Simple Nature - Light and Matter

Applying this to conservation of energy, we haveE = K + U[2]= p22m + U= (h/λ)22m+ UNote that both equation [1] and equation [2] have λ 2 in the denominator.We can simplify our algebra by multiplying both sides ofequation [2] by Ψ to make it look more like equation [1]:E · Ψ = (h/λ)22mΨ + U · Ψ= 1 ( ) h 2 ( ) 2π 2Ψ + U · Ψ2m 2π λ= − 1 ( ) h 2d 2 Ψ2m 2π dx 2 + U · ΨFurther simplification is achieved by using the symbol (h with aslash through it, read “h-bar”) as an abbreviation for h/2π. We thenhave the important result known as the Schrödinger equation:E · Ψ = − 2 d 2 Ψ2m dx 2 + U · Ψ(Actually this is a simplified version of the Schrödinger equation,applying only to standing waves in one dimension.) Physically itis a statement of conservation of energy. The total energy E mustbe constant, so the equation tells us that a change in interactionenergy U must be accompanied by a change in the curvature ofthe wavefunction. This change in curvature relates to a changein wavelength, which corresponds to a change in momentum andkinetic energy.self-check GConsidering the assumptions that were made in deriving the Schrödingerequation, would it be correct to apply it to a photon? To an electron movingat relativistic speeds? ⊲ Answer, p.929Usually we know right off the bat how U depends on x, so thebasic mathematical problem of quantum physics is to find a functionΨ(x) that satisfies the Schrödinger equation for a given interactionenergyfunction U(x). An equation, such as the Schrödinger equation,that specifies a relationship between a function and its derivativesis known as a differential equation.868 Chapter 13 Quantum Physics