Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
Did you try plotting a representative function like this? – anna v May 19 '11 at 3:45
I'm trying to plot it on maple right now. I don't know what to specify the energies as, cause k = sqrt(2mE)/h_bar. Griffiths goes on and says that this wave function is NOT normalizable! So I'm confused. – QEntanglement May 19 '11 at 3:49
Another question that came shortly after this one explores the same math. – dmckee May 19 '11 at 17:46
up vote 10 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.