Quantum Theory - Particle Physics Group

CHAPTER 2. WAVE MECHANICS AND THE SCHRÖDINGER EQUATION 22u(x)√2m(E−V ) 2K =κ =e ±iKxe κx√2m(V −E) 2e −κxVEFigure 2.2: Bound state solutions for the stationary Schrödinger equation.with the Einstein relation E = ω. The stationary solutions ψ(x,t) to the Schrödinger equationthus have the formψ(⃗x,t) = u(⃗x)e −iωt . (2.43)Their time dependence is a pure phase so that probability densities are time independent.In order to get an idea of the form of the wave function u(x) we consider a slowly varyingand asymptotically constant attractive potential as shown in figure 2.2. Since the stationarySchrödinger equation in one dimension− 22m u′′ (x) = (E − V (x))u(x) (2.44)is a second order differential equation it has two linearly independent solutions, which for aslowly varying V (x) are (locally) approximately exponential functions⎧√⎨Ae iKx + Be −iKx = A ′ sin(Kx) + B ′ 2m(E−V )cos(Kx), K = for E > V,u(x) ≈√2⎩Ce κx + De −κx = C ′ sinh(κx) + D ′ 2m(V −E)cosh (κx), κ = for E < V. 2(2.45)In the classically allowed realm, where the energy E of the electron is larger then the potential,the solution is oscillatory, whereas in the classically forbidden realm of E < V (x) we find asuperposition of exponential growth and of exponential decay. Normalizability of the solutionrequires that the coefficient C of exponential growth for x → ∞ and the coefficient D ofexponential decay for x → −∞ vanish. If we require normalizability for negative x and increasethe energy, then the wave function will oscillate with smaller wavelength in the classicallyallowed domain, leading to a component of exponential growth of u(x) for x → ∞, until wereach the next energy level for which a normalizable solution exists. We thus find a sequenceof wave functions u n (x) with energy eigenvalues E 1 < E 2 < ..., where u n (x) has n − 1 nodes(zeros). The normalizable eigenfunctions u n are the wave functions of bound states with adiscrete spectrum of energy levels E n .It is clear that bound states should exist only for V min < E < V max . The lower boundfollows because otherwise the wave function is convex, and hence cannot be normalizable.