Download Quantum Mechanics

Quantum Mechanics
History
• Beginning in 1900, some experiments showed
some disagreement with “classical” mechanics
and understanding of light
• Planck/Einstein first people to propose a key
idea
– Light was made up of particles = photons
Quantum
• Quanta – Latin for little bits
• Light is quantized = comes in little
bits/particles
• Light acts like a wave (interference)
• Light acts like a particle (photoelectric effect)
Photoelectric effect
• Einstein – 1905 – Nobel Prize
• Light comes in and hits a metal
• Electrons leave the metal with a given energy
• Explanation: Photons have a collision with
electrons and transfer energy/momentum to the
electron
• Data matched theory exceptionally well
Light
• Light has energy/momentum
E = hf = hc/l
p = E/c = h/l
h = Planck’s constant = 6.626 x 10-34
Particle-Wave Duality
• If a wave (like light) acts like a particle, can
particles act like waves?
Electrons as waves
• Previous pattern means that electrons
interfere with each other
– They act like waves
– What is their wavelength?
• Pattern consistent with a wavelength
l = h/p
• Same relation as with light
• Called the deBroglie wavelength
Now what?
• If electrons (and other particles) can act like
waves, what else can they do?
• They can setup standing waves
– Waves with particular wavelength/frequency that
appear to not be travelling
Electron standing waves
• Electrons in a box (like strings fixed at both ends)
have a standing wave pattern
• Different harmonics have different frequencies
• Difference frequencies means different energies
Standing waves
• Recall: E = ½ mv2 and p = mv  E = p2/2m
p = h/l and l = 2L/n
• Put it all together: E = n2h2/(8mL2)
• Only certain energies are allowed (n=1,2,3,…)
– Energy is quantized
Electrons in an atom
• Electrons can form standing waves going
around a nucleus
• Only some wavelengths fit  only certain
energy levels
– nl = 2pr
Energy levels
• E = ½ mv2 = h2/(2ml2)
• Use l = 2pr/n and r = a0 = Bohr radius
E = n2h2/(8p2ma02)
Theory: Use n = 1 and a0 = 5.29 x 10-11 m
E1 = 2.18 x 10-18 J
Measured value: 2.176 x 10-18 J
Particles as Waves
• If particles can be described by waves, they are
constantly moving
• Location can be described by a wave function
– Amplitude of wave given by Y(x,t)
– Probability of being at a certain place is given by
Y2(x,t)
• Theory of quantum mechanics is essentially an
equation telling you how to find this
wavefunction
Schrodinger Equation
• Describes where a particle is as a function of
time
Here it is:
Solution is the wavefunction!
Only certain solutions allowed for particular
values of E  Energy is quantized
Schrodinger Equation
• Can use this to solve the electron in the box
and the Hydrogen atom
• Gives the solutions that we had before for the
energies
• What does the wavefunction look like?
– Where is the electron?
Hydrogen atom
• A few solutions for the Hydrogen atom
• Look familiar?
Atomic orbitals
Electron orbitals
• The shape of the electron orbitals is given by
the solution to the Schrodinger equation
• Remember: all came about because energy is
quantized because electrons act like (standing)
waves
Quantum Mechanics
Other Quantum Weirdness
• Heisenberg Uncertainty principle
Can’t measure the position and the
velocity/momentum at the same
• Meaning: The act of measuring the location
an object affects its momentum, so can’t
know both at the same time
Probability
• Everything described by a wavefunction
• Tells you probability of where something is 
can’t tell you for certain
• Has some probability that a particle is here
and there at the same time
– Won’t know until you measure it
Schrodinger’s Cat
• Put cat in box with a radioactive compound
that has some probability of decaying and
close the box
• Cat has some probability that it dies and some
probability that it lives
– We say that cat is in a superposition of dead and
alive states
– Won’t know until we open the box
Tunneling
• Look at the p orbital
• Probability that electron in top half
• Probability that electron in bottom half
• Node in the middle – no probability
• So particle can travel from the top part to the
bottom part without going through the middle 
tunneling