The weapon doesn't change. The damage may increase if your skill gets higher, or if you pick perks that raise damage, but the weapon itself doesn't scale with your level.

Do Skyrim weapons level with you?

Weapons you own do not level with you, however as you level up to certain markers better or different weapons are 'released' into the game world. What will affect your weapon damage is you skill in the weapon type and any perks you take that affect damage. A weapon's damage doesn't level with you.

Do Items scale Skyrim?

The item is scaled in power based on the player's level at the time the reward is given. Some leveled items become stronger in both their physical and magical characteristics, while others only improve their enchantments. The items do not improve in quality when your character levels up.