Chemistry 542 -- Fall 2001 -- Lecturer: Tim Keiderling

Introductory Quantum Mechanics for Chemistry

Monday -- August 20

Review: Syllabus/handout

Fundamentals course - assume you had undergraduate quantum mechanics

expect you to review basics/historical on own as we will go fast at first

Homework essential (in our opinion) for mastery. Expect to spend 10+ hours/week

Exams reflect lectures; problem oriented mostly

NO MAKE-UP EXAMS/--time to be professional--Excused absences

treated on individual basis (minimum: doctor note)

Text: Levine - popular with students, worked out problems

source of much, not all homework (grade for method not answer)

Extra readings are important -- see syllabus and reserve list

Topics: See syllabus p. 3

Our approach will be postulate prop. QM, work out example problems, apply to atoms and molecules.

Survey: Please hand in at end of class -- we schedule an extra session important to give practice in problem solving and provide make-up classes, etc.

HISTORICAL BACKGROUND -- Levine 1, Atkins 1,2, R&S - 1

19th century physics had it under control

Newtonian mechanics explain particle behavior-- eqn of motion to predict (p,x) at t

Maxwell's eqn summarize all E-M radiation -- light seen as having wave properties

Mechanics and deterministic behavior (Levine 1.4, R&S 1.3)

Newton's second law F = ma = m(d2x/dt2) = m(dv/dt) = - dV(x)/dx

Relate force to potential energy and determine x(t), v(t)

Example 1

Total energy--Hamilton's equation: H = T + V = p2/2m + V(x)

dH/dt = (dx/dt)[m(d2x/dt2 )+ (dV/dx)],

function in parentheses => 0 = F - ma, energy conserved (time independent)

Wednesday-- August 22

Wave motion: A(x,t) = Ao cos(kx-wt) k = 2p/l, w = 2p/t

Increase energy, increase amplitude - Ao à KE~ d2A/dt2 PE~A2

Basic property--Waves diffract--picture1--nl = d sin q

Standing waves must fit the box--example2

Wave equation: d2A/dx2 = (1/vp)2 d2A/dt2

Goal of physics -- explain all of nature, so scale should not matter, if theory good Correspondence -- expect microscopic ´ macroscopic

i.e. should be possible to scale up using a consistent set of physical laws

Catch-- scale -- a few things were not working and they tended toward the microscopic

Black body radiation -- Planck postulate energy not continuous, smallest unit -- hn

this extrapolates from Wien law: lmT = k, good at short l: lm from dr/dl = 0

to Rayleigh-Jeans density of states, based on longer l fit fewer oscillators in cavity,

but as shorten l have more options for fit: r(l) = 8pkT/l4,

was good at long at l , n = c/l , but disaster at short l,

high frequency : 8pn2kT/c3 -- blows up (uv catastrophe)

PICTURE2

Planck restricted energy of oscillators to hn, Boltzman relationship model population,

high frequency oscillators have exponential fall off in population with higher frequency,

then formulated BB energy density as:

This quantum hypothesis is also useful to explain low temperature heat capacity

Photo electric effect -- Einstein goes one step further, quantize light, make it particulate

(1/2) mv2 = hn - F = K.E. of photo emitted electron

picture3

K.E. is independent of intensity of light, number of electrons increases with intensity.

F -- work function, property of material, no electrons until hn > F,

independent of the intensity -- photon energy is hn, but light beam energy is classically the square of amplitude, or total energy is the sum of the energies of the photons

 

Wave-particle duality -- deBroglie --postulated particle to behave as waves, have l

l = h/p p = mv but for light p = mc, rationalize: h/mc = hc/mc2 = hc/hn = c/n = l

Davisson and Germer then showed e-diffract in metals and Thompson in polymer

 

 

 

 

Uncertainty Principle (Heisenberg) -- principle of indeterminancy

-- what can/cannot know, fundamental limitation of quantum systems

fundamental difference from classical - no trajectories or predictions with time

consider x and px -- complementary observables - only one can be precisely known

with wave-particle duality problem clear:

know px perfectly: px = h/l à single plane wave, no localization

Dpx = 0 à Dx = •

picture 4

know x perfectly Æ wave must be -function

Fourier analysis (FT) says this corresponds to linear super position of

all -- interference of wavelengths all but x cancel other x value

thus total localization , but

in between -- Dx restricted and Dp restricted -- few l's

IMPORTANT Uncertainty is an intrinsic property of quantum systems

-- not dependent on "gedanken" exp. or measurement conditions or whatever

Correspondence comes with fact that DxDpx „

for macroscopic systems is very small so that

Newtonian trajectories work as well as we can measure them

Note: this is a statement of what can know or what is complete knowledge--

basis for definition of a quantum state

Aside--(parallel development) in Atomic spectra

Atoms when excited emitted line spectra--not classical (which would be continuous)

Balmer, Rydberg, Ritz--numerologists, found patterns based on 1/l and integers

H-atom: n/c = 1/l = R(1/n12 - 1/n22) where : R ~ 105cm-1 is the Rydberg

general (Ritz): 1/l = T1 - T2 Æ light given off depends on differences of

atomic constants since light ´hn energy (Einstein)´T-energy levels

Bohr postulate elect restricted to E-level Æ stationary orbits

Spectra from e- jump between levels-process unknown- but then emit -- hn = DE

also required: angular momentum integer multiple of nh/2p

(ratio of energy of e- to frequency of orbit = hn/2)

put this together by use classical mechanics for e- ´

centrifugal force balance by electrostatic attraction

worked for H atom, failed for all else - especially. molecules

Friday -- August 25

Schroedinger Equation Plausibility -- R&S 1.10

Since particle is a wave, use general wave function:

Y(x,t) ~ exp[i(kx-wt)] -- since complex, Y(x,t)2 is constant--could be probability

recall : p = h/l = (h/2p)k and E = (h/2p)w substitute and get w/f in particle properties

Y(x,t) ~ exp[2pi/h(px-Et)]

Differentiate: d Y(x,t)/dt = -2pi/h E Y(x,t)

d2 Y(x,t)/dx2 = (2pip/h)2 Y(x,t) = 2m(2pi/h)2 E Y(x,t) from E = p2/2m

rearrange to (ih/2p) d Y(x,t)/dt = E Y(x,t) = - (h/2p)2 d2 Y(x,t)/dx2

this is the Schroedinger Equation, shows the E=p2/2m relationship makes natural

the first time derivative to go with the second space derivative

This is not a derivation, just a plausibility demonstration, consistent with all above

AND IT WORKS!