# Characteristic Size

1. Jul 23, 2014

### Bashyboy

Hello Everyone,

I am currently reading about when the Quantum Mechanical Theory becomes relevant. A term that continually emerges is "characteristic length." Could someone provide me with a thorough definition of this concept?

Thank you!

2. Jul 23, 2014

### Staff: Mentor

3. Jul 23, 2014

### WannabeNewton

A characteristic dimension of a system, be it time or length or velocity etc., is a mesure that describes an intrinsic property of the system. It is one of the most important concepts in physics without a doubt. For example in a system of bound charges, the characteristic size would be the maximum distance between any two charges in the system as this tells us how far away we have to be from the system in order for a certain order multipole expansion to be valid. The intrinsic property this characteristic length describes is the system's isolation in space. It also tells us how large in average wavelength the radiation emitted by the charges must be compared to the characteristic length of the system in order for the usual far field approximation of radiation fields to be valid. A characteristic time scale would be for example the time it takes a test particle to orbit a gravitating body or the inverse of the frequency of oscillations of an atom about its lattice site in a crystal. As another example, the thermal de Broigle wavelength of an ideal gas is a characteristic length of the system that describes the extent to which classical Boltzmann statistics is valid for the system by noting that average intermolecular spacings much larger than this characteristic length allow for classical statistics.

4. Jul 23, 2014

### Staff: Mentor

Are you associating it with the EFT approach to QFT where we have a cut-off at some scale determined by the regime of the phenomena we are interested in? I know its widely used in condensed matter physics as well.

Thanks
Bill

Last edited: Jul 23, 2014
5. Jul 24, 2014

### Bashyboy

WannabeNewton,

You say that the characteristic length is an intrinsic property of the system, yet your one example would say otherwise. The de Broglie wavelength of a given particle varies inversely with its momentum, and this can also be related to the temperature. Thus, as the temperature varies, the de Broglie wavelength varies;

An inherent property would be something that does not change even when its environment is changing in every way.

It would seem that the characteristic length would have to be something that varies, for if it did not, we would never observe classical theories being valid in one case, and then, when the characteristic length varies because environmental conditions are varying, valid in another case.

I imagine it would the some parameter of the system that, when varied, causes Quantum Mechanical effects to be manifest; this parameter makes the laws of Quantum Mechanics necessary, if we are to properly describe the behavior of the system. And so, not only does the characteristic length change when the environmental conditions change, but what the characteristic length is, changes when the system is placed in a completely different environment.

For example, suppose we have some particles placed in contact with some heat source. As the temperature varies, so does the de Broglie wavelength; when it changes, certain Quantum effects either manifest or disappear. So, the characteristic length is the de Broglie wavelength. Now, take this same system of particles, and by some means, propel them near the speed of light, certain relativist effects manifest. Thus, for the same system, the characteristic length has changed in two different ways.

I don't intend to cause any dissension, I am simply trying to search out a good definition of characteristic length

EDIT: Now that I have re-read your post, I believe I might have just said the same thing you did.

Last edited: Jul 24, 2014