Presentation on theme: "Introduction and Background"— Presentation transcript:
1Introduction and Background Chapter 1Introduction and Backgroundto Quantum Mechanics
2The Need for Quantum Mechanics in Chemistry Without Quantum Mechanics, how would you explain:• Periodic trends in properties of the elements• Structure of compoundse.g. Tetrahedral carbon in ethane, planar ethylene, etc.• Bond lengths/strengths• Discrete spectral lines (IR, NMR, Atomic Absorption, etc.)• Electron MicroscopyWithout Quantum Mechanics, chemistry would be a purelyempirical science.PLUS: In recent years, a rapidly increasing percentage ofexperimental chemists are performing quantum mechanicalcalculations as an essential complement to interpretingtheir experimental results.
3• Problems in Classical Physics Outline• Problems in Classical Physics• The “Old” Quantum Mechanics (Bohr Theory)• Wave Properties of Particles• Heisenberg Uncertainty Principle• Mathematical Preliminaries• Concepts in Quantum MechanicsThere is nothing new to be discovered in Physics now.All that remains is more and more precise measurement.Lord Kelvin (Sir William Thompson), ca 1900
4Blackbody Radiation Heated Metal Low Temperature: Red Hot IntensityHeated MetalLow Temperature: Red HotIntermediate Temperature: White HotHigh Temperature: Blue Hot
5Rayleigh-Jeans (Classical Physics) Assumed that electrons in metal oscillate about their equilibriumpositions at arbitrary frequency (energy). Emit light at oscillation frequency.IntensityThe Ultraviolet Catastrophe:You are NOT responsible for equations in section on Problems in Classical Physics.
6Max Planck (1900)Arbitrarily assumed that the energy levels of the oscillating electronsare quantized, and the energy levels are proportional to : = h(n)n = 1, 2, 3,...h = empirical constantIntensityExpression matches experimental data perfectly forh = 6.626x10-34 J•s [Planck’s Constant]He derived the expression:
7The Photoelectric Effect A- VS +Kinetic Energy of ejected electrons can bemeasured by determining the magnitude ofthe “stopping potential” (VS) required tostop current.ObservationsLow frequency (red) light: < o - No ejected electrons (no current)High frequency (blue) light: > o - K.E. of ejected electrons K.E.o
8 is the metal’s “work function”: the energy required to eject an K.E.oPhotonsEinstein (1903) proposed that lightenergy is quantized into “packets”called photons.Eph = hSlope = hK.E. = h - = h - hoo = / hPredicts that the slope of the graphof K.E. vs. is h (Planck’s Constant)in agreement with experiment !!Explanation of Photoelectric EffectEph = h= + K.E. is the metal’s “work function”: the energy required to eject anelectron from the surface
9Equations Relating Properties of Light Wavelength/Frequency:Wavenumber:Units: cm-1c must be in cm/sEnergy:You should know these relations between the properties of light.They will come up often throughout the course.
10Atomic Emission Spectra HeatSampleWhen a sample of atoms is heated up, the excited electrons emitradiation as they return to the ground state.The emissions are at discrete frequencies, rather than a continuumof frequencies, as predicted by the Rutherford planetary modelof the atom.
12Outline• Problems in Classical Physics• The “Old” Quantum Mechanics (Bohr Theory)• Wave Properties of Particles• Heisenberg Uncertainty Principle• Mathematical Preliminaries• Concepts in Quantum Mechanics
13The “Old” Quantum Theory Niels Bohr (1913)Assumed that electron in hydrogen-like atom moved in circular orbit,with the centripetal force (mv2/r) equal to the Coulombic attractionbetween the electron (with charge e) and nucleus (with charge Ze).eZerDon’t worry about equations.eps-o is the permittivity (dielectric constant) of free space.He then arbitrarily assumed that the “angular momentum” is quantized.n = 1, 2, 3,...(Dirac’s Constant)Why??Because it worked.
14It can be shown (Bohr Radius) n = 1, 2, 3,... = 0.529 Å Don’t worry about equations.n = 1, 2, 3,...
15nU nL EU EL Lyman Series: nL = 1 Balmer Series: nL = 2 Paschen Series: nL = 3
16nU nL EU EL Close to RH = 108,680 cm-1 Explain v-bar = Eph/hc. Then show calculation to get 109,800Close to RH = 108,680 cm-1Get perfect agreement if replace electron mass (m) by reducedmass () of proton-electron pair.
17The Bohr Theory of the atom (“Old” Quantum Mechanics) works perfectly for H (as well as He+, Li2+, etc.).And it’s so much EASIER than the Schrödinger Equation.The only problem with the Bohr Theory is that it fails as soonas you try to use it on an atom as “complex” as helium.
