A Commentary, Critical, Practical, and Explanatory on the Old and New Testaments, by Robert Jamieson, A.R. Fausset and David Brown  at sacred-texts.com
psa 137:1This Psalm records the mourning of the captive Israelites, and a prayer and prediction respecting the destruction of their enemies. (Psa 137:1-9)
rivers of Babylon--the name of the city used for the whole country.
remembered Zion--or, Jerusalem, as in Psa 132:13.
psa 137:2upon the willows--which may have grown there then, if not now; as the palm, which was once common, is now rare in Palestine.
psa 137:3Whether the request was in curiosity or derision, the answer intimates that a compliance was incongruous with their mournful feelings (Pro 25:20).
psa 137:5For joyful songs would imply forgetfulness of their desolated homes and fallen Church. The solemn imprecations on the hand and tongue, if thus forgetful, relate to the cunning or skill in playing, and the power of singing.
psa 137:7Remember . . . the children of Edom--(Compare Psa 132:1), that is, to punish.
the day of Jerusalem--its downfall (Lam 4:21-22; Oba 1:11-13).
psa 137:8daughter of Babylon--the people (Psa 9:13). Their destruction had been abundantly foretold (Isa 13:14; Jer 51:23). For the terribleness of that destruction, God's righteous judgment, and not the passions of the chafed Israelites, was responsible.