You should also think of how your attitude might change when you have professionally programmed for several years. It is of course good to like programming. But doing real projects that have users other than yourself is very different from just programming for fun. This is the kind of experience everyone is looking for: finishing real projects.
Also, many software companies will see you as overqualified for what they are doing. For example they might assume you will easily get bored.
Try to find a company full of like-minded people (maybe math/physics PhDs) and a job that is actually interesting to you. There will be no "overqualified" questions if interviewers understand you and your motivations well and you are enthusiastic about the job.
Experience is the best way to prove it though. If you have done well in a similar job no one will usually question your motivation.