I will graduate with a physics degree in a few years, since starting I've become more and more interested in programming.
My question is will I be able to compete with Computer Science graduates, assuming I know the relevant programming languages and have some experience?
Moreover, much of what needs to be done with programming doesn't require CS knowledge as such, but rather a lot of ancillary skills that aren't taught in any school; some of them are barely even mentioned in books. There you and the CS majors start out tied.
Conspicuous among that set of skills are business skills: Not MBA stuff, but things like "here is a potential customer: what are their actual problems, can I help solve them, can I prove that to myself, can I prove that to them using words they will understand and like, will they really pay me?" So much of your work will be about these things, in programming or in physics, and CS training doesn't teach them any more than stat mech class does.