So apparently researchgate has its own forums and some of the interactions kind of remind me of quora drama in the sense it's a bunch of third world r-slurs who think too much of themselves.
It starts with a basic question by Halo A. Abdulkhalaq:
Does wave function (in Quantum mechanics) have a unit?
Ulrich Mutze, who from now on is the protagonist of this thread, responds first with the only useful (albeit slightly condescending) contribution to this thread:
The answer is simple. The dimension of the wave function is set such that the scalar product in state space is dimension-less (since probability is dimension-less by definition). This gives different dimensions to wave functions in 'position representation' and in 'momentum representation' (and whatever 'representations' you may use).
Saeid Soheyli of Buali Sina University wanders into the thread, to give another answer which restates our dear Dr. Mutze's answer but does it much worse:
The dimension of probability density is the inverse of length, the inverse of area, and the inverse of volume in one dimensional, two dimensional, and three dimensional space, respectively. Therefore, the dimension of the wave function is the square root of 1/length, 1/length^2, and 1/length^3 for one, two, and three dimensional spaces, respectively.
Saeid, why do you ignore the possibility that wave functions refer to momentum space?
At this point I'm beginning to suspect our dear Dr. Soheyli is kinda r-slurred.
Dear Ulrich My answer is limited to the one, two, or three dimensional spatial space. Therefore, in momentum space, the dimension of the wave function is similar to the above, provided that the length is replaced by the linear momentum.
Ulrich becomes mildly annoyed and passive aggressively says:
OK, but all this is already expressed in the first contribution to this thread. Sometimes reading the precedent contributions is useful.
OP sees it devolving and intervenes.
Thanks for both of you Mr. Ulrich and Mr. Saeid. your reply is useful and appreciated.
It seems Dr. Mutze's bloodlust is interrupted for now and unfortunately we will be getting no more passive aggressiveness, condescension or rslurration from this thread anymore.
Luckily our journey doesn't end here. Himangshu Prabal Goswami of the Max Planck Institute for the Physics of Complex Systems (the first institution here that isn't a random no name) jumps in 4 days later to save the thread with an unironically r-slurred take:
The wave function does have dimensions, depending on what space you use to represent it. Say, the spherical polar co-ordinate system,eg. 1s=1/(sqrt[pi ao^3])Exp[-r/ao]. So its dimensions are L^(-3/2). You can reduce the dimensions as well.
It essentially comes from the interpretation of the wave-function. |Psi(q,t)|^2=1. So the volume element, dq, will determine the dimension and that will be reflected in the normalisation constant. Anyway, although wave-function is a very useful tool, I agree with lot of books which states, it has no physical meaning. That makes it useless to check its dimensions.
Ulrich immediately smells blood in the water:
Himangshu, the logic of physics is complicated enough that it is not always easy to express it unambiguously in our language. Saying that the wave function has no physical meaning may create a wrong impression for some readers. Saying that it has no direct physical meaning is more precise. Think of the four-vector potential A in electrodynamics. In the 19th century it was considered as a merely mathematical tool. Then quantum theory came and it turned out that A is the primary quantity that transmits electromagnetic interaction. So the indirect physical meaning changed into a direct one. Dimensions have to be right for formal consistency. The need for handling them correctly has noting to do with physical meaning. So your last sentence is strongly misleading.
Himangshu reveals he is a chemist. 
This explains everything and if for some incomprehensible reason you at all still cared what he had to say, you can safely know the opposite of anything he says is true.
Dear Harry and Ulrich, I agree to whatever you say. The wavefunction is indeed the most beautiful tool for a chemist. What I fail to comprehend is the utility of adding dimensions to particular quantity, which has no physical essence. Dimensions essentially as per as I believe is the conditional ability to add reality to its mathematical description so as to converge more in to the physical intuition, as in the vector potentials. To the wave function, till this date, association with reality has been denied/debated. Hence I made that statement. If you have an alternative and more convincing description, please free to rectify me. You guys have been into this field more than I have. And I will be happy to correct myself.
Some 
barges in 6 months later to semi-incoherently give an even worse answer than any answer given previously. Side note, his profile itself is rslurred:
I work on High Energy Physics theory. Standard model of particle physics, which is very well established today, has a fairly long list of unknown parameters. The number of unknown parameters is likely to increase once dark matter and dark energy problems are addressed.
Anyways, his contribution:
Actually it is a very good question.
Wikipedia says that
"For example, for a 3-dimensional wave function the amplitude
has a "bizarre" dimension [L^β3/2]."
Interesting point to note is that when you are solving an one
dimensional problem the probability amplitude has a
different dimension from what is quoted above. In a one
dimensional problem psi^2 dx is dimensionless so psi
has a dimension L^-1/2
In a two dimensional problem psi^2 dxdy has to be
dimensionless so psi has a dimension L^-1.
