Lecture 18: Quantum Mechanics. Reading: Zumdahl 12.5, 12.6 Outline. Problems (Chapter 12 Zumdahl 5 th Ed.)

Transcription

1 Lecture 18: Quantum Mechanics Reading: Zumdahl 1.5, 1.6 Outline Basic concepts of quantum mechanics and molecular structure A model system: particle in a box. Demos how Q.M. actually obtains a wave function. Confining potentials: Box and Coulomb s law Problems (Chapter 1 Zumdahl 5 th Ed.) 36, 38, 39, 40, 41, 44(find n also), 45(What should happen when system does work?), 46 (find visually). 1

2 Quantum Concepts The Bohr model was capable of describing the discrete or quantized emission spectrum of H. But the failure of the model for multi-electron systems combined with other issues (the ultraviolet catastrophe, work functions of metals, etc.) suggested that a new description of atomic matter was needed.

3 Wave-Particle Duality This new description is known as wave mechanics or quantum mechanics. Recall, photons and electrons readily demonstrate waveparticle duality. The idea behind wave mechanics is that the existence of the electron in fixed energy levels should be thought of as a standing wave, rather than a particle s trajectory. Think about two ways to describe your travels during the day, one uses position and time (the classical way), and the other is just position and probability (the statistical description). Q.M. Provides a way to describe an electron probabilistically without saying in detail how it moves from place to place (that idea must be discarded). 3

4 n = 1 n = Q.M. Concepts: The Standing Wave Ψ ( x) = A sin ( k x) A k π n π = n= 1,, 3 λ L A standing wave is a motion in which translation of the wave does not occur. (A is the amplitude.) In the guitar string analogy (illustrated), note that standing waves involve nodes in which no motion of the string occurs. n = 3 Note also that integer and halfinteger values of the wavelength correspond to standing waves; n must be an integer. 4

5 Q.M. Concept: The debroglie Relation Louis de Broglie suggests that for the e - orbits envisioned by Bohr, only certain orbits are allowed since they satisfy the standing wave condition. Light has momentum (but no mass) and so for light: Amplitude in a line E = cp= hν = light p light = h λ hc λ If true for light, then true for matter (particles) as well. m v = p= h λ not allowed 5 Amplitude in a circle.

6 Amplitude of a periodic function? Ψ φ = cos m φ = cos φ+ π ( m ) ( ) ( ) ( ) Show periodic function circling around 6 times. Only integer values of m keep the function in phase when circling an indefinite number of times. This is debroglie s idea: If an electron behaved as a standing wave in the phi (φ) direction then only integer values of m could keep the wave in phase indefinitely for any value of φ, not just zero to π. m = 3.1 m = 3.0 6

7 Q.M. Concept: Wavefunction What is a wavefunction? Ψ ( r) is a probability amplitude For Example a wave in one dimension (the guitar string): Prob ( x) A sin ( kx) Ψ = ( ) ( ) * ( ) sin ( ) = P x =Ψ x Ψ x = A kx Probability (density) of finding a particle in space is the absolute square of the amplitude. A proper probability must sum to 1, when one explores over all x. This is a way of saying the particle must be somewhere. Here the standing wave of the string is confined from 0 to L. Therefore N 1= P( x ) Δx 1= P( x) dx i i= 1 x= 0 Probability is a statistical description of where the particle is. Compare to a probability distribution of a student on campus. L 7

8 Particle (electron) in a box or trap This is a salad bowl model for electron confinement: See the corral gallery at IBM/Almaden Particle s position is constrained by any confining potential: The choices of physical solutions are countable (but infinite), so quantized. inf. 0 0 L The possible positions of particle are limited to the dimension of the box. Classically the probability is uniform. x Particle in a Box like the ideal gas molecules; now it is a quantum particle. Inside the box the particle just bounces around (no potential energy inside) but the particle cannot get out of the box. 8

9 Particle (in a Box) Wavefunction What do the wavefunctions look like? Amplitude 0 0 Ψ ( x) Ψ( x) Ψ( x) * Probability n = 3 Z1.46 Estimate Prob n = 0 x L 6 0 n =1 nπ Ψ n ( x) = A sin ( k x) k = n= 1,,3 A= L L A standing wave: n is an integer counter; keeps track of the different functions. 9

10 Q.M. Concept: A Wave Equation Erwin Schrodinger developed a mathematical formalism that describes matter using waves. Why waves: Newton s and Maxwell s formulations did not work. Follow debroglie. Must try something else. The Wave Equation: ĤΨ = EΨ This is an equation of motion, but it does not contain time. It is a statement only about position of a particle in space (in a probabilistic sense) and the energy of the particle (E). The wave function (Ψ) describes the particle position. H is the Hamiltonian, which is the total energy written in terms of momentum and position. Just like the Bohr Model: 1 1 v m H = T + V T = m = p V = V r ( ) 10

