The general form of a wavepacket satisfying the time-dependent Schrödinger (free particle) equation is:
$$ \psi_{w}(x,t)=\dfrac{1}{\sqrt{2\pi\hbar}}\displaystyle\int_{-\infty}^{\infty} e^{i(px-Et)/\hbar}\phi(p)dp. $$
On the other hand, the Expansion Postulate gives us:
$$ \psi_{ep}(x,t) = \sum_{\alpha}c_{\alpha}\psi_{\alpha} e^{-iE_{\alpha}t/\hbar}. $$
I know the following:
- The wavepacket is a Fourier transform in the $p$-space and the Expansion Postulate is a Fourier series in the $E$-space.
- From the Expansion Postulate we can obtain (depending on the Hamiltonian) bound or/and scattering states.
I'm not sure about:
- From the wavepacket we can obtain (depending on the Hamiltonian) bound or/and scattering states.
Question:
It's correct $\psi_{w}=\psi_{ep}$ ?
