From Last time. Exam 3 results. Probability. The wavefunction. Example wavefunction. Discrete vs continuous. De Broglie wavelength

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From ast time Eam 3 results De Broglie wavelength Uncertainty principle Eam average ~ 70% Scores posted on learn@uw D C BC B AB A Wavefunction of a particle Course evaluations: Tuesday, Dec. 9 Tue.

1 From ast time Eam 3 results De Broglie wavelength Uncertainty principle Eam average ~ 70% Scores posted on learn@uw D C BC B AB A Wavefunction of a particle Course evaluations: Tuesday, Dec. 9 Tue. Dec. 008 Physics 08, ecture 5 The wavefunction Particle has a wavefunction Ψ() Ψ Ψ () d Ψ Heads Tails P(heads)= P(tails)= P( heads) + P( tails) = Very small -range " ( )d = probability to find particle in infinitesimal range d about arger -range # " ( )d = probability to find particle between and $ Entire -range % " ( )d = particle must be somewhere Tue. Dec. 008 Physics 08, ecture 5 3 /6 Tue. Dec. 008 Physics 08, ecture P()=/6 P()=/6 etc P( ) + P( ) + P( 3) + P( 4) + P( 5) + P( 6) = Discrete vs continuous 6-sided die, unequal prob /6 Infinite-sided die, all numbers between and 6 Continuous probability distribution 6 " P()d = P() oaded die / Eample wavefunction Ψ What is P(-<<-)? # Ψ = $ " ( )d = fractional area under curve - < < - # = /8 total area -> P = 0.5 What is Ψ(0)? % " ( )d = entire area under Ψ curve = = (/)(base)(height)=ψ (0)= " #( 0) =/ Tue. Dec. 008 Physics 08, ecture 5 5 Tue. Dec. 008 Physics 08, ecture 5 6

2 Wavefunctions Each quantum state has different wavefunction Wavefunction shape determined by physical characteristics of system. Quantum Particle in a bo Particle confined to a fied region of space e.g. ball in a tube- ball moves only along length Different quantum mechanical systems Pendulum (harmonic oscillator) Hydrogen atom Particle in a bo Each has differently shaped wavefunctions Classically, ball bounces back and forth in tube. A classical state of the ball. State indeed by speed, =(mass)(speed), or kinetic energy. Classical: any, energy is possible. Quantum: momenta, energy are quantized Tue. Dec. 008 Physics 08, ecture 5 7 Tue. Dec. 008 Physics 08, ecture 5 8 Classical vs Quantum Classical: particle bounces back and forth. Sometimes velocity is to left, sometimes to right Quantum version Quantum state is both velocities at the same time Superposition waves is standing wave, made equally of Wave traveling right ( p = +h/λ ) Wave traveling left ( p = - h/λ ) Determined by standing wave condition =n(λ/) : Quantum mechanics: Particle represented by wave: p = mv = h / λ Different motions: waves traveling left and right Quantum wavefunction: superposition of both at same time " = One halfwavelength Quantum wave function: superposition of both motions. p = h " = h "( ) = sin % ' # & $ ( * ) Tue. Dec. 008 Physics 08, ecture 5 9 Tue. Dec. 008 Physics 08, ecture 5 0 Different quantum states p = mv = h / λ " = /n Different speeds correspond to different λ subject to standing wave condition integer number of half-wavelengths fit in the tube. "( ) = sin % ' n # & $ ( Wavefunction: * ) Particle in bo question A particle in a bo has a mass m. Its energy is all kinetic = p /m. Just saw that in state n is np o. It s energy levels " = One halfwavelength " = Two halfwavelengths p = h " = h # p o p = h " = h = p o A. are equally spaced everywhere B. get farther apart at higher energy C. get closer together at higher energy. Tue. Dec. 008 Physics 08, ecture 5 Tue. Dec. 008 Physics 08, ecture 5

Particle in bo energy levels Zero-point motion Quantized p = h " = n h = np o Energy = kinetic ( ) E = p m = np o m Or Quantized Energy = n E o Energy n=5 n=4 owest energy is not zero Confined

