Please wait for the animation to completely load.
In one-dimensional position space, the time-independent Schrödinger equation is:
[−(ħ2/2m)d2/dx2 + V(x)] ψ(x) = E ψ(x), (7.5)
where p2/2m = (−ħ2/2m)
d2/dx2
because p = −iħ(d/dx).
This equation describes the total energy of the system as the sum of the kinetic energy and the potential
energy. To determine the wave function 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. (7.5), for this system. In the animation, ħ = 2m = 1.
The algorithm used to calculate the wave function is called the shooting
method. The shooting method calculates the wave function by numerically
solving the time-independent Schrödinger equation. The solution for the
wave function starts with ψ(xleft) = 0 to approximate ψ(x = −∞) = 0, and then numerically solves the time-independent
Schrödinger equation, calculating the wave function 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 wave functions. Recall that in
order to have a probabilistic interpretation for a bound-state wave function, the wave 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 wave function? The curviness6 changes as the energy changes. As the energy value increases, the curviness of the wave function 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 wave function 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 wave function to go to zero at x = xright. Likewise, when the energy value is too big, the curviness is too large to allow the wave function to go to zero at x = xright.
Also note that for the energy eigenvalue of 19 E0, both the amplitude and curviness of the wave function 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 wave function. 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 wave function 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 [3] uses the term wiggliness to qualitatively describe the oscillatory nature of the wave function.
Qualitatively, the curviness can be thought of as the number of
oscillations per length of the wave function.
7The average kinetic energy can be defined in an interval,
ε, centered on a point, xc, as
−(ħ2/2m) ∫x c − ε/2x c + ε/2 ψ*(x)[d2ψ(x)/dx2] dx / ∫x c − ε/2x c + 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).