An H-1B petition is job and employer specific and thus any substantial change in the position such as job title, requirements, duties, requires the change be submitted through H-1B petition to USCIS.
Even if the position is in the same department at Ohio State, a change in previously approved employment H-1B petition must be submitted before the foreign national can start working in the new position.
Since the employee is in the H-1B status, the employee can begin working in the new position after the Office of International Affairs receives the official H-1B receipt notice from USCIS and on or after the requested start date of the petition. The change in employment H-1B petition takes 4-7 months.
See Step by Step for more details.