III Mathematical interpretation of the Schrödinger equation

We can write the stationary Schrödinger equation (29) as

$$E\psi (x)=-\frac{{\mathrm{\hslash }}^2}{2m}{\psi }^{\prime \prime }(x)+V(x)\psi (x),$$

(31)

where ${\psi }^{\prime \prime }(x)$ denotes the second derivative of the wave function. Equation (31) is a homogeneous linear ordinary differential equation of second order. The solutions $\psi (x)$ have the following properties:

Mathematical property I: superposition principle: If ${\psi }_1(x)$ and ${\psi }_2(x)$ are solutions of Eq. (31) and $A$, $B$ are constants, then

$$\psi (x)=A{\psi }_1(x)+B{\psi }_2(x)$$

(32)

is also a solution of Eq. (31). This is a consequence of the linearity of Eq. (31).

Mathematical property II: continuity conditions: Any solution $\psi (x)$ is a continuous function (the function is smooth, i.e., has no jumps). The first derivative ${\psi }^{\prime }(x)$ is also continuous, with the possible exception of points where $|V(x)|=\mathrm{\infty }$. At such points, ${\psi }^{\prime }(x)$ may jump, which translates into a ‘kink’ in $\psi (x)$ [i.e., $\psi (x)$ is still continuous, but suddenly changes its slope].

Mathematical property III: boundary conditions: The wave function must be bounded for $|x|\to \mathrm{\infty }$ (i.e., it is not allowed to increase indefinitely). This condition has a physical origin, since if it were violated we could not interpret ${|\psi (x)|}^2$ as the position probability density (the particle would be sitting at $\pm \mathrm{\infty }$).

In general, $\psi (x)$ is a function, and $\widehat{H}\psi (x)$ is just another function, which usually is of a very different form than $\psi (x)$. In solving the stationary Schrödinger equation, we find functions ${\psi }_n(x)$ such that $\widehat{H}{\psi }_n=E_n{\psi }_n$. Thus, for a solution ${\psi }_n$ of the stationary Schrödinger equation the operation with $\widehat{H}$ is equivalent to the multiplication by a real number $E_n$. Functions ${\psi }_n(x)$ with this property are called eigenfunctions of $\widehat{H}$, and the numbers $E_n$ are known as eigenvalues. Informed by the physical meaning of these mathematical objects, the eigenfunctions of the Hamiltonian are also called eigenstates, and since the Hamiltonian represents energy its eigenvalues are called eigenenergies.

Eigenfunctions and eigenvalues can also be defined for other operators $\widehat{A}$, by requiring $\widehat{A}\psi =a\psi$. In particular, the eigenfunctions of the momentum operator $\widehat{p}=-i\mathrm{\hslash }d/dx$ are

$${\psi }_p(x)=\frac{1}{\sqrt{2\pi \mathrm{\hslash }}}\exp (ipx/\mathrm{\hslash }),$$

(33)

and thus are given by the position-dependent part of the plane waves ${\mathrm{\Psi }}_k$ [see Eq. (13); the stated form expresses $k$ by the eigenvalue $p$, and the stated amplitude ensures the appropriate normalisation of the probability distribution of momentum, introduced in Section VII.1].

The eigenfunctions of $\widehat{x}$ must be very localised in space, at some place $x_0$, and zero at all other places, so that $\widehat{x}{\psi }_{x_0}(x)=x{\psi }_{x_0}(x)=x_0{\psi }_{x_0}(x)$. These highly singular functions can be expressed in terms of Dirac’s $\delta$-function,

$${\psi }_{x_0}(x)=\delta (x-x_0).$$

(34)

The $\delta$-function is so singular that it actually does not constitute a proper function, but a so-called distribution. Its defining property is the following integral:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}𝑑xf(x)\delta (x-x_0)=f(x_0)$$

(35)

for any function $f(x)$ that is continuous at $x_0$. The $\delta$-function is real and symmetric, ${\delta }^{*}(x-x^{\prime })=\delta (x-x^{\prime })=\delta (x^{\prime }-x)$.
Points to remember

• •

  The stationary Schrödinger equation is a linear differential equation, and the ensuing superposition principle, continuity requirements and boundary conditions determine important features of its solutions.

• •

  The stationary Schrödinger equation can also be interpreted as an eigenvalue equation of the Hamiltonian, $\widehat{H}\psi =E\psi$.

• •

  The eigenfunctions of the momentum operator are given by ${\psi }_p(x)=\frac{1}{\sqrt{2\pi \mathrm{\hslash }}}\exp (ipx/\mathrm{\hslash })$.