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Chapter 1 Introduction and Background to Quantum Mechanics Slide 1 The Need for Quantum Mechanics in Chemistry Without Quantum Mechanics, how would you explain: • Periodic trends in properties of the elements • Structure of compounds e.g. Tetrahedral carbon in ethane, planar ethylene, etc. • Bond lengths/strengths • Discrete spectral lines (IR, NMR, Atomic Absorption, etc.) • Electron Microscopy Without Quantum Mechanics, chemistry would be a purely empirical science. PLUS: In recent years, a rapidly increasing percentage of experimental chemists are performing quantum mechanical calculations as an essential complement to interpreting their experimental results. Slide 2 Outline • Problems in Classical Physics • The “Old” Quantum Mechanics (Bohr Theory) • Wave Properties of Particles • Heisenberg Uncertainty Principle • Mathematical Preliminaries • Concepts in Quantum Mechanics There 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 Slide 3 Intensity Blackbody Radiation Heated Metal Low Temperature: Red Hot Intermediate Temperature: White Hot High Temperature: Blue Hot Slide 4 Rayleigh-Jeans (Classical Physics) ( , T ) 8 kT 2 const 2 3 c Intensity Assumed that electrons in metal oscillate about their equilibrium positions at arbitrary frequency (energy). Emit light at oscillation frequency. The Ultraviolet Catastrophe: 0 ( , T ) d Slide 5 Max Planck (1900) Arbitrarily assumed that the energy levels of the oscillating electrons are quantized, and the energy levels are proportional to : = h(n) He derived the expression: ( , T ) Intensity n = 1, 2, 3,... h = empirical constant 8 h 3 1 c3 e h / kT 1 Expression matches experimental data perfectly for h = 6.626x10-34 J•s [Planck’s Constant] Slide 6 The Photoelectric Effect Kinetic Energy of ejected electrons can be measured by determining the magnitude of the “stopping potential” (VS) required to stop current. - VS + A Observations Low frequency (red) light: < o - No ejected electrons (no current) K.E. High frequency (blue) light: > o - K.E. of ejected electrons o Slide 7 Photons K.E. Einstein (1903) proposed that light energy is quantized into “packets” called photons. Slope = h Eph = h Explanation of Photoelectric Effect o Eph = h = + K.E. is the metal’s “work function”: the energy required to eject an electron from the surface K.E. = h - = h - ho o = / h Predicts that the slope of the graph of K.E. vs. is h (Planck’s Constant) in agreement with experiment !! Slide 8 Equations Relating Properties of Light Wavelength/ Frequency: Wavenumber: c 1 (cm) Energy: c c E ph h Units: cm-1 1 hc c c must be in cm/s hc You should know these relations between the properties of light. They will come up often throughout the course. Slide 9 Atomic Emission Spectra Sample Heat When a sample of atoms is heated up, the excited electrons emit radiation as they return to the ground state. The emissions are at discrete frequencies, rather than a continuum of frequencies, as predicted by the Rutherford planetary model of the atom. Slide 10 Hydrogen Atom Emission Lines UV Region: (Lyman Series) Visible Region: (Balmer Series) 1 1 108, 680 1 2 cm 1 n n = 2, 3, 4 ... 1 1 108, 680 0.25 2 cm 1 n n = 3, 4, 5, ... Infrared Region: 1 108, 680 0.111 1 cm 1 n 2 (Paschen Series) n = 4, 5, 6 ... General Form (Johannes Rydberg) 1 1 1 R H 2 2 cm 1 n1 n 2 n1 = 1, 2, 3 ... n2 > n 1 RH = 108,680 cm-1 Slide 11 Outline • Problems in Classical Physics • The “Old” Quantum Mechanics (Bohr Theory) • Wave Properties of Particles • Heisenberg Uncertainty Principle • Mathematical Preliminaries • Concepts in Quantum Mechanics Slide 12 The “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 attraction between the electron (with charge e) and nucleus (with charge Ze). e v2 1 Ze e f ma m r 4 0 r 2 r Ze He then arbitrarily assumed that the “angular momentum” is quantized. L rxp m vr n n = 1, 2, 3,... h (Dirac’s Constant) 2 Why?? Because it worked. Slide 13 v2 1 Ze e m r 4 0 r 2 m vr n It can be shown 4 0 a0 me 2 4 0 2 n 2 2 a0 n r 2 Z me Z 2 = 0.529 Å (Bohr Radius) 2 Ze 1 E K .E . P .E . 