PChem C362 Exam 1 Material

Vectors are fully described by their

magnitude and direction (scalar only has magnitude)

the magnitude of the vector can be calculated using

the dot product (or inner product)

the dot product tells us

the magnitude of a times the projection of b on a

dot product formula

ab=|a||b|cos(theta)

orthonormal

orthogonal by being 90 deg apart, and normal by the vector being normalized when its magnitude is 1 (ex: i, j, k)

the s and p orbital are

orthogonal

when doing matrix multiplication, the dot product is in this form

bra-ket (2nd place of 1st matrix has to match 1st place of 2nd matrix) to produce a scalar

do matrices when multiplied commute?

no (hence in QM, observables may not commute because quantities are matrices)

in quantum mechanics, a wavefunction is…

an orbital, represented as a vector

complex number i

sqrt(-1)

euler’s formula (represents complex number on a 2D plane, not exponential)

exp(itheta) = cos(theta) + isin(theta)

complex conjugate

a + ib, a - ib

the schrodinger equation is an…

eigenvalue problem, provides us with the orbitals in any system

eigenvalue

the constant number that pulls out of an equation when an operator acts on a function, leaving the original function completely unchanged

pictorially what is an integral

area under a curve, shows how functions can be orthogonal

dual space analogue

complex conjugated transpose of a vector

dyad vs dot product

outer product vs inner product, produces matrix vs scalar, ket bra vs bra ket

black body radiation and ultraviolet catastrophy

frequency comes down at higher temps, planck postulated light has a wave nature and particle nature, discrete packets have energy E = hv

photoelectric effect

einstein reaffirms planck, experiment involving detachment of electrons from a metal surface, implies light is made of discrete packets (KE electron = hv - phi)

davisson germer experiment

wave nature of electrons, interference patterns with beam of electrons hitting nickel foil and showing pattern on screen demonstrating wave nature (where is electron on the screen? everywhere, it is a wave), size of slit needs to be smaller than the wavelength (larger? no diffraction), diffraction shows wave bending around object (1 slit) where interference shows wave overlap pattern (2 slit)

stern-gerlach experiment

leads to a detailed understanding of the behavior of the spin of an electron (1/2) and that electrons can have only two spin states

light can be made to behave as a particle by the _____ and behave as a wave by the _____

photoelectric effect, two slit experiment

debroglie

dual nature idea of wave-particle duality of MATTER (aka electrons), wavelength = h/p = h/mv = 2L/n, p = h/wavelength = nh/2L

1D PIB boundary conditions

box edges have infinite repulsive potentials to keep the particle inside the molecular framework, but into the molecular framework the electrons are completely free to move as they should be on account of resonance therefore V=0 inside and V=infinity outside

the wavefunction in PIB must be continuous everywhere therefore the value of psi must by zero at…

0 and L

why does the wavefunction have to be normalized

because you cant get a zero, therefore it wouldn’t exist, so the probability of finding the particle in the box equals exactly one

to normalize, you need to

do the integration of the probability distribution

n cannot equal 0, why?

n = 1 is the zero point energy, lowest state physically possible, n=0 is at the bottom of the trough

what do nodes mean

the solution to PIB is oscillatory, therefore it contains points inside the box where the probability is zero (these are called nodes)

energy in PIB is quantized therefore

cant be infinity, must be some discrete value(s) (BC enforce quantization)

Case I: No delocalization (no resonance)

higher energy state, less stable seen in PIB treatment

Case II: Delocalized (resonance)

two electrons per state, look at HW

Spectroscopy of dyes

as the size of the molecule increases, they emit red color whereas the smaller ones are blue, as the size of the box increases the energy levels decrease, as the size of the polyene increase the gap between the occupied and unoccupied states decrease

quantum dots

size of particles dictating wavelength of light it emits and therefore color, when quantum dots are hit with a light source each emits a color of a specific bandwidth, larger dots emit light that is skewed toward red, semiconductor material