Additional SQL Server features and topics not covered by specific categories
The delete statement will not be able to seek the index to find the rows, since the purchase_date isn't the first column in the index key. It might scan the index, you'd have to consult the execution plan to see what it decided.
You can execute that create index command, but it will be blocked by the delete operation since it requires a very restrictive locked - and the delete also has conflicting locks. When the delete has finished and released it locks, the create index can proceed.