A harmonic oscillator has its minimum possible energy. What is the probability of finding...

A harmonic oscillator has its minimum possible energy. What is the probability of finding it in the classically forbidden region? (Note: At some point, a computer or calculator able to do numerical integration will be needed.)

Computational Exercises

Guidelines for application of equation (5-35). With length, time, and mass at our disposal, we choose our units so that the particle mass m and the value of h are both I. Now, let x be a point Ax to the right of the origin, Equation (5-35) becomes

After choosing initial values ψ(0) and ψ(Δx:), and assuming U is known and a value for Δx selected, we need only pick an E, and ψ(2Δx) can be found, Thereafter, taking x to be a point 2Δx to the right of the origin, we have

ψ(3Δx) = 2ψ (2Δx) - ψ(Δx) + 2(Δx)2(U(2Δx) - E)ψ(2Δx)

and the process can be repeated indefinitely. Choosing ψ(0) and ψ(Δx) is simplified if U(x) is symmetric about x = 0, for this implies a symmetric probability density, which in turn requires that ψ(x) be either an even or odd function of x. An odd function is 0 at the origin, so ψ(0) must be 0. Since the slope is, in general, nonzero, ψ(Δx) must be nonzero, and we define it to be l (affecting only the vertical scale of ψ(0) and ψ(Δx) must be of equal value, which we can define to be 1.