1Mon. Mar. 27, 2006 Phy107 Lecture 251From Last Time…• Particle can exist in different quantum states, having— Different energy— Different momentum— Different wavelength• The quantum wavefunctiondescribes wave nature of particle.• Square of the wavefunctiongives probability of finding particle.• Zero’s in probabilityarise from interference of the particle wave with itself.No office hours TuesdayGuest lecturer Wed: Entanglement & Quantum ComputingMon. Mar. 27, 2006 Phy107 Lecture 252Particle in a boxLowest energystateNext higherenergy state3rd energystateWavefunction ProbabilityLMon. Mar. 27, 2006 Phy107 Lecture 253Wavefunction of pendulumn=2n=3n=1ground stateHere are quantumwavefunctions of apendulum. Which hasthe lowest energy?Mon. Mar. 27, 2006 Phy107 Lecture 254Classical vs quantumLow classical amplitude, lowenergyHigher classical amplitude,higher energyMon. Mar. 27, 2006 Phy107 Lecture 255Probability density of oscillatorMoves fast here,low prob of finding in a‘blind’ measurementMoves slow here,high prob offindingClassical probMon. Mar. 27, 2006 Phy107 Lecture 256Wavefunctions in two dimensions• Physical objects often can move in more thanone direction (not just one-dimensional)• Could be moving at one speed in x-direction,another speed in y-direction.• From deBroglie relation, wavelength related tomomentum in that direction• So wavefunction could have differentwavelengths in different directions.! "=hp2Mon. Mar. 27, 2006 Phy107 Lecture 257Two-dimensional (2D) particle in boxGround state: samewavelength (longest) inboth x and yNeed two quantum #’s,one for x-motionone for y-motionUse a pair (nx, ny)Ground state: (1,1)Probability(2D)Wavefunction Probability = (Wavefunction)2One-dimensional (1D) caseMon. Mar. 27, 2006 Phy107 Lecture 2582D excited states(nx, ny) = (2,1)(nx, ny) = (1,2)These have exactly the same energy, but theprobabilities look different.The different states correspond to ball bouncingin x or in y direction.Mon. Mar. 27, 2006 Phy107 Lecture 259The classical versionSame velocity (energy),but details of motion are different.Motion in x directionMotion in y directionMon. Mar. 27, 2006 Phy107 Lecture 2510Particle in a boxWhat quantum state could this be?A. nx=2, ny=2B. nx=3, ny=2C. nx=1, ny=2Mon. Mar. 27, 2006 Phy107 Lecture 2511Next higher energy state• The ball now has same bouncing motion inboth x and in y.• This is higher energy that having motion onlyin x or only in y.(nx, ny) = (2,2)Mon. Mar. 27, 2006 Phy107 Lecture 2512Three dimensions• Object can have different velocity (hencewavelength) 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 aprobability associated with it.• Not enough dimensions to plot probability• But can plot a surface of constant probability.3Mon. Mar. 27, 2006 Phy107 Lecture 25133D particle in box• Ground statesurface of constantprobability• (nx, ny, nz)=(1,1,1)2D caseMon. Mar. 27, 2006 Phy107 Lecture 2514(211)(121)(112)All these states have the sameenergy, but different probabilitiesMon. Mar. 27, 2006 Phy107 Lecture 2515(221)(222)Mon. Mar. 27, 2006 Phy107 Lecture 2516Hydrogen atom• Hydrogen a little different, in that it hasspherical symmetry• Not square like particle in a box.• Still need three quantum numbers, but theyrepresent ‘spherical’ things like– Radial distance from nucleus– Azimuthal angle around nucleus– Polar angle around nucleus• Quantum numbers are integers (n, l, ml)Mon. Mar. 27, 2006 Phy107 Lecture 2517Hydrogen atom:Lowest energy (ground) state• Lowest energy state issame in all directions.• Surface of constantprobability is surface ofa sphere. ! n = 1, l = 0, ml= 0Mon. Mar. 27, 2006 Phy107 Lecture 2518n=2: next highest energy ! n = 2, l = 1, ml= 0 ! n = 2, l = 1, ml= ±1 ! n = 2, l = 0, ml= 02s-state2p-state2p-stateSame energy, but different probabilities4Mon. Mar. 27, 2006 Phy107 Lecture 2519 ! n = 3, l =1, ml= 0 ! n = 3, l =1, ml= ±13s-state3p-state3p-state ! n = 3, l = 0, ml= 0n=3: two s-states, six p-states and…Mon. Mar. 27, 2006 Phy107 Lecture 2520…ten d-states ! n = 3, l = 2, ml= 0 ! n = 3, l = 2, ml= ±1 ! n = 3, l = 2, ml= ±23d-state3d-state3d-stateMon. Mar. 27, 2006 Phy107 Lecture 2521Back to the particle in a box• Here is the probability of finding the particlealong the length of the box.• Can we answer the question: Where is the particle?WavefunctionProbability = (Wavefunction)2Mon. Mar. 27, 2006 Phy107 Lecture 2522Where is the particle?• Can say that the particle is inside the box,(since the probability is zero outside the box),but that’s about it.• The wavefunction extends throughout the box,so particle can be found anywhere inside.• Can’t say exactly where the particle is,but I can tell you how likely you are to find at aparticular location.Mon. Mar. 27, 2006 Phy107 Lecture 2523How fast is it moving?• Box is stationary, so average speed is zero.• But remember the classical version• Particle bounces back and forth.– On average, velocity is zero.– But not instantaneously– Sometimes velocity is to left, sometimes to rightLMon. Mar. 27, 2006 Phy107 Lecture 2524Quantum momentum• Quantum version is similar. Both contributions• Ground state is a standing wave, made equally of– Wave traveling right ( positive momentum +h/λ )– Wave traveling left ( negative momentum - h/λ )! "= 2LOne half-wavelength! p =h"=h2LmomentumL5Mon. Mar. 27, 2006 Phy107 Lecture 2525Particle in a boxWhat is the uncertainty of the momentum in theground state?A. ZeroB. h / 2LC. h / L! "= 2LOne half-wavelength! p =h"=h2LmomentumLMon. Mar. 27, 2006 Phy107 Lecture 2526Uncertainty in Quantum MechanicsPosition uncertainty = LMomentum ranges from ! "h# to +h# : range = 2h#=hL(Since λ=2L)Reducing the box size reduces position uncertainty,but the momentum uncertainty goes up!L! "= 2LOne half-wavelengthThe product is constant:(position uncertainty)x(momentum uncertainty) ~ hMon. Mar. 27, 2006 Phy107 Lecture 2527Heisenberg Uncertainty Principle• Using– Δx = position uncertainty– Δp = momentum uncertainty• Heisenberg showed that the product ( Δ x ) • ( Δp ) is always greater than ( h / 4π )Planck’sconstantIn this case we found:(position uncertainty)x(momentum uncertainty) ~ hMon. Mar. 27, 2006 Phy107 Lecture 2528Zero-point energy• In all cases, wave represents
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