Am I correct in assuming that if I convert my 40 compact with a conversion barrel to 9mm, that I also need 9mm compact magazines? Also, how about recoil springs? Thanks.

You can use factory S&W barrels which you can buy for $65.

The magazines will work, but the last shell may not feed because the follower may hit the slide lock before the magazine is actually empty.

The recoil spring is the same.

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The 9mm magazines are optional.