I have the one dimensional free particle Schrödinger equation
$$i\hbar \frac{\partial}{\partial t} \Psi (x,t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \Psi (x,t), \tag{1}$$
with general solution
$$\psi(x,t) = A e^{i(kx-\omega t)} + B e^{i(-kx-\omega t)}. \tag{2}$$
I'd expect that the solution is normalized:
$$ \int_{-\infty}^\infty |\psi(x,t)|^2 dx = 1. \tag{3}$$
But
$$|\psi(x,t)|^2 = \psi(x,t)\psi^* (x,t) = A^2 + B^2 + AB ( e^{2ikx} + e^{-2ikx} ), \tag{4}$$
and the integral diverges:
$$ \int_{-\infty}^\infty |\psi(x,t)|^2 dx = \frac{AB}{2ik} (e^{2ikx} – e^{-2ikx})\biggl|_{-\infty}^\infty + (A^2+B^2) x\biggl|_{-\infty}^\infty. \tag{5}$$
What is the reason for this? Can it be corrected?
Best Answer
Schroedinger's equations may have both normalizable and non-normalizable solutions. The function
$$ \psi_k(x,t) = A e^{i(kx-\omega t)} + B e^{i(-kx-\omega t)}. \tag{2} $$
is a solution of the free-particle Schroedinger equation for any real $k$ and $\omega = |k|/c$.
As a rule, if the equation has a class of solutions parameterized by continuous parameter ($k$), these solutions are not normalizable to infinite space.
One purpose of wave function is to use it to calculate probability of configuration via the Born rule; the probability that the particle described by $\psi$ has $x$ in the interval $(a,b)$ of the line is
$$ \int_a^b|\psi(x)|^2\,dx. $$
For this to work, $\psi$ has to be such that it has finite integral
$$ \int_S |\psi(x)|^2\,dx $$
where $S$ is region where it does not vanish.
Plane wave (or sum of such waves) cannot be normalized for $S=\mathbb R$ (or higher-dimensional versions of whole infinite space), but it can be normalized for finite intervals (or regions of configuration space which similarly have finite volume).
People deal with this situation in several ways:
instead of $\mathbb R$, they describe system by functions that are limited to an imaginary finitely-sized box, so all regular functions are normalizable (delta distributions will remain non-normalizable even there). The exact size of the box is assumed to be very large but it is almost never fixed to definite value, because it is assumed as it gets expanded to greater sizeits influence on the result becomes negligible.
retain infinite space, but use only normalizable functions to calculate probability (never use non-normalizable function with the Born rule);
retain infinite space, retain plane waves, use Dirac formalism and be aware of its drawbacks. Never work with $\langle x|x\rangle$ as with something sensible, do not think $|x\rangle$,$|p\rangle $ represent physical states (people call them states to simplify the language), mind that $\langle x|$ is a linear functional that is introduced to act on some ket, not a replacement notation for $\psi^*$.