# Why do we define a particle as a 'wave packet' but use 'waves' in application problems?

In QM we define the particle as a superposition of waves i.e a wave packet,yes?But as I was reading a few problems like the Harmonic oscillator or the particle in a box,I found out that we were denoting the wave function of the particle as a single wave.

Now I want to know:

1.How do we make this transition from wave packets to waves?

2.Is there any good reference book which can explain it in a comprehensive way?

• While Gennaro told you all the right things I would add that the "waves" in QM are not physical waves. They are an accounting procedure to calculate all possible futures of a quantum system. You can calculate the exact same thing without waves using matrices (linear operators) or path integrals. Commented Dec 24, 2015 at 19:01

In QM we define the particle as a superposition of waves

That is not entirely true. In QM particles are states $|\psi\rangle$ in a Hilbert space (actually they are rays, namely equivalence classes of states) with a finite expectation value of the position and momentum operator, i. e. $\langle \hat{p}\rangle_{\psi}$ as well as $\langle \hat{x}\rangle_{\psi}$ must not be infinite.

Now it can be argued that, because of how the position and momentum operators are defined, one needs an infinite superposition of plane waves in order for the expectation values (and the norms) to be finite, which translates to requiring the most general solution to be a Fourier transform.

1.How do we make this transition from wave packets to waves?

Since the Schrödinger equation is linear, whatever is true for a single plane wave is also true for an (infinite) superposition of waves, hence one can simply prove the results in the simple one wave case and then extend making superpositions (taking care of all possible infinities eventually). Moreover, for some simple domains, position and momentum are perfectly normalisable with no need of an infinite superposition.

2.Is there any good reference book which can explain it in a comprehensive way?

I believe every standard textbook does so: I have particularly liked A. Messiah and K. Gottfried, although all the others would do as well.

• All right but it does not answer why do we use a single wave and not wave packets in these simple problems??? Commented Dec 24, 2015 at 16:49
• As I said under point 1.: if the result holds for a single wave, then it holds for the entire superposition (by linearity). Since a single wave is much more handble than a complex integral, one uses the former and then extends to the lattet instead of directly plugging in the whole Fourier transform. Commented Dec 24, 2015 at 17:17