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