THE SCHRODINGER EQUATION
The Schrodinger equation, developed by the German physicist Erwin Schrodinger (1887-1961), is the basic relationship for determining wave functions and energy levels. We will apply it to several systems, including the particle in a box (which we’ve
already discussed), the harmonic oscillator, and the hydrogen atom. We won’t pretend that we can derive this equation from known principles. We can’t; it is a new principle.
But we can show how it is related to the de Broglie equations, and we can make it seem plausible. The ultimate test of the Schrodinger equation is to compare its predictions with experimental observations. As we will see, it passes such tests with flying colors.We’ll begin with a one-dimensional problem, a particle moving freely along the x-axis between two walls like a particle in a box. From Eqs. (42-1) and (42-2) the wave functions for a particle in a box have the general form By using the de Broglie relation,p = hlA, the energy E of the corresponding level can be expressed as notice that the second derivative ofEq. (42-11) is-klA sin 10:, that is, the original function multiplied by _kl. So taking the second derivative of ‘If and then multiplying.
We invite you to verify that Eq. (42-11) satisfies this equation, no matter what the value of k, if E is given hy Eq. (42-12).
Equation (42-13) is the simplest form of the Schrodlnger equation. We obtained it in a somewhat contrived way, but we can see that it is consistent with what we know about wave properties of particles and about the wave functions for a particle in a box.
We can avoid writing a lot of factors of 21r in later discussions by using the abbreviation tr (pronounced “h-bar”) for Planck’s constant divided by 21r: So far, we’ve been talking a ut afree particle, a particle that moves along the x-axis .th no force acting on it (exce for the forces that act at the instant a particle in a box . es a wall). Now suppose the article is acted on by a force with an x-component F. depends on x (but not on tim ). We’ll assume the force is conservative so that there a corresponding potential ener y U. In guessing how to generalize the Schrodinger tion for this more general p oblem, we start with the classical energy relation for
.on under a conservative for e: K + U = E, or equation, ‘” and U are, in general, functions of x, while E is a constant for a given
level. How do we know that this equation is correct? Because it works. ions made using this equation agree with experimental results and thus confirm . is indeed the correct way to include conservative, time-independent interactions Schrodinger equation. We’ll apply this equation to several problems in the fol sections. -orking with this more general form of the Schrodinger equation, we will insist wave functions be normalized, that is, J’:11fIl2 dx = I. Note that if ‘” is a solution Schriidinger equation, C”, is also a solution, where C is any constant. We can lm. dlOOsethe value of C to satisfy the normalization requirement. we state that the function ‘” must be continuous. Its first derivative, dvldx. ~30 be continuous except where the potential energy becomes infinite (as at the ‘::::!~,aflhebox). These requirements ensure that the wave function is a mathematically solution to the Schrodinger equation. They are analogous to the require the wave functions for a vibrating string, and their derivatives, should be continuous at points where the linear mass density or tension in the string changes, An exception occurs at the end points of the string, where only the function itself, not its derivative, needs to be continuous.