Note that for 2D systems, as the effective potential takes the form V eff ( r ) = V ( r ) − − ℏ ℏ 2 ( ℓ ℓ 2 − − 1 4 ) 2 m r 2 , so Langer correction goes: ( ℓ ℓ 2 − − 1 4 ) → → ℓ ℓ 2 . This manipulation is also equivalent to insert a 1/4 constant factor whenever ℓ ℓ 2 appears.