From Last time… De Broglie wavelength Uncertainty principle Wavefunction of a particle

Exam 3 results Exam average ~ 70% Scores posted on learn@uw AB B BC C D Exam average ~ 70% Scores posted on learn@uw Course evaluations: Tuesday, Dec. 9 Tue. Dec. 1 2008 Physics 208, Lecture 25

The wavefunction 2  Particle has a wavefunction (x) x x dx Very small x-range = probability to find particle in infinitesimal range dx about x Larger x-range = probability to find particle between x1 and x2 Entire x-range particle must be somewhere Tue. Dec. 1 2008 Physics 208, Lecture 25

Probability P(heads)=0.5 P(tails)=0.5 P(1)=1/6 P(2)=1/6 etc Heads 0.0 P(heads)=0.5 P(tails)=0.5 1 Probability 0.5 0.0 2 3 4 5 6 1/6 P(1)=1/6 P(2)=1/6 etc Tue. Dec. 1 2008 Physics 208, Lecture 25

Discrete vs continuous 6-sided die, unequal prob 0.5 Loaded die Probability 1/6 0.0 1 2 3 4 5 6 Infinite-sided die, all numbers between 1 and 6 “Continuous” probability distribution 1 P(x) 0.5 0.0 2 3 4 5 6 1/6 x Tue. Dec. 1 2008 Physics 208, Lecture 25

Example wavefunction 2  What is P(-2<x<-1)? What is (0)? = fractional area under curve -2 < x < -1 = 1/8 total area -> P = 0.125 What is (0)? = entire area under 2 curve = 1 = (1/2)(base)(height)=22(0)=1 Tue. Dec. 1 2008 Physics 208, Lecture 25

Wavefunctions Each quantum state has different wavefunction Wavefunction shape determined by physical characteristics of system. Different quantum mechanical systems Pendulum (harmonic oscillator) Hydrogen atom Particle in a box Each has differently shaped wavefunctions Tue. Dec. 1 2008 Physics 208, Lecture 25

Quantum ‘Particle in a box’ Particle confined to a fixed region of space e.g. ball in a tube- ball moves only along length L Classically, ball bounces back and forth in tube. A classical ‘state’ of the ball. State indexed by speed, momentum=(mass)x(speed), or kinetic energy. Classical: any momentum, energy is possible. Quantum: momenta, energy are quantized L Tue. Dec. 1 2008 Physics 208, Lecture 25

Classical vs Quantum Classical: particle bounces back and forth. Sometimes velocity is to left, sometimes to right L Quantum mechanics: Particle represented by wave: p = mv = h /  Different motions: waves traveling left and right Quantum wavefunction: superposition of both at same time Tue. Dec. 1 2008 Physics 208, Lecture 25

Quantum version Quantum state is both velocities at the same time Superposition waves is standing wave, made equally of Wave traveling right ( p = +h/ ) Wave traveling left ( p = - h/ ) Determined by standing wave condition L=n(/2) : One half-wavelength momentum L Quantum wave function: superposition of both motions. Tue. Dec. 1 2008 Physics 208, Lecture 25

Different quantum states p = mv = h /  Different speeds correspond to different  subject to standing wave condition integer number of half-wavelengths fit in the tube. Wavefunction: n=1 momentum One half-wavelength n=1 n=2 n=2 Two half-wavelengths momentum Tue. Dec. 1 2008 Physics 208, Lecture 25

Particle in box question A particle in a box has a mass m. Its energy is all kinetic = p2/2m. Just saw that momentum in state n is npo. It’s energy levels A. are equally spaced everywhere B. get farther apart at higher energy C. get closer together at higher energy. Tue. Dec. 1 2008 Physics 208, Lecture 25

Particle in box energy levels Quantized momentum Energy = kinetic Or Quantized Energy n=quantum number Tue. Dec. 1 2008 Physics 208, Lecture 25

Zero-point motion Lowest energy is not zero Confined quantum particle cannot be at rest Always some motion Consequency of uncertainty principle p cannot be zero p not exactly known p cannot be exactly zero Comment here about right-hand-side of uncertainty relation. Tue. Dec. 1 2008 Physics 208, Lecture 25