11 Potentials and Quantization Any confining or binding potential will lead to quantized states (distinct energies and wave functions) Aids in understanding how the wave functions are determined. Potential Energy (V): (three examples) Free Particle Particle in Box Hamiltonian Ĥ = T + V Kinetic Energy (T): (True for every problem) V r ( ) = 0 (not confined) V( x) = 0 0 x L V( x) = otherwise ez Hydrogen Atom V( r) = Coulomb's Law πε r 4 o T = 1 m p 11

12 Implementing Schrödinger s Equation Schrodinger s recipe (called the correspondence principle ) is to substitute for the momentum of the particle the derivative process (operator): p Schrodinger Recipe Ψ pˆ Ψ = d Ψ dx This turns the energy equation into a second order differential equation, which can be solved (by look up/inspection), to give both the wavefunction and the energy. Show how this works: Take two derivatives of the wavefunction (for the guitar string): d sin dx ( kx) ( kx) Ψ = A k Asin = ( kx) = k Ψ d dx Ψ= pˆ Asin ( k) Ψ= Ψ 1

13 Demonstrate The wave function satisfies the Schrodinger Equation and Gives the Energies Use the guitar string wave function and demonstrate that it is a (the) solution to the Schrodinger Equation: Find the energy of each wave function (for different values of k). d Ψ If: Ψ= Asin ( kx) and = k Ψ dx ˆ ˆ ˆ 1 HΨ= EΨ V = 0 H = T = pˆ m d Ψ d Ψ m = EΨ and = k Ψ dx dx ( ) m k p h k Ψ= EΨ = E = p= k = (Just what debroglie said) m m λ 13

14 Summarize: The Schrödinger Equation Assume the wave function is the same as the amplitude wave for a string, and k is some constant, to be determined. Using the result that Hˆ Ψ = k Ψ = EΨ 1 m ( ) Now we connect the constant k to the energy E (which is just a number): 1 m k = E ( ) If the problem is like a string tethered at the two ends (x=0 and x=l) then we use the amplitude wave for that case, π π n which gives: k = = n= 1,, 3 λ L Because n is an integer, we have an infinite number of different possible results and a different energy for each result, but the energies are quantized. π E = En = n ml ( ) 14

15 Q.M. Concept: Energy is quantized E E n n = ( k ) m h = n 8mL = 1,, 3 Energy Show some wave functions with the energy associated with each wave function. The classical limit (Z1.45) Amplitude Ψ n ( x) 15

16 QM Concept: Uncertainty Another error of the Bohr model was that it assumed we could know both the position and momentum of an electron exactly. Werner Heisenberg s development of quantum mechanics leads to the understanding that there is a fundamental limit to how well one can know both the position and momentum of a particle at the same time. There are tradeoffs. Uncertainty in position 1 1 ΔxiΔp = h π Uncertainty in momentum, which is mass times velocity 16

17 Uncertainty (on the small scale) Z1.41a Example: What is the uncertainty in velocity for an electron in a 1Å radius orbit (orbital) in which the positional uncertainty is 1% of the radius. Δx = (1 Å)(0.01) = 1 x 10-1 m Δp = ( 34 J.s) h 4πΔx = 6.66x10 4π 1x10 1 m ( ) = 5.7x10 3 kg.m /s 3 Δp 5.7x10 kg. m/ s 7 Δ v= = = 5.7x10 m Huge 31 m 9.11x10 kg s Near C So we really have to give up the idea we know (even roughly) the position and velocity at the same time. 17

18 Uncertainty (on the large scale) Z1.41b Example (quantum description of large objects): What is the uncertainty in position for a 80 kg student walking across campus at 1.3 m/s with an uncertainty in velocity of 1%. Δp = m Δv = (80kg)(0.013 m/s) = 1.04 kg.m/s Δx = ( 34 J.s) h 4πΔp = 6.66x10 4π 1.04kg.m /s ( ) = 5.07x10 35 m Very small you know where you are. 18

19 Optical Spectra Z using Particle in Box Energy Levels Consider the following dye molecule, the length of which can be considered the length of the box an electron is limited to: hν + N N L = 8 Å=0.8 nm What wavelength of light corresponds to ΔE from n=1 to n=? (A photon/energy is absorbed by the molecule) h ( ) h ( ) 19 Δ E = n final ninitial = = J 8mL 8 m(8 Å) hc λ 700nm Ephoton = λ (should be 680 nm, orange) 19

20 Potentials and Quantization (cont.) One effect of a constraining potential is that the energy of the system becomes quantized. Back to the hydrogen-like atom: e- 0 r r P + Z V() r = e r Z constraining potential 0

21 H Atom Potential Also in the case of the hydrogen atom, energy becomes quantized due to the presence of a confining/constraining potential. 0 r 0 V() r = e Z r Schrödinger Equation Above zero (positive energy states) the energy is not quantized; any K.E. is O.K. Recovers the Bohr behavior 1