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The animation depicts seven bound states in a finite square well.1 You can use the slider to change the energy level, n, and see the corresponding wave function. 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)
and
[d2/dx2 − κ2] ψ(x) = 0 → ψ(x) = A eκx + B e−κx (7.4)
where k2 ≡ 2m(E − V0)/ħ2 and κ2 ≡ 2m(V0 − E)/ħ2, so that both k2 and κ2 are positive.3 For an oscillatory solution, the larger the k value, the larger the curviness of the wave function at that point.4
How does this analysis help us understand the wave functions depicted in the animation? In the region where E > V0, the wave function oscillates. In the region that is classically forbidden, E < V0, which corresponds to the far right and far left of the animation, the wave function must be exponentially decaying, ψleft(x) is proportional to eκx and ψright(x) is proportional to e−κx, in order for the wave function to be normalizable.
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1This problem is discussed in
detail in Chapter 11.
2Usually, the third case E =
V0 is not
considered in bound-state wave functions 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 wave function is such
that the wave function 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 wave function 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.