Are hydrogen lamps coherent?

Question

Those hydrogen or mercury tubes should not be coherent, but when looking at them through a grating without using a small aperture in front of the tube, you still see the discrete lines. Or just any random ceiling light really, without using any aperture you still see rainbow through a grating. How could that be?

Answer 1

First consider what happens when you look at a lamp directly.
You "see" the lamp which means that an optical system (your eye) has produced a real image on your retina.
Every point, $a'$, on the image has a one to one correspondence with a point, $a$, on the object and furthermore adjacent points, $b,\,c,\,d\ldots$, to $a$ on the object produce image points, $b',\,c',\,d'\ldots$ on the retina in the same relative positions to $a'$.
Furthermore waves which leave point $a$ in all directions and are focused by the eye onto point $a'$ on the retina all take the same time to travel from point $a$ to point $a'$ and hence the waves arrive in phase with one another.
So that is what an image is.

If you have done a spectrometer with a diffraction grating experiment as a preliminary to taking readings you set up the spectrometer so that when looking through the telescope you saw a sharp image of the slit which meant that an image of the slit was formed on your retina with the same one-to-one correspondence between slits and its image that I described before.

Now what dies the grating do?
It splits light from each point on the slit into component colours and so the spectrum that you see are images of the first slit in the component colours of the light that passes through the slit.
If you made the slit wider the image is wider and the images of adjacent coloured images overlapped one another and so the resolving power of the grating was diminished.

Now consider the system, light source and eye.
An image of the light source is produced on the eye and remember the one-to-one correspondence between points on the object and point on the retina of the eye which produce the image and the idea that the time of travel for each ray one point on object to corresponding point on the retina is the same.

Inserting a grating produced the zero order image and then higher order images which are displaced from the zero order image dependent on number of lines on the grating, distance between the grating/eye and the source of light, wavelength of the light.

So here are some photographs which I have produced using a $500$ lines per mm grating between the light source and an iPhone.

3 white leds (reading light) and you will note three distinct streaks (spectra) but the images of the leds for each of the wavelengths which form white light overlap so it loos as an (imperfect) continuous spectrum.

A white liquid crystal clock display with some colours more intense than others, eg the first order blue image on the left is clearly seen.

A red led clock display which the red light looking fairly monochromatic.

Are hydrogen lamps coherent?

Each point in a hydrogen lamp is a source of light but each of those sources are not coherent with another source, there is no constant phase relationship between the sources.
This is easily shown by having the light from a hydrogen lamp hit a screen after passing through a diffraction grating.
No interference pattern would be seen on the screen.
This is because the light from each point in the hydrogen lamp produces an interference pattern which is displayed over the whole screen and the light from those points in the hydrogen lamp are not coherent.

This is radically different from what happens when light from the hydrogen lamp is focused onto the screen. Then each object point in the hydrogen lamp produces an interference pattern at a series of corresponding image points (zero order, first order etc) on the screen.

Here are some photographs to illustrate this.