I am writing a motivation letter where I have to write why I am directly applying for a Ph.D. position without masters. What can I write so that it seems valid and convincing to the professor? (The Ph.D. will take 4 years anyway for people who will apply from bachelor, the 1-year preliminary course then 3 years Ph.D.)
For any application, for school or a job, the key thing is to make sure that everything in your application indicates a successful future. In Germany, where a masters is mostly "expected" it might be hard to convince an advisor that you are ready unless you have done some exceptional things otherwise. Having the master's gives the PI a better chance to evaluate your future, based on your past. So, there, and in some similar places, the hurdle is pretty high.
In the US, by contrast, less is expected since the program is longer and involves coursework that can be used to help make predictions. Also, in the US, you choose the advisor later, giving everyone a better chance to evaluate.
But, what you can write to convince people must involve listing your past successes and other suitability for the proposed position. Just being a good student in a classroom situation probably isn't enough. A different situation if you have some research behind you and some publications.
Convince with deeds, not words, but get letters of recommendation that back you up.