Simple questions

1. Mar 25, 2007

Saketh

I'm trying to learn QM on my own, and I've encountered some conceptual issues. The textbook that I'm working from just glazes over everything.

1. What is a wave packet? I know how to calculate wave packets, but how does a wave packet relate to the wave function?
2. What is the conceptual difference between the probability density and the probability amplitude? (I know $|\psi|^2$ vs. $\psi$...but what do they represent?)
3. What is the difference between the group velocity and the phase velocity? Moreover, where does the relation $$v_g = \frac{\partial \omega}{\partial k}$$ come from?
4. How can I determine a particle's momentum from wavepackets/wavefunctions?
5. 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.)

I appreciate all help in trying to get over the learning curve for QM.

Last edited: Mar 25, 2007
2. Mar 25, 2007

Staff: Mentor

A wave packet is a wave function. It's the wave function of a free particle that is localized to some region of space.

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.

For a visual illustration, see the animations on the following page:

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.

Fourier analysis. (Sorry, it's late at night and all my books are at the office.)

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.

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.

3. Mar 26, 2007

Saketh

Thanks! You have cleared up most of my doubts.

The only one that is remaining is why is group velocity $$\frac{\partial \omega}{\partial k}$$?

(I'm using Merzbacher, and he doesn't explain things very well.)

4. Mar 29, 2007

erwin_the_kat

I would suggest finding Sakurai's Modern Quantum Mechanics, preferbly the newer edition (it has a mainly red cover) at a library. Also, if you are interested in a foundation for where quantum mechanics is going, I suggest looking at the many admirable presentations of quantum information, the standard text of which is Nielsen and Chuang's Quantum Computation and Quantum Information.