Well first of all, congratulations on your quit!! This is a HUGE decision and I'm so glad you made it. When I quit, I also had trouble breathing and my chest felt tight and heavy like you say. This lasted a few days, so in my opinion this is normal, lol. This certainly doesn't happen to everyone though, so the word "normal" becomes relative.
Try not to let it panic you because every day will get better, you'll see. It takes 72 hours for the nicotine to leave your system so hang in there.
Keep posting and let us know how you are.
Take care.........
I noticed several posts that talked about similar feelings and I wonder if you have read those posts?
I believe it is common but perhaps you should see your doctor.
I smoked for so long that I was always feeling short of breath. How long have you been smoking? You might be feeling a little anxious?
Sorry I couldn't be of more help but I'm glad you have taken the first step and I wish you good luck : )
Kathy Jo