MSE 494/598 Materials for Energy Production and Storage Homework Assignment/Project # 3 Due: Feb. 16th 1. Two equally intense beams of 5 eV electrons travel at 45o with respect to each other meet creating an interference pattern. Assume the electrons can be described as free electrons.

MSE 494/598 Materials for Energy Production and Storage Homework Assignment/Project # 3 Due: Feb. 16th 1. Two equally intense beams of 5 eV electrons travel at 45o with respect to each other meet creating an interference pattern. Assume the electrons can be described as free electrons. a) Determine the numerical values of the wavelengths, wavenumbers, frequencies and momenta of the two beams before they intersect. [4] b) Using complex exponential notation, write down algebraic expressions for the wavefunctions of these two beams before they intersect (Hint: choose an arbitrary origin and remember that wavevector is a vector). [4]. c) Determine the wavefunctions of the new wave formed when the electrons interact during crossing. Show that the new wavefunction can be decomposed into two orthogonal components representing standing waves and travelling waves. (Just work with algebraic expressions – don’t bother putting in the numbers). [10] d) How would the probability of finding an electron at a point space vary in this interference pattern? [4] The next question requires the use of the applets that I use in class. These applets can be found at: http://falstad.com/mathphysics.html You should spend a little time playing with these applets to develop some familiarity with them. The Ripple Tank (2D Wave) one is good to demonstrate many of the fundamental properties of waves and interference. I will ask you to hand in printouts of some of the applet results. The easiest way to accomplish this is to use the Print Screen option on your computer. Try the following. When you have an image you want to record, press the printscreen button on your computer (usually labeled PrtSc). In Powerpoint, paste this object into the display. You can use the Powerpoint cropping tool to (Select View Toolbar Drawing. Hold the mouse over the Drawing tools to identify the tool labeled Crop. Now select the object that you have just pasted into Powerpoint, select cropping tool and you will see vertical and horizontal bars at the side of the pasted object. You can move these bars to crop the image. To change the size of the image, right click and select format picture size (make sure you click the Lockaspect ratio button so that you do not distort the object). 2. Use the Ripple Tank (2D Wave) simulator to qualitatively illustrate the interference situation that you described algebraically in question 1c. Use the intersecting planes option to set up two plane waves travelling at approximately 45o with respect to each other. Make printouts of the interference pattern that is established when the two waves cross. On your printout draw an arrow on the pattern illustrating the direction of the travelling wave components of the interference patterns. Along which direction do you see the standing wave component of the interference patterns (mark this on your printout). Can you relate the animation to the algebra of question 1? Both are equivalent. [20] 3. a) A simple model for the electronic structure of metals assumes that the valence electrons can be considered to be free electrons propagating in opposite directions. By considering only plane waves propagating along all symmetrically equivalent <100> directions in an infinite 3D cubic crystal, determine the wavefunction for the valence electrons and discuss the form of the resulting wavefunction. [10] b) At what points in 3D space is the probability of finding an electron maximum? [4] 4. The standing modes on a 2D circular membrane have features in common with 3D orbitals around the hydrogen atom. Identify and run the Circular Membrane Wave Applet and the Hydrogen Atom applet (this is best accomplished by having two separate browser windows open at the same time). a) With the Circular Membrane applet identify the non-circularly symmetric single mode that shows 3-fold symmetry (i.e. it is invariant under rotations of 120o ). (Notice that double clicking on the square array will excite a single mode whereas single clicking will allow you to excite many modes simultaneously – this is what happens when you strike the surface of a drum. Each mode is an eigenfunction of the system). Display the mode in 3D and 2D representation and print it out. [5] b) Identify the corresponding hydrogen wavefunction that is most closely related to the drum mode of part a). Print it out and label it with the corresponding quantum numbers. [5] Comment- Notice that you get a standing wave (a stationary state) if you only excite one mode in the system. If you excite more than one mode the wave moves around on the surface of the drum. This is not a standing wave. This state would be described with a linear combination of eigenfunctions. In general, the quantum state of an electron will be a linear combination of eigenstates. You can now see that the stationary states corresponding to hydrogen orbitals are quite special since they involve one eigenfunction.

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