3 Particle in bo energy levels Zero-point motion Quantized p = h " = n h = np o Energy = kinetic ( ) E = p m = np o m Or Quantized Energy = n E o Energy n=5 n=4 owest energy is not zero Confined quantum particle cannot be at rest Always some motion Consequency of uncertainty principle ""p > h/ Δp cannot be zero p not eactly known p cannot be eactly zero E n = n E o n=quantum number n=3 Tue. Dec. 008 Physics 08, ecture 5 3 Tue. Dec. 008 Physics 08, ecture 5 4 Question A particle is in a particular quantum state in a bo of length. The bo is now squeezed to a shorter length, /. The particle remains in the same quantum state. The energy of the particle is now A. times bigger B. times smaller C. 4 times bigger D. 4 times smaller E. unchanged Tue. Dec. 008 Physics 08, ecture 5 5 Quantum dot: particle in 3D bo Decreasing particle size Energy level spacing increases as particle size decreases. i.e E n + " E n = ( n +) h " n h 8m 8m CdSe quantum dots dispersed in heane (Bawendi group, MIT) Color from photon absorption Determined by energylevel spacing Tue. Dec. 008 Physics 08, ecture 5 6 Interpreting the wavefunction Higher energy wave functions interpretation The square magnitude of the wavefunction Ψ gives the probability of finding the particle at a particular spatial location n p E n=3 3 h 3 h 8m Wavefunction Wavefunction = (Wavefunction) h h 8m h h 8m Tue. Dec. 008 Physics 08, ecture 5 7 Tue. Dec. 008 Physics 08, ecture 5 8 3

of finding electron Quantum Corral Classically, equally likely to find particle anywhere QM - true on average for high n D. Eigler (IBM) Zeroes in the probability!

The ripples inside the ring reflect the electron quantum states of a circular ring (interference effects). Tue. Dec.

a pair (n, n y ) Ground state: (,) Motion in direction Motion in y direction Same velocity (energy), but details of motion are different. Wavefunction = (Wavefunction) One-dimensional (D) case Tue.

4 of finding electron Quantum Corral Classically, equally likely to find particle anywhere QM - true on average for high n D. Eigler (IBM) Zeroes in the probability! Purely quantum, interference effect Tue. Dec. 008 Physics 08, ecture 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. 008 Physics 08, ecture 5 0 Particle in bo again: dimensions Quantum Wave Functions (D) Ground state: same wavelength (longest) in both and y Need two quantum # s, one for -motion one for y-motion Use a pair (n, n y ) Ground state: (,) Motion in direction Motion in y direction Same velocity (energy), but details of motion are different. Wavefunction = (Wavefunction) One-dimensional (D) case Tue. Dec. 008 Physics 08, ecture 5 Tue. Dec. 008 Physics 08, ecture 5 D ecited states Particle in a bo What quantum state could this be? (n, n y ) = (,) (n, n y ) = (,) A. n =, n y = B. n =3, n y = C. n =, n y = These have eactly the same energy, but the probabilities look different. The different states correspond to ball bouncing in or in y direction. Tue. Dec. 008 Physics 08, ecture 5 3 Tue. Dec. 008 Physics 08, ecture 5 4 4

Net higher energy state Ball has same bouncing motion in and in y.

(n, n y ) = (,) Three dimensions Object can have different velocity (hence wavelength) in, y, or z

Need three quantum numbers to label state (n, n y, n z ) labels each quantum state (a triplet of

Not enough dimensions to plot probability But can plot a surface of constant probability. Tue. Dec.

008 Physics 08, ecture 5 6 Particle in 3D bo Ground state surface of constant probability (n, n y, n z

5 Net higher energy state Ball has same bouncing motion in and in y. Is higher energy than state with motion only in or only in y. (n, n y ) = (,) Three dimensions Object can have different velocity (hence wavelength) in, y, or z directions. Need three quantum numbers to label state (n, n y, n z ) 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. 008 Physics 08, ecture 5 5 Tue. Dec. 008 Physics 08, ecture 5 6 Particle in 3D bo Ground state surface of constant probability (n, n y, n z )=(,,) () () () D case All these states have the same energy, but different probabilities Tue. Dec. 008 Physics 08, ecture 5 7 Tue. Dec. 008 Physics 08, ecture 5 8 () () 3-D particle in bo: summary Three quantum numbers (n,n y,n z ) label each state n,y,z =,, 3 (integers starting at ) Each state has different motion in, y, z Quantum numbers determine p = h h = n " n Momentum in each direction: e.g. Energy: Some quantum states have same energy E = p m + p y m + p z m = E n o + n ( y + n z ) Tue. Dec. 008 Physics 08, ecture 5 9 Tue. Dec. 008 Physics 08, ecture

Hydrogen atom energies. Friday Honors lecture. Quantum Particle in a box. Classical vs Quantum. Quantum version

Hydrogen atom energies. Friday Honors lecture. Quantum Particle in a box. Classical vs Quantum. Quantum version Friday onors lecture Prof. Clint Sprott takes us on a tour of fractals. ydrogen atom energies Quantized energy levels: Each corresponds to different Orbit radius Velocity Particle wavefunction Energy Each