## Section 1.6: Exploring the Input of Data: Complex Expressions

Please wait for the animation to completely load.

Complex functions are absolutely necessary to describe quantum-mechanical phenomena. Quantum-mechanical time evolution is governed by the Schrödinger equation^{1} which is itself complex, thus yielding complex solutions. In many exercises you will be expected to enter a formula to control the animation **(position and time are given in arbitrary units)**. Restart. In this Exploration, you are to enter the real (the blue curve on the graph) and imaginary (the pink curve on the graph) parts of a function, *f*_{Re}(*x,t*) and *f*_{Im}(*x,t*), for *t* = 0. Once you have done this, the time evolution of the function is governed by the form of the function you have chosen and the "Resume" and "Pause" buttons.

Please wait for the animation to completely load.

Besides entering [*f*_{Re}(*x,t*), *f*_{Im}(*x,t*)], the real and imaginary components of the function, you will also be asked to enter the function in its magnitude and phase form, *f*(*x,t*) = *A*(*x,t*)e^{iθ(x,t)} where *A* and θ are real functions. The default function for this Exploration is [cos(*x − t*), sin(*x − t*)] or ψ(*x,t*) = e^{i(x − t)} which is called a plane wave.^{2} In the text box you can enter a complex function in magnitude and phase form. Try it for the plane wave, exp(*i**(*x−t*)), to see if you get the same picture as above.

Input the following functions for the real and imaginary parts of f(*x,t*) in the first animation, then determine what amplitude and phase form you have to enter into the text box of the second animation to mimic the results you saw in the first animation.

**real**= exp(-0.5*(x+5)*(x+5))*cos(pi*x) |**imaginary**= exp(-0.5*(x+5)*(x+5))*sin(pi*x)**real**= sin(2*pi*x)*cos(4*t) |**imaginary**= sin(2*pi*x)*sin(4*t)

Try some other complex functions for practice.

^{1}By the Schrödinger equation we mean what is often called the **time-dependent** Schrödinger equation since this is ** the** Schrödinger equation.

^{2}For example, the complex function

*z*(

*x*) = e

*= cos(*

^{ix}*x*) +

*i*sin(

*x*) and

*z*(

*x*) = 1/(

*x*+

*i*) =

*x*/(

*x*

^{2 }+ 1) −

*i*/(

*x*

^{2}+ 1).

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