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Section 7.3: Eigenfunction Shape for Spatially-varying Potentials
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In one-dimensional position space, the time-independent Schrödinger equation is:
[−(ħ2/2m)d2/dx2 + V(x)] ψ(x) = E ψ(x), (7.7)
where p2/2m = (−ħ2/2m) d2/dx2 because p = −iħ(d/dx). To determine the energy eigenfunction we must solve the time-independent Schrödinger equation for ψ(x) given a V(x). Restart .
In the animation a particle is confined to a one-dimensional potential energy well V(x), proportional to x2, which describes a harmonic oscillator potential.5 You may change the energy value by varying the sliders (the top slider changes the energy value by 1 E0 while the bottom slider changes the energy value by 0.1 E0 and then examine the solutions to the one-dimensional time-independent Schrödinger equation, Eq. (7.7), for this system. In the animation, ħ = 2m = 1.
The algorithm used here to calculate energy eigenfunctions is called the shooting method. The shooting method calculates the energy eigenfunction by numerically solving the time-independent Schrödinger equation. The solution for the energy eigenfunction starts with ψ(xleft) = 0 to approximate ψ(x = −∞) = 0, and then numerically solves the time-independent Schrödinger equation, calculating the energy eigenfunction from left to right. Note that all energy values solve the time-independent Schrödinger equation, but only some of these are referred to as energy eigenvalues (eigen is German for proper or characteristic) which yield valid (proper) bound-state energy eigenfunctions. Recall that in order to have a probabilistic interpretation for a bound-state energy eigenfunction the function must be finite everywhere and must go to zero at ±∞.
Use both sliders to change the energy to the eigenvalue of 19 E0. As you change the energy, what do you notice about the energy eigenfunction? The curviness6 changes as the energy changes. As the energy value increases, the curviness of the energy eigenfunction increases. As a consequence, the number of x-axis crossings increases. The ground state always has zero crossings, the first-excited state has one crossing, the second-excited state has two crossings, etc. Now move the lower slider so that the energy value is 18.9 E0. Now increase the energy value to 19.1 E0. What do you notice about what happens to the solution of the time-independent Schrödinger equation? If the energy value is either too small or too large, the energy eigenfunction either undershoots or overshoots (hence the name shooting method) the boundary condition of ψ(x → ∞) = 0 which is approximated by ψ(xright) = 0. When the energy value is too small the curviness is too small to allow the energy eigenfunction to go to zero at x = xright. Likewise, when the energy value is too big, the curviness is too large to allow the energy eigenfunction to go to zero at x = xright.
Also note that for the energy eigenvalue of 19 E0, both the amplitude and curviness of the energy eigenfunction change as a function of x. This is because the potential energy function, V(x) is proportional to x2, changes with position. As a consequence, E − V(x) changes as well. In classical mechanics, E − V(x) can be easily associated with the kinetic energy, T(x). In quantum mechanics we cannot make this association directly, as it would imply knowing both the position and the momentum at the same time which violates the uncertainty principle. Instead, we calculate expectation or average values of quantities like the kinetic energy. We may however talk about the average kinetic energy in a finite interval, which we refer to qualitatively as curviness.7 Where E − V(x) is larger, there is a larger curviness in the energy eigenfunction. Also note that for the energy eigenvalue of 19 E0, you are less likely to find the particle near the origin (since the amplitude of the energy eigenfunction is related the probability density). For regions where E − V(x) is larger, there is a smaller probability of finding a particle in that region as compared to another region where E − V(x) is smaller. This determination agrees with the time classical spent arguments8 for classical probability distributions which was discussed in Section 6.2.
5This problem is discussed in detail in Chapter 12.
6We use the term curviness much like Robinett  uses the term wiggliness to qualitatively describe the oscillatory nature of the wave function (energy eigenfunction). Qualitatively, the curviness can be thought of as the number of oscillations per length of the wave function (energy eigenfunction).
7The average kinetic energy can be defined in an interval, ε, centered on a point, xc, as
−(ħ2/2m) ∫xc − ε/2xc + ε/2 ψ*(x)[d2ψ(x)/dx2] dx / ∫xc − ε/2xc + ε/2 ψ*(x) ψ(x) dx , (7.6)
where upon letting ε → ∞ we recover the usual expectation value of the kinetic energy. In quantum mechanics E − V(x) can be negative, and hence the average kinetic energy in a finite interval can be negative. The expectation value of kinetic energy, however, will always be positive.
8See for example: R. W. Robinett, Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples, Oxford, New York (1997).