18Outline• Problems in Classical Physics• The “Old” Quantum Mechanics (Bohr Theory)• Wave Properties of Particles• Heisenberg Uncertainty Principle• Mathematical Preliminaries• Concepts in Quantum Mechanics
19Wave Properties of Particles The de Broglie WavelengthLouis de Broglie (1923): If waves have particle-like properties (photons,then particles should have wave-like properties.Photon wavelength-momentum relationandde Broglie wavelength of a particle
20What is the de Broglie wavelength of a 1 gram marble traveling at 10 cm/sh=6.63x10-34 J-s = 6.6x10-30 m = 6.6x10-20 Å (insignificant)What is the de Broglie wavelength of an electron travelingat 0.1 c (c=speed of light)?c = 3.00x108 m/sme = 9.1x10-31 kg = 2.4x10-11 m = 0.24 Å(on the order of atomic dimensions)
21Reinterpretation of Bohr’s Quantization of Angular Momentumn = 1, 2, 3,...(Dirac’s Constant)The circumference of a Bohrorbit must be a whole numberof de Broglie “standing waves”.Bd: Show picture of circle with whole number of standing waves
22Outline• Problems in Classical Physics• The “Old” Quantum Mechanics (Bohr Theory)• Wave Properties of Particles• Heisenberg Uncertainty Principle• Mathematical Preliminaries• Concepts in Quantum Mechanics
23Heisenberg Uncertainty Principle Werner Heisenberg: 1925It is not possible to determine both the position (x) and momentum (p)of a particle precisely at the same time.p = Uncertainty in momentumx = Uncertainty in positionExplain that this is not due to experimental errors, but is an inherent property.Often see as >= h.Discuss: Accurate position - short lambda = high pAccurate momentum - low p = long lambdaText has “derivation” based on lambda of particle - not responsible.There are a number of pseudo-derivations of this principle in various texts,based upon the wave property of a particle. We will not give one ofthese derivations, but will provide examples of the uncertainty principleat various times in the course.
24Calculate the uncertainty in the position of a 5 Oz (0.14 kg) baseball traveling at 90 mi/hr (40 m/s), assuming that the velocity can bemeasured to a precision of 10-6 percent.h = 6.63x10-34 J-sħ = 1.05x10-34 J-s1x = 9.4x10-28 mCalculate the uncertainty in the momentum (and velocity) of anelectron (me=9.11x10-31 kg) in an atom with an uncertainty inposition, x = 0.5 Å = 5x10-11 m.p = 1.05x10-24 kgm/sv = 1.15x106 m/s (=2.6x106 mi/hr)
25Outline• Problems in Classical Physics• The “Old” Quantum Mechanics (Bohr Theory)• Wave Properties of Particles• Heisenberg Uncertainty Principle• Mathematical Preliminaries• Concepts in Quantum Mechanics
26Math Preliminary: Trigonometry and the Unit Circle x axisy axisxy1sin(0o) =cos(180o) =-1sin(90o) =1cos(270o) =cos() = xsin() = yFrom the unit circle,it’s easy to see that:cos(-) = cos()sin(-) = -sin()
27Math Preliminary: Complex Numbers Real axisImag axisRxyComplex PlaneEuler RelationsComplex number (z)orwhereComplex conjugate (z*)or
28Math Preliminary: Complex Numbers Real axisImag axisRxyComplex PlaneorwhereMagnitude of a Complex NumberWork out the 2 forms of |z|^2 on boardor
29Outline• Problems in Classical Physics• The “Old” Quantum Mechanics (Bohr Theory)• Wave Properties of Particles• Heisenberg Uncertainty Principle• Mathematical Preliminaries• Concepts in Quantum Mechanics
30Concepts in Quantum Mechanics Erwin Schrödinger (1926): If, as proposed by de Broglie, particles displaywave-like properties, then they should satisfy awave equation similar to classical waves.He proposed the following equation.One-Dimensional Time Dependent Schrödinger Equation is the wavefunctionm = mass of particle||2 = * is the probability offinding the particle betweenx and x + dxV(x,t) is the potential energy
31Wavefunction for a free particle +-V(x,t) = const = 0whereandClassical Traveling WaveUnsatisfactory becauseThe probability of finding the particle at any position(i.e. any value of x) should be the sameFor a particle:is satisfactoryNote that:
32“Derivation” of Schrödinger Eqn. for Free Particle whereandonboardon boardSchrödinger Eqn.for V(x,t) = 0
33Note: We cannot actually derive Quantum Mechanics or the Schrödinger Equation.In the last slide, we gave a rationalization of how, if aparticle behaves like a wave and is given by the de Broglierelation, then the wavefunction, , satisfies the wave equationproposed by Erwin Schrödinger.Quantum Mechanics is not “provable”, but is built upona series of postulates, which will be discussed in thenext chapter.The validity of the postulates is based upon the fact thatQuantum Mechanics WORKS. It correctly predicts the propertiesof electrons, atoms and other microscopic particles.