I have had a bit of a non-standard career trajectory. I started out with an undergrad in engineering (aerospace), with a masters in applied physics, then a PhD in theoretical physics. Then not wanting to bounce between post-docs, I decided to try out industry and someone was kind enough to give me a chance at a full-stack role which was interesting but got a little boring after a year so I tried quantitative research at a bank for a couple years which ended up being mostly writing code and not much modeling. I craved more research so I looked elsewhere and found a role doing High Performance Computing for an Oil and Gas company's research center for a few years which was fun and I learned a lot but there wasn't much career growth in that role and due to some other family/financial constraints, I ended up leaving that to come back to NY to work at a bank as a quant developer.
At this point in my life, I really do enjoy being a developer (primarily C++ but I enjoy more theoretical/drier aspects as well and I often dabble in Haskell and functional programming and incorporate that in my work and personal projects). I have about 7 years of work experience after 6 years of grad school.
Lately, I have been battling with anxiety and depression and some of that has to do with me being not completely satisfied with my job. Although I enjoy programming, I was never the hacker teenager and didn't grow up with friends or family of coders. I have primarily seen myself as a problem solver throughout my life and I enjoy that. The main thing that I am lacking at this point and have missed throughout my life is a purpose. I would like to have a job where I can solve problems while making an impact of some kind, whether it be through writing code or something else. But I don't know how to go about finding what that might be. I am also feeling a little burnt out but I feel that it's a bit silly to be burnt out so early in my career.
Any advice is appreciated.
I'm not one of those people.
Tried therapy. Working with coaches. Experimented with drugs, both prescribed by psychopharmacologists, and ... the other kind.
Don't let this discourage you: most people don't seem to have this problem, finding their purpose.
If you truly and actually want a job solving problems and making an impact of some kind, as you say you do, then start with yourself: how could you solve this problem? What experiments might you design and execute? What questions would you seek answers to? What are your success criteria? Are there experts you could seek out for assistance?
Ultimately, you might find that you're only fooling yourself, saying you want a purpose because you've formed that opinion through life experience or some other influence, either internal or external.
It's also perfectly okay to just "be" and not have a purpose: this is probably the hardest thing for me, personally, to recognize as true and fully accept, too. But, it is.
In the end, my personal advice would be to spend some time, whether it's minutes, hours, or days, and write down a list of every single thing you like to do, no matter how trivial or whether you think it's "wrong" for you to like doing it or not. Don't focus on all the things you don't like doing, don't create a "cons" list of those things. Only focus on the "pros" and write them ALL down.
Then, once you've got a fairly extensive list, start creating groups or clusters (permutations, with reuse) of the things on the list, and look for patterns that emerge that suggest possible professions or at least things you could develop and get good at.
Then, start doing those things. The things you truly, genuinely like doing. Then, eventually, you can figure out how to monetize those activities. Almost everything in life is monetizable if you're willing to.
The challenge is finding the things you truly like doing and going out and doing them.