How to know I have done enough work in one semester?

Everyone works at their own pace; moreover, the pace can fluctuate a lot depending on the time of year, your personal life, and "position of stars in the sky". Even though the semester seems like a decent amount of time to average out those fluctuations, I don't think it is.

I would try to use the following criteria to estimate the term success:

• feedback from your advisor
• how did the term go compared to the original plan (yep, here I assume that you make the plans for your term/month/week and correct them accordingly. Hopefully, some milestones are discussed with the advisor as well)
• feedback from your committee (in some universities, Ph.D. students meet with their committee regularly or send them the progress report to hear their feedback)

I would certainly not recommend comparing your progress with other students because they are different, they have different goals, and you don't have complete information about their progress either. It is very easy to get discouraged for no reason.

To sum it up, I would stress the importance of initial planning and correcting the plans throughout the term. Then, you will have a very good measure of your success. You will end up with a different question of "how to plan", but that is a totally different problem.

Quality is more important than quantity in research.

One solid proof that proves an important result likely matters far more than a bunch of smaller, less significant proofs. Similarly, one big research advance that shows long-term promise will matter more than a bunch of small experiments that don't open any new doors research-wise.

Also remember that most research does not succeed at first! I tell my students when they're getting started that research will likely be a bunch of failures punctuated by the occasional success. This is not to discourage them, but rather to get them to realize that the process is slow and winding. At this early stage, I want to get them familiar with the tools they need and the skills they must develop, rather than focusing on early breakthroughs.

So don't try to plan your research too closely—you should have some goalposts and markers in mind so that you're not a perpetual grad student, but you don't need to micromanage your work down to the day or week (other than keeping track of external deadlines). Focus on doing the best work you can, and talk to your advisor about your progress.

Worrying about whether you've been productive enough is not going to make you more productive.

I suggest that you diversify your goals for yourself. You mentioned two prongs to your work: reading papers, and writing proofs. See if you can find some additional prongs to add to your definition of "being productive." Here are a few ideas to get you started:

• Get to know the publications in your area, and figure out what parts of each are of most interest to you.

• Regularly browse the most recent issues of those publications. Gradually get better at skimming articles to get a general idea what they're about.

• Develop a system for cataloguing your notes about your journal reading.

• Attend department seminars. Try to gradually increase your level of understanding of these talks. When you feel ready, start to ask questions, either during the Q&A part at the end, or afterwards, more informally.

• Form a study group with some fellow students.

• Attend some thesis defenses.

• Attend a conference.

• Start figuring out what you think makes a good talk.

• Start working on your writing and powerpoint skills.

I don't mean that you should necessarily try to do all of the above, and of course, the papers and the proofs do need to continue being in center stage. But a bit of diversification may help you worry less and produce more.