Simple Nature - Light and Matter

l / A typical wavefunction of an electron in an atom (heavy curve)and the osculating sine wave (dashed curve) that matches its curvatureat point P.to energy, so it can’t be infinite. A physically realistic wavefunctionmust always “tail off” gradually, k/2. In classical physics, a particlecan never enter a region in which its interaction energy U would begreater than the amount of energy it has available. But in quantumphysics the wavefunction will always have a tail that reaches intothe classically forbidden region. If it was not for this effect, calledtunneling, the fusion reactions that power the sun would not occurdue to the high electrical energy nuclei need in order to get closetogether! Tunneling is discussed in more detail in the followingsubsection.13.3.6 The Schrödinger equationIn subsection 13.3.5 we were able to apply conservation of energyto an electron’s wavefunction, but only by using the clumsy graphicaltechnique of osculating sine waves as a measure of the wave’scurvature. You have learned a more convenient measure of curvaturein calculus: the second derivative. To relate the two approaches, wetake the second derivative of a sine wave:d 2dx 2 sin(2πx/λ) = d ( )2π 2πxcosdx λ λ( ) 2π 2= − sin 2πxλ λTaking the second derivative gives us back the same function,but with a minus sign and a constant out in front that is relatedto the wavelength. We can thus relate the second derivative to theosculating wavelength:[1]d 2 ( )Ψ 2π 2dx 2 = − ΨλThis could be solved for λ in terms of Ψ, but it will turn outbelow to be more convenient to leave it in this form.Section 13.3 Matter As a Wave 867