There is a professor with whom I've taken two courses so far, and who has also agreed to write me a reference letter for graduate school when I complete my undergraduate program. I usually write to all professors "Dear Prof. [...]" or "Hello Prof. [...]", but some professors reply to me as "Hi [...]", which is OK. However, I'm wondering if it is also okay to reply "Hi Prof. [...]" in North America?
In this type of situation I always take the lead from the person I am writing to if they are more senior than me. This means that I start off the conversation using "Dear Professor X Y" and only switch to using "Hi X" once they do.
I use the same rule of thumb for working out when to switch from using titles and surnames to first names. In most email communications I find this is normally after only one or two emails.
In other words, formality and respect are nice, but so is being friendly and saying hi.
Note: academic cultures do vary, and I am basing this on my experiences in the UK and EU. My impression is that in the USA there is a greater social distance between students and academics and therefore more formality is expected.
To be honest, I don't pay attention to the salutation. Say whatever you want, and I won't care. However, some people do. Some people think that faculty-student communication should be formal business communication. Some of those people will get upset with you or think less of you if you don't meet that formal expectation.
If you're concerned then I would ask the person how they would like to be addressed. Doing so is not considered rude- it is especially understandable if English is not your first language. Most professors are happy to help.
Being curious, I just looked through the last 20 emails I got from students. Their introductions were:
No salutation - 6 "Dr. X," - 4 "Professor X," - 3 "Hello Dr. X," - 2 "Hi Professor," "Dear Dr. X," "Hello," "Hello Dr. X!" "Hello again Dr. X!"
So based on this small and unscientific sample, it would seem that starting your emails with "Hi Professor X" is an uncommon thing to do.
Rather than answer your question directly, which other answers have already done quite adequately, I'll address it at a higher level of abstraction. Sociologists define the notion of closure, which is the tendency of groups of people to restrict access to the group and its resources using various mechanisms. One such mechanism is the use of unwritten norms of speech and behavior, which have the effect of making outsiders to the group appear as uncomfortable and "clueless" when they try to operate within the confines of the group.
While I am not an expert in sociology, it seems to me that your question about email salutations is a small example of this phenomenon. The truth is that, in North American academia at least, there is no importance whatsoever to whether one uses "hi" or "hello" in an email greeting. When professors talk about whether a given greeting seems "professional", or "clueless", or some such nonsense, what they are really thinking (at a subconscious level that is) is whether the greeting signals to them that you are a legitimate member of the group - someone who has been around long enough to learn the unwritten codes of behavior. If you are perceived as a member of the group, your email and you will receive a more favorable treatment as a result.
The moral of looking at things at this level of generality is that in my opinion it's best not to waste too much time and energy about trivial things like this. It saddens me that academia uses subtle mechanisms like salutations and titles (which also get discussed a lot here on academia.se) to make people who are new to academia feel uncomfortable and ill-at-ease, but I think the best way to fight such phenomena is to not care - we would all be much better off spending our time thinking about the substance of what we want to say in our emails rather than whether "hi" is an appropriate greeting.
For what it's worth, I tend to interpret "informality/familiarity" from very young people as a sign of naivete/cluelessness. As in "oh, we're all over 21, so are peers." Or, sometimes, this can arise from a spamming effort (of students to faculty), in which case I am less inclined to be tolerant. E.g., when the email is bcc... or, even more silly, cc, with 50 names on the list.
Forms of address inevitably are an opportunity to "demonstrate formal respect". Yes, one can argue on ideological grounds that honorifics are misguided, and so on, and I have no powerful counter-argument... except on my own behalf, that if a very young person sees themself as my peer in a professional sense (as opposed to "moral" sense, etc.), then I wonder why they'd bother emailing me at all...
That is, either my (decades-in-the-acquisition) expertise is potentially helpful to them, in which case our relationship cannot possibly be intellectually "equal", or, if "intellectual equality" is an axiom, well, ok, then why involve me?
That is, I do not view my seniority merely as "gate-keeping", but as having given me the possibility of having thought about many things for a long time, and, thus, "being more expert" than people who've not had that time. That's all I have to offer.
I don't know about North America, but here in the UK I have always just used
'Hi [First Name]'
If it's my very first time contacting someone I might use 'Dear' first but I never bothered with titles apart from my first year at university where I was too concerned with proper formality as a naïve fresher.
It hasn't caused me any trouble yet, or perhaps I have just been very oblivious to subtle requests to change my tone.
You can probably tell from the other answers that it really depends. It depends on the the environment of the school, the culture of the department, the level of the student, and the crotchetiness of the particular professor.
Strictly speaking, there is nothing rude about "Hi Prof. [...]," but you can't go wrong with being more formal. If Prof. [...] wants to be less formal, they'll say so.
Typically you want to convey both some familiarity and some respect. The way this happens varies by culture.
In the UK I expect to usually be addressed in an email as
until I am on more familiar terms with the sender. 'Dear' shows politeness, while 'Jessica' shows friendliness.
In North America I found I was most often addressed by students as
As a Brit I found this very annoying, as from my British viewpoint it is bordering on rude, and I am not a Prof as I use the term. But I could see that this was the local equivalent of 'Dear Jessica', which the respect and friendliness denoted in different ways.
So for North America I would say 'Hi Prof' is fine (indeed, fairly standard).
What I would warn against is going too far either towards informality or formality. I've often had
and sometimes no greeting or sign-off at all, just (typically) a demand. These suggest that the sender has forgotten either the nature of our relationship or the need to modify their behaviour accordingly.
Going the other way, I sometimes get
Dear Dr. X
accompanied by full name and number for the student. Mostly this seems to be some degree of 'I pay your salary so do what I want', but sometimes it seems the student thinks I don't remember who they are, even though I address them by name when I talk to them each week.
Depends, if they're human a simple greeting will suffice. If they're morally inclined by their title, you need to treat your message like you would your resume.
Best observation don't take the lead, wait for them to make or engage or send the first email. Look at the message's header, body and footer, the presentation of how their message is delivered. Then compose a reply based on that analysis. Just don't treat the message like you're talking to your best pal, treat it with an emphasis of student/teacher conversation. Know the bounds of moral presentations, don't just drop illegible or short case words. Use proper sentences and English to convey your point/message.
Also keep the email short, unless you're including attachments or extra edits.