For a photon, the momentum is Planck's constant divided by wavelength. The uncertainty in momentum is therefore directly related to the uncertainty in wavelength. Does this mean that for purely monochromatic light, the uncertainty in position is very large? Perhaps related to very long coherence length?
Thoughts?
plasma_guy writes:
> Does this mean that for purely monochromatic light, the uncertainty in position is very large? Perhaps related to very long coherence length?
Uncertainty of position is given by
delta_poaition delta_momentum = h / 2 pi
So yes, a narrow band of wavelengths has a large uncertainty of position. Any narrow-band wave train has the same property -- since the spatial frequencies are close together, they can only heterodyne ( beat ) slowly. Conversely, a highly localized wave packet requires a larger spread of frequency ( or wavelength ) components which 'add up' within the specified locale, and cancel each other out everywhere else.
Coherence length refers to the beam as a whole, not to uncertainty-of-position of individual photons. A beam can lose its coherence without any change in the individual photons' localizations. Consider a beam crossing through an inhomogeneous medium of randomly varying index of refraction. The beam then passes into a vacuum. The electromagnetic wavelets of the beam no longer march in step -- the beam's coherence length is now minimal. Yet any given photon still has the same uncertainty-of-position, since its frequency spread and wavelength spread haven't changed since it left the emitter.
Note however that "monochromatic" is a weaker adjective than "coherent", and pre-dates lasers. The Michelson-Morley experiment, for example, used sodium D light from an electric arc, which was "monochromatic" ( consisted of a narrow band of frequencies ), but had very limited coherence lengtb of a few centimeters. A single-mode laser produces "coherent" lighy matching a single resonant frequency of the laser cavity. However "coherent" can connote particle proprties, e.g. many photons sharing a single degenerate boson state.
"Coherence" can mean either that the Schrodinger wave packets march in step ( physics parlance ) or that the electromagnetic wavelets march in step ( engineering parlance ). The former is 'grainy'; the latter ignores the graininess and treats the beam as simply an electromagnetic wave. Photons whizzing around don't just 'add up' to a classical electromagnetic wave, because the EM field itself is quantized. ( and doesn't only follow Maxwell's equations for classical wave propagation ). For the general case of photons whizzing around in different directions (, the field analysis is complicated enough to have just won Glauber the Nobel Prize for physics.
The differences matter in newer engineering fields such as quantum cryptography and photonic computing devices, where mere handfuls of photons are causing field effects which excite other photons or cause some other change of state.
Coherence length per se doesn't 'translate back' into fundamental quantum properties. Coherence length is roughly speaking the maximum path length over which you can do interferometry. Beyond that distance, differing wavelength components fuzz out the interference effects. For example, a cheap laser may have more than one resonant mode active at once, or may switch randomly between modes. You might have a long enough short-term coherence length to get an experiment done. Sometimes a poor coherence length can be improved by filtering out spurious components.
> Does this mean that for purely monochromatic light, the uncertainty in position is very large? Perhaps related to very long coherence length?
Uncertainty of position is given by
delta_poaition delta_momentum = h / 2 pi
So yes, a narrow band of wavelengths has a large uncertainty of position. Any narrow-band wave train has the same property -- since the spatial frequencies are close together, they can only heterodyne ( beat ) slowly. Conversely, a highly localized wave packet requires a larger spread of frequency ( or wavelength ) components which 'add up' within the specified locale, and cancel each other out everywhere else.
Coherence length refers to the beam as a whole, not to uncertainty-of-position of individual photons. A beam can lose its coherence without any change in the individual photons' localizations. Consider a beam crossing through an inhomogeneous medium of randomly varying index of refraction. The beam then passes into a vacuum. The electromagnetic wavelets of the beam no longer march in step -- the beam's coherence length is now minimal. Yet any given photon still has the same uncertainty-of-position, since its frequency spread and wavelength spread haven't changed since it left the emitter.
Note however that "monochromatic" is a weaker adjective than "coherent", and pre-dates lasers. The Michelson-Morley experiment, for example, used sodium D light from an electric arc, which was "monochromatic" ( consisted of a narrow band of frequencies ), but had very limited coherence lengtb of a few centimeters. A single-mode laser produces "coherent" lighy matching a single resonant frequency of the laser cavity. However "coherent" can connote particle proprties, e.g. many photons sharing a single degenerate boson state.
"Coherence" can mean either that the Schrodinger wave packets march in step ( physics parlance ) or that the electromagnetic wavelets march in step ( engineering parlance ). The former is 'grainy'; the latter ignores the graininess and treats the beam as simply an electromagnetic wave. Photons whizzing around don't just 'add up' to a classical electromagnetic wave, because the EM field itself is quantized. ( and doesn't only follow Maxwell's equations for classical wave propagation ). For the general case of photons whizzing around in different directions (, the field analysis is complicated enough to have just won Glauber the Nobel Prize for physics.
The differences matter in newer engineering fields such as quantum cryptography and photonic computing devices, where mere handfuls of photons are causing field effects which excite other photons or cause some other change of state.
Coherence length per se doesn't 'translate back' into fundamental quantum properties. Coherence length is roughly speaking the maximum path length over which you can do interferometry. Beyond that distance, differing wavelength components fuzz out the interference effects. For example, a cheap laser may have more than one resonant mode active at once, or may switch randomly between modes. You might have a long enough short-term coherence length to get an experiment done. Sometimes a poor coherence length can be improved by filtering out spurious components.
Thanks for the very informative tutorial! 
Does this mean, then, that for a monochromatically illuminated area, the arrival of photons is random, since their positions are unknown due to the large uncertainty and low localization?
Does this mean, then, that for a monochromatically illuminated area, the arrival of photons is random, since their positions are unknown due to the large uncertainty and low localization?
A plane wave of the form exp(ikx) has precise momentum hk/(2pi) and its position uncertainty is infinite. If you take |amplitude|^2, it is one everywhere.
A finite pulse, by Fourier transform, has a wide spread of momentum. The narrower the pulse, the wider the spread of momentum, and the narrower your uncertainty of position (photon position random within pulse).
A pulse train will be bimodal in momentum distribution, which really widens your momentum spread.
A finite pulse, by Fourier transform, has a wide spread of momentum. The narrower the pulse, the wider the spread of momentum, and the narrower your uncertainty of position (photon position random within pulse).
A pulse train will be bimodal in momentum distribution, which really widens your momentum spread.
Following on from qp..
Looking at a single pulse - Fourier analysis suggests a pulse can be regarded as being made up number of frequencies which, when summed, will reproduce your pulse to any desired acuracy. An alternative would be to view your pulse as being made up of the sum of a large number of indivisible events (photon emmissions) each with their own probability of detection - analysis of a pulse in this way is not easy - perhaps someone has already done it - is the answer the same as the method suggested by qp?
Looking at a single pulse - Fourier analysis suggests a pulse can be regarded as being made up number of frequencies which, when summed, will reproduce your pulse to any desired acuracy. An alternative would be to view your pulse as being made up of the sum of a large number of indivisible events (photon emmissions) each with their own probability of detection - analysis of a pulse in this way is not easy - perhaps someone has already done it - is the answer the same as the method suggested by qp?