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pde waves: string, quantum packet, and e&m 499Figure 18.10 The probability density as a function of position and time for an electronincident upon and passing through a slit.18.9 Unit III. E&M Waves via Finite-DifferenceTime Domain ⊙Problem: You are given a 1-D resonant cavity with perfectly conducting walls. Aninitial electric pulse with the shape[ ( ) ] 21 40 − tE x (z =3)=expcos(2πft), 0 ≤ t ≤ T, (18.56)2 12is placed in this cavity. Determine the motion of this pulse at all later times forT =10 −8 s and f = 700 MHz.Simulations of electromagnetic waves are of tremendous practical importance. Indeed,the fields of nanotechnology and spintronics rely heavily upon such simulations. The basictechniques used to solve for electromagnetic waves are essentially the same as those we usedin Units I and II for string and quantum waves: Set up a grid in space and time and thenstep the initial solution forward in time one step at a time. For E&M simulations, thistechnique is known as the finite difference time domain (FDTD) method. What is newfor E&M waves is that they are vector fields, with the variations of one generating the other,so that the components of E and B are coupled to each other. Our treatment of FDTD doesnot do justice to the wealth of physics that can occur, and we recommend [Sull 00] for amore complete treatment and [Ward 04] (and their Web site) for modern applications.18.10 Maxwell’s EquationsThe description of electromagnetic (EM) waves via Maxwell’s equations is givenin many textbooks. For propagation in just one dimension (z) and for free spacewith no sinks or sources, four coupled PDEs result:⃗∇·E =0 ⇒⃗∇·H =0 ⇒∂E x(z,t)∂x∂H y(z,t)∂y=0 (18.57)=0, (18.58)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 499