Question A particle is in a particular quantum state in a box of length L. The box is now squeezed to a shorter length, L/2. The particle remains in the same quantum state. The energy of the particle is now A. 2 times bigger B. 2 times smaller C. 4 times bigger D. 4 times smaller E. unchanged Tue. Dec. 1 2008 Physics 208, Lecture 25

Quantum dot: particle in 3D box CdSe quantum dots dispersed in hexane (Bawendi group, MIT) Color from photon absorption Determined by energy-level spacing Decreasing particle size Energy level spacing increases as particle size decreases. i.e Video of making CdSe nanocrystals: http://www.youtube.com/watch?v=MLJJkztIWfg&feature=related Tue. Dec. 1 2008 Physics 208, Lecture 25

Interpreting the wavefunction Probability interpretation The square magnitude of the wavefunction ||2 gives the probability of finding the particle at a particular spatial location Wavefunction Probability = (Wavefunction)2 Tue. Dec. 1 2008 Physics 208, Lecture 25

Higher energy wave functions L n p E Wavefunction Probability n=3 n=2 n=1 Tue. Dec. 1 2008 Physics 208, Lecture 25

Probability of finding electron Classically, equally likely to find particle anywhere QM - true on average for high n Zeroes in the probability! Purely quantum, interference effect Tue. Dec. 1 2008 Physics 208, Lecture 25

Quantum Corral D. Eigler (IBM) 48 Iron atoms assembled into a circular ring. The ripples inside the ring reflect the electron quantum states of a circular ring (interference effects). Tue. Dec. 1 2008 Physics 208, Lecture 25

Particle in box again: 2 dimensions Motion in x direction Motion in y direction Same velocity (energy), but details of motion are different. Tue. Dec. 1 2008 Physics 208, Lecture 25

Quantum Wave Functions Probability (2D) Ground state: same wavelength (longest) in both x and y Need two quantum #’s, one for x-motion one for y-motion Use a pair (nx, ny) Ground state: (1,1) Wavefunction Probability = (Wavefunction)2 One-dimensional (1D) case Tue. Dec. 1 2008 Physics 208, Lecture 25

2D excited states (nx, ny) = (2,1) (nx, ny) = (1,2) These have exactly the same energy, but the probabilities look different. The different states correspond to ball bouncing in x or in y direction. Tue. Dec. 1 2008 Physics 208, Lecture 25

Particle in a box What quantum state could this be? A. nx=2, ny=2 B. nx=3, ny=2 C. nx=1, ny=2 Tue. Dec. 1 2008 Physics 208, Lecture 25

Next higher energy state Ball has same bouncing motion in x and in y. Is higher energy than state with motion only in x or only in y. (nx, ny) = (2,2) Tue. Dec. 1 2008 Physics 208, Lecture 25

Three dimensions Object can have different velocity (hence wavelength) in x, y, or z directions. Need three quantum numbers to label state (nx, ny , nz) labels each quantum state (a triplet of integers) Each point in three-dimensional space has a probability associated with it. Not enough dimensions to plot probability But can plot a surface of constant probability. Tue. Dec. 1 2008 Physics 208, Lecture 25

Particle in 3D box Ground state surface of constant probability (nx, ny, nz)=(1,1,1) 2D case Tue. Dec. 1 2008 Physics 208, Lecture 25

All these states have the same energy, but different probabilities (121) (112) (211) All these states have the same energy, but different probabilities Tue. Dec. 1 2008 Physics 208, Lecture 25

(222) (221) Tue. Dec. 1 2008 Physics 208, Lecture 25

3-D particle in box: summary Three quantum numbers (nx,ny,nz) label each state nx,y,z=1, 2, 3 … (integers starting at 1) Each state has different motion in x, y, z Quantum numbers determine Momentum in each direction: e.g. Energy: Some quantum states have same energy Tue. Dec. 1 2008 Physics 208, Lecture 25