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?
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.
