Saketh said:
What is a wave packet? I know how to calculate wave packets, but how does a wave packet relate to the wave function?
A wave packet
is a wave function. It's the wave function of a free particle that is localized to some region of space.
What is the conceptual difference between the probability density and the probability amplitude?
The probability density |\psi|^2 describes the probability of finding the particle at various places. The probability amplitude \psi has no direct physical meaning in and of itself, as far as I know.
What is the difference between the group velocity and the phase velocity?
For a visual illustration, see the animations on the following page:
http://physics.usask.ca/~hirose/ep225/animation/dispersion/anim-dispersion.html
and note in particular the first one. The phase velocity is the speed of the individual "humps" in the packet, whereas the group velocity is the speed of the packet as a whole. Observe how the humps appear to enter the packet at the left and travel "through" the packet to the right, because the phase velocity is greater than the group velocity in this example.
Moreover, where does the relation v_g = \frac{\partial \omega}{\partial k} come from?
Fourier analysis. (Sorry, it's late at night and all my books are at the office.)
How can I determine a particle's momentum from wavepackets/wavefunctions?
You can find the momentum probability amplitude \phi(p) by performing a Fourier transform on the position probability amplitude \psi(x). You can also find expectation values involving momentum by using the momentum operator.
In which direction do a particle's matter waves propagate? (I need to know this because I am not sure why Bohr was able to calculate the standing waves in the hydrogen orbit until I understand in which direction the matter waves move.)
You should note that Bohr himself didn't deal with "matter waves" in his original model. de Broglie's idea of "matter waves" came about ten years afterward. Bohr basically assumed that the electron's orbital angular momentum was quantized, and derived the energy levels from that starting point. De Broglie attempted to explain why the angular momentum was quantized by using his idea of "matter waves." I don't remember if de Broglie actually used traveling waves or standing waves. His key point was that in order to avoid destructive self-interference, the circumference of the orbit had to be an integer multiple of the wavelength. As far as I can see, that would apply to traveling waves as well as to standing waves.
But you really shouldn't spend much time trying to puzzle out the details of the Bohr / de Broglie model, because it was superseded by Schrödinger's QM, and the two models have very little in common, conceptually. The atomic electrons don't have circular "orbits" to speak of. Rather, the wave functions for the energy states have the form of three-dimensional standing waves which are somewhat analogous to standing waves of sound in a spherical cavity. In that case, the waves are best thought of as "pulsations" of pressure rather than as sound waves traveling from one place to another inside the cavity.
