Lecture 25 Exam 2 Details Reminders Reading for this week TZD through 7 Homework 8 due today Homework 9 due next Wednesday is posted Exam 2 on Friday October 30 in class More to say on the exam on next slide Today s topic Wave Packets Heisenberg Uncertainty Principle Wave Equation Exam 2 Details On the class web page there is a posted practice exam The last page is the formula sheet for our exam as well The second exam 2 will take place in class on Friday October 30 2009 Please make sure to attend class on time so everyone will have the full 50 minutes to take the exam As last time there will be assigned seating rows The exam will cover all the material on atoms waves photons quantum mechanics as presented in the assigned textbook reading lectures and homeworks 4 5 6 7 8 This includes material in Chapters 3 through Chapter 7 section 7 5 The exam is closed book and closed notes Calculators will be allowed but no use of stored equations or algebraic solvers may be used A formula sheet will be provided with the exam A preview of the formula sheet is included at the end of the practice exam linked below Plane Waves vs Wave Packets Plane wave x t Aei kx t Wave packet x t Anei kn x nt n 0 Matter waves are more like wave packets than plane waves Mathematically we obtain wave packets by adding up plane waves Method of adding up sine waves to obtain an arbitrary function like a wave packet is called Fourier Analysis Excellent PHeT Simulation The blob in Quantum Wave Interference is a 2D wave packet In this simulation intensity is represented by brightness Electron Photon 1 Clicker question Plane Waves vs Wave Packets Plane wave Plane wave x t Aei kx t x t Aei kx t This wave represents a single k and Therefore energy momentum and wavelength are well defined Wave packet x t A ei kn x nt n 0 Amplitude is the same over all space so position is undefined n For which type of wave is the momentum and position most well defined A p is well defined for plane wave x is well defined for wave packet B p is well defined for wave packet x is well defined for plane wave C p is well defined for one but x is equally well defined for both D p is equally well defined for both but x is well defined for one E Both p and x are well defined for both Heisenberg Uncertainty Principle One version of the Heisenberg uncertainty principle is written as Wave packet x t Anei kn x nt n 0 This wave is composed of many different k and waves Thus it is composed of many different energies momenta and wavelengths and so these quantities are not well defined The amplitude is non zero in a small region of space so the position is constrained to be in that region Well defined position Heisenberg Uncertainty Principle x p h 2 When we write x or p we are really referring to the uncertainty on the measurement Sometimes this is described as the spread of values x small p only one wavelength What does this uncertainty principle mean The position and momentum cannot both be determined precisely The more precisely one is determined the less precisely the other is determined x medium p wave packet made of several waves This is a fundamental limitation which has nothing to do with the actual equipment used to measure x or p This is a pretty weird concept in the particle view but makes a lot more sense in the wave view Once at the end of a colloquium I heard Debye saying something like Schr dinger you are not working right now on very important problems why don t you tell us some time about that thesis of deBroglie which seems to have attracted some attention So in one of the next colloquia Schr dinger gave a beautifully clear account of how deBroglie associated a wave with a particle and how he could obtain the quantization rules by demanding that an integer number of waves should be fitted along a stationary orbit When he had finished Debye casually remarked that he thought this way of talking was rather childish To deal properly with waves one had to have a wave equation Felix Bloch x large p wave packet made of lots of waves Clarification of de Broglie Relations I stated that the de Broglie relation is p h hk Originally came from an analysis of massless photons but also works for massive particles like electrons neutrons and atoms In fact there is another relation which is derived from the photon results E hf h Note the momentum relation deals with the space part of a wave wavelength and wave number while the energy relation deals with the time part of the wave frequency For light the space and time quantities are related by c f For massive particles vwave f but vwave vparticle so it is not very useful in practice My advice avoid using velocity Stick with E p k T f 2 t E h 2 Heisenberg Uncertainty Principle Clicker question Which of the two particles A or B can you locate more precisely t small E only one period A A B B C Same precision for A and B t medium E wave packet made of several waves t large E wave packet made of lots of waves Where we go from here Heisenberg Uncertainty Principle x p h 2 There are two Heisenberg uncertainty relations We are going to spend some time thinking about classical waves t E h 2 Waves on a string like on a guitar What does this uncertainty principle mean Electromagnetic waves The wave nature of things prevents a precise determination of both momentum and position or of both energy and time Classical waves obey the wave equation This is a fundamental limitation which has nothing to do with the actual equipment used to measure things Another way of seeing why this makes sense is Heisenberg s microscope Microscopes are limited to resolutions wavelength of light Smaller wavelengths allow a better measurement of x but the photons have larger momentum giving larger kicks to the particle making the momentum more uncertain Electromagnetic waves x 2E 1 2E x 2 c 2 t 2 2 y 1 2 y x 2 v 2 t 2 c speed of light v speed of wave Solutions E x t Magnitude is non spatial Strength of Electric field Finally we will derive the time independent Schr dinger equation 2m x2 V ih t 2 2 h V E 2m x 2 Which of the following is a true statement about standing waves Vibrations on a string y x Once we understand these waves we will go back to matter waves which obey a different wave equation called the time dependent Schr dinger equation h 2 2 Clicker Question Example Wave Equations You Have Seen E 2 y …