -2, -1, 0,+1,+2
n = 4 0 -1, 0,+1 -2, -1, 0,+1,+2 -3,-2, -1, 0,+1,+2,+3
s p d f
1 orbital2 states
3 orbitals6 states
5 orbitals10 states
7 orbitals14 states
(only two possibilities)
Levels, Sublevels of Atomic Orbitals http://en.wikipedia.org/wiki/
Atomic_orbital
Electrons fill the lowest energy states first
Atomic Number
Example 1: Silicon Atom
1 s
2 s
2 p
3 s
3 p
4 s
3 d
Silicon has 14 electrons
1s22s22p63s23p2
Empty States
Occupied States
Example 2: Titanium Atom
1 s
2 s
2 p
3 s
3 p
4 s
3 d
Titanium has 22 electrons
1s22s22p63s23p63d24s2
Empty States
Occupied States
Atomic Orbitals
Hydrogen
Larger Atoms
s (l=0) p (l = 1) d (l = 2) f (l = 3)
Video - Orbitals
http://www.youtube.com/watchv=drCg4ruJCfA&list=PLREtcqhPesTcTAI6di_ysff0ckrjKe83I&index=9
Electromagnetic Waves
EM waves can travel in vacuum Mechanical waves (e.g. sound) need a
medium (e.g. air) EM Waves travel at the speed of light, c
c = 299,792,458 m/s
frequency
wavelength
Electromagnetic Spectrum
How EM Waves are made?
1. Electric field around the electron accelerates
2. The field nearest to the electron reacts first
3. Outer field lags behind4. Electric field is distorted – bend in
the field5. The bend moves away from the
electron6. The bend carries energy
Charges often accelerate and decelerate in an oscillatory manner – sinusoidal waves
Energy is Quantized
Always a positive integer, n = 1, 2, 3, …
E = nhfOrbital’s energy level Principa
l Quantu
m Numbe
r
Planck’s Constant6.626 X 10-34 m2/ kg-s
Frequency at which the atom vibrates
Only certain (discrete) energy levels are allowed!
Photons & Electrons
Atoms gain and lose energy as electrons make transitions between different quantum states
A photon is either absorbed or emitted during these transitions
n=1
n=2
n=3
A photon is absorbed for this
transition
Bohr’s radius correspond to distance from the nucleus where the probability of finding the electron is highest in a given orbital.
Hydrogen Atom
n = 1, 2, 3, …
As n approaches infinity, energy approaches zero.
E1 = 13.6 eV - Ground energy of the electron in the hydrogen atom
If you provide this much energy to the electron, it can leave the hydrogen atom
Photon & Electrons
The momentum of a photon (or an electron) is given by
This relationship is true for all particles Even large particles…
This equation was postulated for electrons by de Broglie in 1924
Video - Double Slit Experiment This is the experiment that confirmed
the wave nature of electrons http://www.youtube.com/watchv=Q1Yqg
PAtzho&list=PLREtcqhPesTcTAI6di_ysff0ckrjKe83I&index=1
Bullets Thru Double-Slit
P12 = P1 + P2
Waves Thru Double-Slit
P12 ≠ P1 + P2
I12 ≠ I1 + I2 + 2sqrt(I1I2) cos(Phi)
Electrons Thru Double-Slit
P12 ≠ P1 + P2
Electrons Thru Double-Slit
Electrons Observed Thru Double-Slit
No device can determine which slit the e- passes thru, w/o changing the interference.
Photon has momentum – after the collision between the photon and the electron, the electron’s momentum is no longer the same and we do not know what it is althought we know electron’s location rather precisely.
Heisenberg Uncertainty Principle
“Accepting quantum mechanics means feeling certain that you are uncertain”…a great statement from your textbook
Video – Heisenberg’s Uncertainty Principle
http://www.youtube.com/watch?v=Fw6dI7cguCg&list=PLREtcqhPesTcTAI6di_ysff0ckrjKe83I&index=3
Erwin Schrodinger
1887-1961 Austrian Physicist Formulated the wave
equation in quantum physics
1933 Nobel Prize 1937 Max Planck
Medal
Schrodinger’s Equation
Schrodinger’s Equation is one of the most important equations in modern physics.
E = Energy
Wave Function – Physical Meaning A wave function is a complex quantity of
the form
The probability of finding an electron at a given location is given by
This is the ONLY physical meaning attached to the wave function
where
Complex Numbers – A Brief Review
a
ib
Free Particle
A free particle is not bound to anything It can freely move and go anywhere… Its energy must be purely kinetic energy
Schrodinger’s Equation
The solution of this equation is…
Free Particle – Continued
What does this mean?
A free particle has kinetic energy only…
But we found…
This mean, the electron momentum is given by
Infinite Potential Well
A single electron is placed in an infinite potential well The walls are
infinitely high The electron is
trapped The probability of
finding the electron outside is…
Which implies…
∞∞
- L/2 + L/20
Infinite Potential Well – Continued The solutions are of the form
Verify:
Inside the well, the electron’s energy is purely kinetic (the potential is zero)
Infinite Potential Well – Solutions The solution was
Applying the boundary conditions
Adding and subtracting the equations:
Infinite Potential Well – Solutions We must satisfy
We have two options
Allowed Momenta
Only discrete momentum values are allowed!
Momentum is quantized…
Allowed Wavelengths
Only certain wavelengths are allowed!
Allowed Energy Levels
Only discrete energy levels are allowed!
Energy is quantized…
Particle in a Well
The result is strikingly similar to atomic orbitals in atoms
Recall:For a hydrogen atom,
En = - 13.6 / n2 eV
Finite Potential Well
A particle has a finite number of allowed energy levels in the potential well
A particle with E > Vo is not bound to the potential well
A particle with E < Vo has a finite probability of escaping the well
- L/2 + L/20
Vo Vo
Infinite vs. Finite Potential WellWave Functions
Vo
The wavefunctions are decaying exponentially outside the potential well
There is a finite probability of finding the electron outside the potential well
Particle (e.g. electron) Tunneling
An electron can tunnel through a potential barrier even though its initial kinetic energy is smaller than the potential barrier.
Electron tunneling is an important topic in nanoelectronics
The frequency of the wave is related to the momentum and the kinetic energy of the particle.
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