A change of initial amounts to a "name change" as far as the federal government is concerned.
If your name has been legally changed due to marriage or a court order within one year from the date of issuance of passport, then you would simply submit a completed
DS-5504, "Application for a U.S. Passport: Name Change, Data Correction, and Limited Passport Book Replacement Form", a certified copy of your marriage certificate or your name change court decree, two recent color photographs (2” by 2”), and your current valid passport to the address listed on the form. There is not, I believe, any fee for this (but the form should tell you).
If your name has changed by other means and/or it has been more than one year from the date of issuance of the passport, then you must submit a
DS-82 renewal application and pay all applicable fees.
Photocopies and notarized copies are not acceptable for passport purposes.