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 ψ(x,t) for the free particle,
ψ(x,t)=Aeik(x−(ℏk/2m)t)+Be−ik(x+(ℏk/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±vt (for some constant v) represents a wave of fixed profile, traveling in the ±x-direction, at speed v.
What does this sentence mean?
Answer
It means there are many possible shapes for waves, not just pure sine waves.
For example,
ψ(x,t)=Ae−k2(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 ψ 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.
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