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I'm an undergraduate who is going to apply for scholarships and to graduate school in physics in the coming months. At the start of last semester in August, I tried to join a group of a respected professor, where I was assigned a theoretical project. I've met with him maybe three times during the few coming weeks, before we mutually agreed that the assigned problem is pretty much a dead end. The professor had no idea for another suitable project, so I decided to join a different group, with which I stayed until very recently, when an analogous issue occured (but in this case I have actually done some work until that happened).

I was debating whether this "project" deserves being mentioned in my resume, at least for the sake of continuity, and so far have received contrary pieces of advice. Currently, that project is listed on my resume as follows:

Worked with […] at […] on a project attempting to reformulate the definition of quantum-mechanical work for autonomous Hamiltonians.

I am certainly not going to ask for a letter of recommendation from that professor since there are others who know me much better. I only want to know whether the inclusion of the above description in my resume would have a positive or negative influence on my application.

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I think for an undergraduate student, it is quite valuable to show that they have experience working on a diverse range of problems with different people, even if some of such projects do not lead to anything substantial.

Usually for grad school application and other fellowships there is an essay component. It may also be useful to mention such projects (briefly), and how such experience clarified your research interest/taught you how to do research/provided cross-disciplinary/cultural experience, etc. The exact messaging of course depends on what you think the program is looking for from the applicants.

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There is, or should be, no shame in not completing an undergraduate research project assuming that it was real research, not just an exploration into a relatively known space. Real research = exploring the unknown. Real research results can't be scheduled to meet with academic scheduling. It is, in its very nature, open ended, both in time and in results. This is why, in fact, it is impossible to even give a firm deadline in graduate studies. It also explains why some academics produce few papers with a long interval in between.

So, yes, list them. Given the constraints, it is a good thing that you worked on tough problems, rather than a bad thing that you didn't complete them.

And, when you say failed, I hope you just meant "didn't complete" rather than received a failing grade. Note that your professor also "failed" in some sense by suggesting a project that had a very uncertain probability of success.

It may well be that this professor would be a good reference, though a face to face conversation would help you know whether such a letter would be supportive. I wouldn't dismiss the possibility out of hand.

As a doctoral student, I worked on three problems in succession. It was a "three bears" situation. The first was too easy, and I produced theorems every day. It was fun, but my advisor and I agreed there was no real substance. The second was too hard and nothing I could think of (nor my advisor) could but the slightest dent in the shell of it. Again, abandonment was the right path. The third problem was just right and I was able to complete the research with good results in about a year or so. But, it wasn't over until it was over.

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