Simple Quantum Mechanics question about the Free particle, (part1)

Question

I am reading Introduction to Quantum Mechanics by David Griffiths and I am in Ch2 page 59. He starts out writing the time dependent Schrödinger equation and the solution for $\psi(x,t)$ for the free particle,

$$\psi(x,t) = A e^{ik(x-(\hbar k/2m)t)} + B e^{-ik(x + (\hbar k/2m)t)}$$

Then he goes and says the following,

Now, any function of $x$ and $t$ that depends on these variables in the special combination $x \pm vt$ (for some constant $v$) represents a wave of fixed profile, traveling in the $\pm x$-direction, at speed $v$.

What does this sentence mean?

Answer 1

It means there are many possible shapes for waves, not just pure sine waves.

For example,

$$\psi(x,t) = A\textrm{e}^{-k^2(x-vt)^2}$$

is a possible wavefunction. It represents a Gaussian wave packet that travels down the x-axis in the positive direction at speed $v$. The important part is that you can make the substitution $u = x-vt$ into $\psi$ and get a function of a single variable $u$.

So, start with any function $f$ of a single variable $u$. Now make the substitution $u = x - vt$. $f$ has now become a wave that travels down the x-axis at speed $v$ with some funky shape.

The mathematically-important thing is that such functions can be represented as a superposition of sinusoidals of continuously-varying frequencies all traveling in tandem down the x-axis (by "traveling" I mean "have phase velocity"). The sinusoidals that go with a given $f$ are found through fourier analysis. This is important because the sinusoidals are the eigenfunctions of the Hamiltonian for a free particle.