Luckily our protagonist is still monitoring the thread, passive aggression at the ready:
@Biswajoy
Actually not only the question is good. Good are also some answers. These render your's superfluous.
Ulrich: There is no harm in appreciating a good question or
a good answer. Many people might have identical thoughts.
I did not find a note in this thread which points to the fact that
the dimension of wave function depends on the type of problem
you are solving. Such as 1-D, 2-D or 3-D. This is a simple fact but
requires to be stressed specially for non-experts. That was the reason
behind my posting.
At this point this guy makes poor Ulrich miss even Saeid:
@Biswajoy
For your information: the second contribution (by Saeid Soheyli) gives the relation to space dimension explicitely.
Fast forward 4 years:
A few boring posts have happened but nothing interesting. Luckily a new lolcow has come to save the day, James Robinson:
szabo and ostlund - modern quantum chemistry, 1989, p42, wavefunction shown as a0^3/2, 2.5978E15 m^-3/2.
What did he mean by this? 
Ulrich is on the case!
Dear James,
unless you tell us how a comma separated pair of values can stand for a wavefunction and whether m stands for a particle mass or for the SI unit of length your contribution makes no sense to me.
a0 is 0.52918 Angstroms (1 bohr if memory is correct) and the m is meters, there was a typing error, the conversion factor for psi to si units is a0^-3/2, which is 2.5978E15 and the units are meters^(-3/2). Apologies.
James,
in addition to the dimension (in 3-dimensional position space --- as opposed to momentum space) you also give numerical values. As far as dimension is concerned, many of the previous answers gave this already. Hopefully you are aware of the fact that the numerical value is irrelevant (or meaning-less) in the context of the original question.
At this point this r-slur deserves to be beat to death with a stick and if anything our protagonist goes too easy on him.
Hi Ulrich, I meant to provide a numerical simple example.
In a physics examination the professor pointed out of the window to the sky and asked the student: You see the clouds all around --- why they don't fall down, given that they consist of water which undoubtedly is heavier than air? The student plays for time and asks: Which of the many clouds do you refer to?
The idea that a 'simple numerical example' would enlighten any aspect of our question looks to me not more reasonable than this student's question.
Waleed Alawsey
Thanks for the question
To this day (a year ago) people are still occasionally giving worse variations of the same answers in this 9 year old thread.
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Yes, the key is to give the correct answer to the question without any examples or equations so that when others include them in their answers, you can passive aggressively state that their reply is superfluous and nit pick it to death.
As any teacher will know, pedantry is the surest sign of erudition, and students must learn without specific examples which only serve to narrow the mind.
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I didn't even notice he didn't give any examples, adds a whole extra layer to our protagonist. His answer is still the best until Himangishu's until Himangishu ends his answer with rslurration.
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it wouldn't really be meaningful to give an example
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I think an example is useful for making things more concrete. βIt depends on the representationβ is true but doesnβt show how to go from representation to units, and most of the time people posting on these forums and students really just want answers to their problem sets/exams lol.
Our class solved the particle in a box in one dimension problem over and over until this shit clicked.
https://en.wikipedia.org/wiki/Particle_in_a_box
For rslurs, it spits out an equation that just says the particle has to be in the box (integral of the probability density over the length L of the box is 1).
https://wikimedia.org/api/rest_v1/media/math/render/svg/47d0d6b3a4882e7f4a6358c2bb8ac12490566109
The probability density is the same thing as the scalar product in this case is the wave function absβd and squared thing. Itβs akin to squaring a real number or getting the squared magnitude of a vector but accounts for fancier things like complex numbers. And for units, it just squares them.
Integrating over a length for units just means multiply by length.
You can pretty much read the units off that equation with that info:
wave_function_units^2 * length = 1
which gives:
wave_function_units = sqrt(1/length)
And now Ulrich would get on my butt for not including the two dimensional case or the momentum representation and not being precise enough about scalar products.
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If you left it at that without mentioning the general case I'd be team Ulrich tbh.
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Depends on the level. Some r-slur 2nd year probably doesn't know what a scalar product even is.
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Old Ulrich triggered me lol. I took a QM course through edX where there was someone like him on the class forums. Heβd give the right answer, but itβd never be specific or include an example that would make it actually useful to us rslurred students. Ulrichβs answer is the type that is an epiphany moment for students near end of the class when the big ideas finally snuggle into their brains, and they can actually reason about what theyβre doing instead of doing it rote.
Being a TA in college showed me that using many, many examples is king for understanding, and students are like babies. Putting yourself in their shoes when youβre teaching is like trying to remember what it was like to not know how to read and then explaining what sentences are.
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Theyre all nerds
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