Questioning Wavepacket: Seeking Clarification

[Alice Jin]

Dear everyone,

I'm wondering about why we need the wavepacket?
Is there anyone to clarify my stupidity?
Thanks in advance.

 

[Ken G]

By "wavepacket", I believe you mean a wave function that does not correspond to a particular momentum for the particle it describes, correct? Or do you just mean the more generic term "wavefunction"? If the latter, then we need wavefunctions to get the concept of interference, and experiments show that particles do exhibit interference patterns. If you mean why do we need wavefunctions that don't correspond to definite momenta, the answer to that is that there are other measureables that are not compatible with momentum measurements. So if a quantum system is prepared using a process that does not yield a definite momentum (which is pretty much anything that isn't expressly designed to yield a definite momentum) then we will need to treat it as a wavepacket over all the possible different momenta, with all the correlations and capacity for interference preserved.

 

[Alice Jin]

Thank you for your reply.

I understand the reason why we use the wavepacket is there are undefined momenta in a quantum system. So since the wavepacket is composed of different momentum, it can describe the quantum system. Is that your saying, right?

Here's the another question. Then, if we have just one particle in a free space or particle in a box, do you need the wavepacket? A particle in a box have one wavefunction, right?

 

[Ken G]

Then, if we have just one particle in a free space or particle in a box, do you need the wavepacket? A particle in a box have one wavefunction, right?

You would in general need a wavepacket in either case. There is really no fundamental difference between the words wavepacket and wavefunction, except that the former explicitly implies you are dealing with a range of possible values for some measurable (often momentum), and the latter is often used even when you do have a definite value for some measurable. When you talk about a particle in a box, for example, you might be interested in the definite values that you can get out of an energy measurement. If you have done such a measurement and know the energy, you can say the particle has a wavefunction that is an "energy eigenfunction", which would therefore not be considered a wavepacket in regard to energy measurements. However, if you look at some incompatible measurement, like position, then an energy eigenstate for a particle in a box is still a "wavepacket" in regard to a subsequent position measurement.

To me, the word wavepacket just means you have a wave function and you are using it to tell you something about a measurement that is incompatible with the way that wave function was previously prepared. Typically, the wave function was prepared by a natural process, and the source of the incompatibility is that you are now probing it with a much more precise instrument, and that is the origin of the need for the wavepacket concept.

 

[Alice Jin]

Thanks again, Ken G.

"However, if you look at some incompatible measurement, like position, then an energy eigenstate for a particle in a box is still a "wavepacket" in regard to a subsequent position measurement."
From this, you mean I need different kind of wavefunctions for each measurement like momentum and position?? wavefunction for energy and wavepacket for position??
I don't get it. Could you explain more fundametally?

 

[olgranpappy]

I don't get it. Could you explain more fundametally?

Physical states must be square-integrable.

For example, although plane-waves are eigenstates of the free Schrödinger equation they do not describe physical states because they are not square integrable.

 

[Ken G]

From this, you mean I need different kind of wavefunctions for each measurement like momentum and position?? wavefunction for energy and wavepacket for position??

It's not that the wavefunction is different, it is that it can be expressed in different ways, using different bases. When you want to explicitly express the fact that the particle is spatially confined, you might tend to use a position basis, so that you can think of the wavefunction as a complex amplitude at every location. A wavepacket is normally such a picture, but to me the main point is simply that the measuring device you are going to use on the system is usually capable of a much more precise determination of some observable than was previously engendered into the system by whatever physical process prepared it. That is the normal state of affairs when we talk about a "superposition" state, so I would tend to think of a "wavepacket" as simply being a "superposition state" in the context of a position basis, where there is also a stress on the physical confinement of the state. It tends to also have a connotation that the wavefunction is propagating. But the bottom line is, once you understand the concepts of superpositions and position bases, there doesn't seem to me to be anything to add to that in order to understand wavepackets.