Wavefunction Curvature - Introduction to Relativity and Quantum Mechanics - Problem Sets, Exercises of Quantum Mechanics

Download Wavefunction Curvature - Introduction to Relativity and Quantum Mechanics - Problem Sets and more Exercises Quantum Mechanics in PDF only on Docsity! Group Problems #26 - Solutions Monday, October 31 Problem 1 Wavefunction curvature in classically forbidden region A finite potential energy function U(x) allows ψ(x), the solution of the time-independent Schrödinger equation, to penetrate the classically forbidden region. Without assum- ing any particular function for U(x), show that ψ(x) must have an inflection point at any value of x where it enters a classically forbidden region. The time-independent Schrödinger equation is given by: − ~ 2 2m d2ψ(x) dx2 + U(x)ψ(x) = Eψ(x) (1) (2) =⇒ d 2ψ(x) dx2 = 2m ~2 [U(x)− E]ψ(x), (3) where E is the total energy of a particle with mass m. The left side of Eqn. 2 represents the curvature of the wavefunction ψ(x) (remember your calculus - the first derivative is the slope, the second derivative is the curvature). An inflection point is a point where the curvature changes sign, that is where d2ψ(x)/dx2 = 0. Inspection of Eqn. 2 shows that this happens when E = U(x) and/or when ψ(x) = 0. In general, ψ(x) 6= 0, so we must have E = U(x), which is exactly the boundary between the classically allowed (E > U(x)) and forbidden (E < U(x)) regions. Problem 2 Penetration depth, I A 50 eV electron is trapped between electrostatic walls 200 eV high. How far does its wavefunction extend beyond the walls? The penetration depth, δ, is given by: δ ≡ 1 α = ~√ 2m(U0 − E) = ~c√ 2mc2(U0 − E) . (4) 1