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Section 7.2: Eigenfunction Shape for Piecewise-constant Potentials
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The animation depicts seven bound energy eigenfunctions in a finite square well.1 You can use the slider to change the energy level, n, and see the corresponding energy eigenfunction. Restart. In regions where the potential energy function does not change too rapidly with position, and can therefore be considered a constant, the time-independent Schrödinger equation is just:
[−(ħ2/2m)d2/dx2 + V0] ψ(x) = E ψ(x), (7.1)
which we can write as
[d2/dx2 − 2mV0/ħ2 + 2mE/ħ2] ψ(x) = 0. (7.2)
In this situation, as in general, there are two cases:2 E > V0 which is classically allowed and E < V0 which is classically forbidden. In these two cases the time-independent Schrödinger equation reduces to:
[d2/dx2 + k2] ψ(x) = 0 → ψ(x) = A cos(kx) + B sin(kx)
or ψ(x) = A' eikx + B' e−ikx (7.3)
[d2/dx2 − κ2] ψ(x) = 0 → ψ(x) = A eκx + B e−κx (7.4)
k2 ≡ 2m(E − V0)/ħ2 (7.5)
κ2 ≡ 2m(V0 − E)/ħ2, (7.6)
so that both k2 and κ2 are positive.3 For an oscillatory solution, the larger the k value, the larger the curviness of the energy eigenfunction at that point.4
How does this analysis help us understand the energy eigenfunctions depicted in the animation? In the region where E > V0, the energy eigenfunction oscillates. In the region that is classically forbidden, E < V0, which corresponds to the far right and far left of the animation, the energy eigenfunction must be exponentially decaying, ψleft(x) is proportional to eκx and ψright(x) is proportional to e−κx, in order for the energy eigenfunction to be normalizable.
1This problem is discussed in detail in Chapter 11.
2Usually, the third case E = V0 is not considered in bound-state energy eigenfunctions except at the classical turning point. There are, however, bound states in which it naturally occurs. For these cases, the time-independent Schrödinger equation becomes: d2ψ(x)/dx2 = 0, which has a straight-line solution ψ(x) = Ax + B. For more examples, see Refs. [29-31].
3Even though E < 0 and V0 < 0, E − V0 > 0. Thus, k2 > 0.
4You may be wondering why we use curviness instead of curvature. Mathematically, the curvature of a (wave) function is defined by d2ψ(x)/dx2 which can change magnitude and sign as a function of position, even when the (wave) function's curviness is constant. For example, when E < V0, the curvature of the energy eigenfunction is such that the energy eigenfunction curves away from the axis (positive curvature for ψ(x) > 0 and negative curvature for ψ(x) < 0). For E > V0 the curvature of the energy eigenfunction is such that the wave function is oscillatory (negative curvature for ψ(x) > 0 and positive curvature for ψ(x) < 0). Even sin(kx), which we think of as having a constant curviness, has a curvature that depends on position, −k2 sin(kx). In reality, the only curve that has a constant curvature is a circle.