## Section 7.2: Eigenfunction Shape for Piecewise-constant Potentials

**Choose a level:**

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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:

[−(*ħ*^{2}/2*m*)*d*^{2}/*dx*^{2 }+ *V*_{0}] ψ(*x*) = *E* ψ(*x*), (7.1)

which we can write as

[*d*^{2}/*dx*^{2 }− 2*mV*_{0}/*ħ*^{2} + 2*mE*/*ħ*^{2}] ψ(*x*) = 0. (7.2)

In this situation, as in general, there are two cases:^{2} *E* > *V*_{0} which is *classically allowed* and *E* < *V*_{0} which is *classically forbidden*. In these two cases the time-independent Schrödinger equation reduces to:

[*d*^{2}/*dx*^{2 }+ *k*^{2}] ψ(*x*) = 0 → ψ(*x*) = *A* cos(*kx*) + *B* sin(*kx*)

or ψ(*x*) = *A*' e^{ikx} + *B*' e^{−ikx } (7.3)

and

[*d*^{2}/*dx*^{2 }− κ^{2}] ψ(*x*) = 0 → ψ(*x*) = *A* e^{κx} + *B* e^{−κx} (7.4)

where

*k*^{2} ≡ 2*m*(*E* −* V*_{0})/*ħ*^{2} (7.5)

and

κ^{2} ≡ 2*m*(*V*_{0} −* E*)/*ħ*^{2}, (7.6)

so that both *k*^{2} and κ^{2} 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* > *V*_{0}, the energy eigenfunction oscillates. In the region that is classically forbidden, *E* < *V*_{0}, which corresponds to the far right and far left of the animation, the energy eigenfunction must be exponentially decaying, ψ_{left}(*x*) is proportional to e^{κx} and ψ_{right}(*x*) is proportional to e^{−κx}, in order for the energy eigenfunction to be normalizable.

^{1}This problem is discussed in detail in Chapter 11.

^{2}Usually, the third case *E* = *V*_{0} 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: *d*^{2}ψ(*x*)/*dx*^{2 }= 0, which has a straight-line solution ψ(x) = *Ax* + *B*. For more examples, see Refs. [29-31].

^{3}Even though *E* < 0 and *V*_{0} < 0, *E* − *V*_{0} > 0. Thus, *k*^{2} > 0.

^{4}You may be wondering why we use *curviness* instead of *curvature*. Mathematically, the curvature of a (wave) function is defined by *d*^{2}ψ(*x*)/*dx*^{2} which can change *magnitude* and *sign* as a function of position, even when the (wave) function's *curviness *is constant. For example, when E < *V*_{0}, 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* > *V*_{0} 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, −*k*^{2} sin(*kx*). In reality, the only curve that has a constant curvature is a circle.

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