Minimum length in quantum gravity -- modified commutator versus modified operators

Quantum gravity suggests that we~should~modify the Heisenberg uncertainty relation to obtain a minimum length scale.~Previously modifying the~commutator was considered sufficient to lead to a minimum length scale. However, we~show~that the modification of the position and momentum operators is~more~crucial~than~the modified commutator when specifying the existence~of the minimum length scale~We do this by using well behaved trial wave functions to examine the modified uncertainty connected with the modified commutator and operators.