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

real part of the function =

imaginary part of the function =

Complex functions are absolutely necessary to describe quantum-mechanical phenomena. Quantum-mechanical time evolution is governed by the Schrödinger equation1 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, fRe(x,t) and fIm(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.

f(x,t) =

Besides entering [fRe(x,t), fIm(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)eiθ(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) = ei(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.

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

Try some other complex functions for practice.

1By the Schrödinger equation we mean what is often called the time-dependent Schrödinger equation since this is the Schrödinger equation.
2For example, the complex function z(x) = eix = cos(x) + i sin(x) and z(x) = 1/(x + i) = x/(x2 + 1) −i/(x2 + 1).

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