can i get reassurance that this is going to work and there is nothing wrong with it?

First, just a little about me, I"m patrified of talking to girls. I have social anxiety bad, and the best thing i can compare it to is freezing up before getting on a roller coaster or trying to
go sky diving but you freeze up before jumping out.I've been working on fixing this most of my life. Anyhow there is this hot girl in my math class in college. After class we were both
waiting for the bus that will take us to the dorms. It was raining but we waited outside anyhow we talked for like 40 minutes cause the bus didn't come. Just small talk about random things
working out, majors, food etc. She is so down to earth, and i didn't feel fearful talking to her.The next day i saw her with her back turned waiting for the bus again, i didn't approach her out
of fear but i wish i would of at least said hi, pretty sure she eventually saw me. I want a girlfriend more then anything in life at this point, I"m 21 and never had one because of my fear. This girl seems nice and she is seriously hot. There are only 2 weeks left of this semester though so i have to make my move quickly or chances are i wont see her again. I noticed she eats at the student buffet usually around the time i go to eat there. Would there be anything weird about me approaching her while she is eating and saying " Hey ___ , mind if i join you?" I think it sounds like an okay thing to say...but I"m not really familiar with her, aside from that one conversation we had and i'm afraid she will think I"m a creep or something. Also what if she already has a boyfriend? i tried searching for her on facebook but could not find her... What if when I"m sitting with her there is an awkward silence? What if i don't even see her at the student buffet again!!? i can't approach her after math class because she almost always waits for the bus with her girlfriend .. Please advice :upset:

To put it bluntly, a girl doesn't want a guy who is afraid of them.
You're going to have to put yourself out there and, yes, it's always possible that she will reject you. She will reject you anyways if she sees you are more like a scared rodent then a strong man. I've been in your shoes man. I used to be afraid to truly be myself but once I eventually opened up I found more people liked me/were interested in me and to be honest I didn't care regardless because I don't need to feel like I need to impress other people to be who I am.

You can either put yourself out there and find out if you could possibly date her, or you'll just end up wondering what if until she disappears or you see her with someone else.

Yes, you should ask to join her for lunch. It's just sharing a table, not having a kid.

The thing about your skydiving analogy, is yes you are scared but once you do it, you will find it's amazing. Ask the girl if she's like to get together to study for finals. Buy her some coffee or a coke to help study.

Some food for thought:

Asking girls out is a lot like playing the lottery: You can't win if you don't play the game... and if you play and lose, what are you out? Not much. What's the worst she can do? Say no? So what?

Seriously, you need to rationalize your fears and get over them. They do nothing for you.

I dont think you have a fear of talking to girls, youve a fear of being rejected. Rejection is a part of life and you need to learn to deal with. You coukd send out 100 cvs and not get one imterview, go to 5interviews and not get one job, your friends could cancel plans last minute, you could be told no you cant have that day off-you have to work or 9/10 girls could turn you down.

It doesnt mean theres something wrong with you or your not good enough.

So what if she turns you down or has a bf? You can just say "hes a lucky guy" smile and walk away. Its only a big deal if you make it one

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I would definetly go for it.

A classic case of 'nothing ventured, nothing gained'. There's no harm in trying. You've been having small talks with her, which is a great start. You can slowly gauge whether or not she's interested and/or taken by simply talking to her. So yeah, go for it!