Quantum Mechanics. Particle in a box All were partial answers, leading Schrödinger to wave mechanics

Transcription

1 Chemistry 4521 Time is flying by: only 15 lectures left!! Six quantum mechanics Four Spectroscopy Third Hour exam Three statistical mechanics Review Final Exam, Wednesday, May 4, 7:30 10 PM

2 Quantum Mechanics Overall goals: Introduce QM and qualitatively solve limited problems in 1D; extend to 3D Particle in a box (translation quantization) Harmonic oscillator (molecular l vibration) Rigid rotor (molecular rotation) Atomic and molecular electronic energy levels Apply these results to spectroscopic analyses and statistical equilibrium Lecture 1 (today) Classical mechanics, optics, wave motion, thermodynamics Failures when extending to short distances, small masses Blackbody radiation ( existence of photons) Wave particle duality Diffraction, photoelectric t effect Matter (debroglie) waves, electron diffraction Qualitative QM Bohr atom Particle in a box All were partial answers, leading Schrödinger to wave mechanics

3 Classical Physics/Mechanics Predicts precise trajectory for particles with precisely specified locations and momenta at each instant t Allows the translational, rotational and vibrational modes of motion to be excited to any energy, simply by controlling forces that are applied Considers matter and energy as distinct Shattered by three observations involving matter and light, all indicating the presence of discrete energy states Blackbody radiation Photoelectric t effect Atomic line spectra

4 Quantum Mechanics Circumference = 2r=n r Describes the "Wave-Particle" Duality Light is an electromagnetic wave, described by Maxwell s equation - but it can also behave like a particle Particles - also have wavelike nature (manifest only when mass is tiny) Th ti f ti l b d ib d b The wave properties of particles can be described by a modified form of Maxwell s equations for wave motion, known as the Schrödinger equation.

5 Pre-Quantum Mechanics 1890's I. Classical Mechanics General Equations (F= ma on steroids) LaGrange Hamilton II. Electricity & Magnetism Maxwell's Equations Electromagnetic Waves: Central theses of the time: No real conceptual issues remain unresolved. Computations ti on real systems were unbelievably bl hard, however!

6 Traveling Wave Wave amplitude is orthogonal to the direction of propagation Single vertical pulse moves on horizontal string With only five lectures, we must ignore time dependence, and restrict ourselves es to those states characterized by standing waves. Time-independent quantum mechanics 2 Y xt, Asin xvt v with frequency

7 Wave Motion in Restricted Systems One-half wavelength (/2) is the quantum of the guitar string s vibration X X

8 Pre-Quantum Mechanics II. Electricity & Magnetism Maxwell's Equations Light: III. Thermodynamics (you know all about this!) IV. Optics Wave Diffraction ( ~ object size) Two slit diffraction: Geometrical Optics ( >> object size)

9 Failures of Classical Physics (waves behaving as particles) 1. Blackbody radiation

10 Blackbody Radiation xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Smoldering coal 1000 K Electric heating coil 1500 K Light bulb filament 2000 K

11 Radiation from a Cavity is Blackbody Radiation Atoms in solid vibrate and produce radiation Radiation at in cavity has range of frequencies, Treated rigorously with classical thermodynamics and E & M

12 Rayleigh-Jeans Blackbody Calculation (rigorous thermo and E&M) 2 8 kt d 3 B c 8 kt d 4 B How well did it work? Correct at large The Ultraviolet Catastrophe

13 Blackbody Radiation Radiation emitted from object increases and shifts to shorter wavelengths at higher temperatures Not quantitatively explained by classical physics is energy at a frequency, per unit volume, and per unit frequency range

14 Planck's Idea that Oscillator Energy is Quantized Max Planck "fixed it up" by trial and error, in the process incorporating a "non-physical" assumption: Photons can behave like particles under some conditions, and have a relationship between their frequency and energy given by E = h = h c/, where h is a constant. Substituting i this hypothesis h into the Rayleigh - Jeans theory, he obtained ( T, ) 8 hc 1 5 hc kt e 1 Planck simply adjusted h, and found one value, 6.66 x erg s, that fit the experimental data! It was later known as Planck's constant. No one took it seriously.

15 Comparison of Planck and Rayleigh-Jeans equations Rayleigh Jeans (non-quantized energy): 2 8 kt 3 B c Planck (quantized energy): 2 8 h 3 c exp(h / k T) 1 A constant (later called h) reappeared in the photoelectric A constant (later called h) reappeared in the photoelectric effect and in the diffraction of matter waves, in each case with the same numerical value! A clue! B

16 Failures of Classical Physics (waves behaving as particles) 2. Photoelectric effect Light Electron

17 Photoelectric Effect Light Electron, mass m e speed u Electron emission depends on frequency of light, NOT on intensity of light Light acts as if it is a beam of particles that has energy h One-to-one relationship between photon absorbed and e emitted h = 1/2 mu 2 + w where w = e= work function (E required to remove e ) and 1/2 m e u 2 = KE of ejected e

18 Experimental Characteristics of the Photoelectric Effect h = 1/2 mu 2 + e 1. There is a threshold frequency for electron ejection. Neither the threshold nor KE depends on intensity of the light. 2. KE of ejected electrons increases with frequency. 3. There is no time lag for electron ejection 4. e is the work function (e - binding energy) of the metal 5. Nobel prize for Einstein!

19 ConcepTest # 1 The kinetic energy of the photo- electron is plotted versus the frequency of incident radiation for potassium, rubidium and sodium. From left to right, identify the lines. A. K, Rb, Na B. Rb, Na, K C. Rb, K, Na D. Na, K, Rb

20 Failures of Classical Physics (particles behaving as waves) 3. X-ray and Electron Diffraction d e- me Led to debroglie x ray d V d matter h p p matter p = momentum = mv Au 4. Atomic Line Emission seems to indicate discrete energy states Dispersion of visible light Au Dispersion of light from excited atoms

21 Photons as waves and particles Light particles with energy h known as photons Dual theory of radiation Diffraction and interference depends on WAVE properties of light Photoelectric effect depends on PARTICLE properties of light

22 Matter: particles & waves Matter with mass m and velocity v seems to have a wave of wavelength D =h/p associated with it. Matter behaves normally (F=ma) when it interacts with objects with dimensions >> D Matter displays diffraction and interference effects when it interacts with objects with dimensions i D. (constructive and destructive interference leading to the concept of stationary states) D The Schrödinger equation is essentially a wave equation applied to matter waves. h p