Minimal amount of bending of the wave function necessary for it to be zero at both walls but nonzero in between - thisĬorresponds to half a period of a sine or cosine (depending on the choice of Rather, the lowest energy state must have the
Since it cannot have a discontinuity, and must be zero just inside theĪn immediate consequence is that the lowest state cannotĬonstant ψ ( x ). The wave function ψ ( x ) necessarily goes to zero right at the walls, The wave function has the form ψ ( x ) = A sin ( k x + δ ). Well, that is, between infinitely high walls a distance L apart.įor that case, the potential between the walls is identically zero so Wave functions and energies for a particle trapped in an infinitely deep square In an earlier lecture, we considered in some detail the allowed One Dimensional Infinite Depth Square Well This wave function is only relevant for positive x , and the coefficients A, B are functions of theĮnergy - for certain energies it turns out that A = 0 , and the wave function converges. Shall see below, there are situations with spatially varying potentials where Of course, this wave function will diverge in Ψ ( x ) tends to zero, the curvature tends to zero,įor a constant potential V 0 > E , the wave function is ψ ( x ) = A e α x + B e − α x, Will the curvature be just right to bring the wave function to zero as x goes to infinity. Only with exactly the right initial conditions Means that ψ ( x ) tends to diverge to infinity. Potential V ( x ) = V 0 E , the curvature is always away from the axis. The simplest example is that of a constant In bothĬases, ψ ( x ) is always curving towards the x -axis - so, for E > V ( x ), ψ ( x ) has a kind of stability: its curvature isĪlways bringing it back towards the axis, and so generating oscillations. This means that if E > V ( x ) , for ψ ( x ) positive ψ ( x ) is curving negatively, for ψ ( x ) negative ψ ( x ) is curving positively.
Rate of change of slope, is the curvature – so the curvature of the function is Curvature of Wave Functionsĭ 2 ψ ( x ) d x 2 = 2 m ( V ( x ) − E ) ℏ 2 ψ ( x )Ĭan be interpreted by saying that the left-hand side, the
\[\nabla^2 \Psi(x,y,z,t) -\dfrac argues that the time-dependent (i.e.Previous index next PDF Schrödinger’s Equation in 1-D: Some Examples In classical electromagnetic theory, it follows from Maxwell's equations that each component of the electric and magnetic fields in vacuum is a solution of the 3-D wave equation for electronmagnetic waves: Nonetheless, we will attempt a heuristic argument to make the result at least plausible. Any rule that might be capable of predicting the allowed energies of a quantum system must also account for the wave-particle duality and implicitly include a wave-like description for particles. Moreover, it relies heavily on classical ideas, clumsily grafting quantization onto an essentially classical picture, and therefore, provides no real insights into the true quantum nature of the atom. While the Bohr model is able to predict the allowed energies of any single-electron atom or cation, it by no means, a general approach. The Schrödinger Equation: A Better Approach There is no rigorous derivation of Schrödinger’s equation from previously established theory, but it can be made very plausible by thinking about the connection between light waves and photons, and construction an analogous structure for de Broglie’s waves and electrons (and, later, other particles). This was a direct challenge to Schrödinger, who spent some weeks in the Swiss mountains working on the problem and constructing his equation.
Schrödinger gave a polished presentation, but at the end Debye remarked that he considered the whole theory rather childish: why should a wave confine itself to a circle in space? It wasn’t as if the circle was a waving circular string, real waves in space diffracted and diffused, in fact they obeyed three-dimensional wave equations, and that was what was needed. Shortly after it was published in the fall of 1925 Pieter Debye, Professor of Theoretical Physics at Zurich and Einstein's successor, suggested to Erwin Schrödinger that he give a seminar on de Broglie’s work.