وَإِذَا كَانَ الْأَمْرُ مُتَوَقَّعَ الْحُصُولِ، فَإِنَّ تَرَقُّبَهُ يُسَمَّى تَرَجِّيًا
When an outcome is expected, anticipating it is called rajāʾ (hope).
Tamannī concerns a desired but unexpected result. Tarajjī concerns a desired result that remains reasonably anticipated. Both may involve something favorable; the difference lies in the speaker’s expectation.
لَيْتَ الشَّبَابَ يَعُودُ
If only youth would return.
لَعَلَّ الْمَرِيضَ يَشْفَى
It is hoped that the patient recovers.
The first recalls an unavailable past; the second looks toward a possible recovery.