1 mv 2 4 0 r 2 me 4 Z 2 1 18 1 2.181 x 10 Joules E 2 2 2 2 2 n 8 0 h n n n = 1, 2, 3,... Slide 14 nU EU nL EL E ph E E U E L E 2 nU 2 nL 1 1 E 2 2 nL nU 2.181x10 18 Lyman Series: nL = 1 1 1 2 2 Joules nL nU Balmer Series: nL = 2 Paschen Series: nL = 3 Slide 15 nU EU nL EL 1 1 E 2 2 nL nU 1 1 2.181x10 18 2 2 Joules nL nU 1 E 1 109, 800 2 2 cm 1 hc hc n L nU E ph Close to RH = 108,680 cm-1 Get perfect agreement if replace electron mass (m) by reduced mass () of proton-electron pair. Slide 16 The 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 soon as you try to use it on an atom as “complex” as helium. Slide 17 Outline • Problems in Classical Physics • The “Old” Quantum Mechanics (Bohr Theory) • Wave Properties of Particles • Heisenberg Uncertainty Principle • Mathematical Preliminaries • Concepts in Quantum Mechanics Slide 18 Wave Properties of Particles The de Broglie Wavelength Louis de Broglie (1923): If waves have particle-like properties (photons, then particles should have wave-like properties. Photon wavelength-momentum relation E h hc and E mc2 hc hc h h E mc 2 mc p de Broglie wavelength of a particle h h p mv Slide 19 What is the de Broglie wavelength of a 1 gram marble traveling at 10 cm/s h=6.63x10-34 J-s = 6.6x10-30 m = 6.6x10-20 Å (insignificant) What is the de Broglie wavelength of an electron traveling at 0.1 c (c=speed of light)? c = 3.00x108 m/s me = 9.1x10-31 kg = 2.4x10-11 m = 0.24 Å (on the order of atomic dimensions) Slide 20 Reinterpretation of Bohr’s Quantization of Angular Momentum L rxp m vr n n = 1, 2, 3,... mvr h (Dirac’s Constant) 2 nh 2 h h 2 r n n mv p The circumference of a Bohr orbit must be a whole number of de Broglie “standing waves”. 2 r n Slide 21 Outline • Problems in Classical Physics • The “Old” Quantum Mechanics (Bohr Theory) • Wave Properties of Particles • Heisenberg Uncertainty Principle • Mathematical Preliminaries • Concepts in Quantum Mechanics Slide 22 Heisenberg Uncertainty Principle Werner Heisenberg: 1925 It is not possible to determine both the position (x) and momentum (p) of a particle precisely at the same time. p x 2 p = Uncertainty in momentum x = Uncertainty in position 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 of these derivations, but will provide examples of the uncertainty principle at various times in the course. Slide 23 Calculate 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 be measured to a precision of 10-6 percent. h = 6.63x10-34 J-s -34 J-s1 ħ = 1.05x10 x = 9.4x10-28 m Calculate the uncertainty in the momentum (and velocity) of an electron (me=9.11x10-31 kg) in an atom with an uncertainty in position, x = 0.5 Å = 5x10-11 m. p = 1.05x10-24 kgm/s v = 1.15x106 m/s (=2.6x106 mi/hr) Slide 24 Outline • Problems in Classical Physics • The “Old” Quantum Mechanics (Bohr Theory) • Wave Properties of Particles • Heisenberg Uncertainty Principle • Mathematical Preliminaries • Concepts in Quantum Mechanics Slide 25 Math Preliminary: Trigonometry and the Unit Circle y axis y 1 x sin(0o) = 0 x axis cos(180o) = -1 sin(90o) = 1 cos(270o) = 0 cos() = x sin() = y From the unit circle, it’s easy to see that: cos(-) = cos() sin(-) = -sin() Slide 26 Math Preliminary: Complex Numbers i 1 Imag axis Euler Relations e i c o s ( ) i s in ( ) y e i c o s ( ) i s in ( ) R x Real axis Complex number (z) z x iy where or z R e i x R c o s ( ) a n d y R s in ( ) Complex Plane Complex conjugate (z*) z * x iy or z * R e i Slide 27 Math Preliminary: Complex Numbers z x iy Imag axis x R c o s ( ) a n d y R s in ( ) where y R x z R e i or Magnitude of a Complex Number Real axis z z 2 z z * z ( x iy )( x iy ) x 2 y 2 2 or Complex Plane z z * z (R e i )(R e i ) R 2 2 Slide 28 Outline • Problems in Classical Physics • The “Old” Quantum Mechanics (Bohr Theory) • Wave Properties of Particles • Heisenberg Uncertainty Principle • Mathematical Preliminaries • Concepts in Quantum Mechanics Slide 29 Concepts in Quantum Mechanics Erwin Schrödinger (1926): If, as proposed by de Broglie, particles display wave-like properties, then they should satisfy a wave equation similar to classical waves. He proposed the following equation. One-Dimensional Time Dependent Schrödinger Equation 2 2 i V ( x, t ) 2 t 2 m x h ( D irac ' s C ons tan t ) 2 m = mass of particle V(x,t) is the potential energy is the wavefunction ||2 = * is the probability of finding the particle between x and x + dx Slide 30 Wavefunction for a free particle - + V(x,t) = const = 0 C cos(kx t) k where Classical Traveling Wave 2 and For a particle: Unsatisfactory because 2 E h h p C 2 co s 2 ( kx t ) co n stan t The probability of finding the particle at any position (i.e. any value of x) should be the same C e i ( kx t ) Note that: is satisfactory 2 * C 2 co n stan t Slide 31 “Derivation” of Schrödinger Eqn. for Free Particle Ce i ( kx t ) where k 2 E h and h p on board E i Ce i ( px Et ) t on board 2 p2 2 2m 2 m x 2 2 2 i t 2 m x 2 Schrödinger Eqn. for V(x,t) = 0 Slide 32 Note: We cannot actually derive Quantum Mechanics or the Schrödinger Equation. In the last slide, we gave a rationalization of how, if a particle behaves like a wave and is given by the de Broglie relation, then the wavefunction, , satisfies the wave equation proposed by Erwin Schrödinger. Quantum Mechanics is not “provable”, but is built upon a series of postulates, which will be discussed in the next chapter. The validity of the postulates is based upon the fact that Quantum Mechanics WORKS. It correctly predicts the properties of electrons, atoms and other microscopic particles. Slide 33