40.2.1 Restrictions and Boundary Conditions

Not all functions $\psi \left(x\right)$ make acceptable solutions to the Schrödinger equation. That is, some functions may satisfy the Schrödinger equation but not be physically meaningful. We have previously encountered restrictions in our solutions of algebraic equations. We insist, for physical reasons, that masses be positive rather than negative numbers, that positions be real rather than imaginary numbers, and so on. Mathematical solutions not meeting these restrictions are rejected as being unphysical.

Because we want to interpret $\left|\psi \left(x\right)\right|{}^2$ as a probability density, we have to insist that the function $\psi \left(x\right)$ be one for which this interpretation is possible. The conditions or restrictions on acceptable solutions are called the boundary conditions. You will see, in later examples, how the boundary conditions help us choose the correct solution for $\psi \left(x\right)$. The primary conditions the wave function must obey are:

1. $\psi \left(x\right)$ is a continuous function.

2. $\psi \left(x\right)=0$ if $x$ is in a region where it is physically impossible for the particle to be.

3. $\psi \left(x\right)\to 0$ as $x\to +\infty$ and $x\to -\infty$.

4. $\psi \left(x\right)$ is a normalized function.

The last is not, strictly speaking, a boundary condition but is an auxiliary condition we require for the wave function to have a useful interpretation. Boundary condition 3 is needed to enable the normalization integral ${\displaystyle \int {\left|\psi \left(x\right)\right|}^{2}dx}$ to converge.

Once boundary conditions have been established, there are several approaches to solving the Schrödinger equation: Use general mathematical techniques for solving second-order differential equations, solve the equation numerically on a computer, or make a physically informed guess.

More advanced courses make extensive use of the first and second approaches. However, we are not assuming a knowledge of differential equations, so you will not be asked to use these methods. The third, although it sounds almost like cheating, is widely used in simple situations where we can use physical arguments to infer the form of the wave function. The upcoming examples will illustrate this third approach.

A quadratic algebraic equation has two different solutions. Similarly, a second-order differential equation has two independent solutions ${\psi }_1\left(x\right)$ and ${\psi }_2\left(x\right)$. By “independent solutions” we mean that ${\psi }_2\left(x\right)$ is not just a constant multiple of ${\psi }_1\left(x\right)$, such as $3{\psi }_1\left(x\right)$, but that ${\psi }_1\left(x\right)$ and ${\psi }_2\left(x\right)$ are totally different functions.

Suppose that ${\psi }_1\left(x\right)$ and ${\psi }_2\left(x\right)$ are known to be two independent solutions of the Schrödinger equation. A theorem you will learn in differential equations states that a general solution of the equation can be written as

where $A$ and $B$ are constants whose values are determined by the boundary conditions. Equation 40.8 is a powerful statement, although one that will make more sense after you see it applied in upcoming examples. The main point is that if we can find two independent solutions ${\psi }_1(x)$ and ${\psi }_2(x)$ by guessing, then Equation 40.8 is the general solution to the Schrödinger equation.