The spatial gradient of momentum p with respect to position x is given by the relation dp dx = ( x^2 h ) , where and are constants, and h is Planck's constant. The dimensions of are

Step-by-step solution

Dimensions of momentum [p] = [ MLT ⁻¹] Dimensions of spatial gradient of momentum [ dp dx ] = [ MLT ⁻¹] [ L ] = [ MT ⁻¹] Dimensions of Planck's constant [h] = [ ML ^2 T ⁻¹] The argument of the sine function must be dimensionless: [ x^2 h ] = [ M ^0 L ^0 T ^0] [ ] = [h] [x^2] = [ ML ^2 T ⁻¹] [ L ^2] = [ MT ⁻¹] Now, equating the dimensions of the LHS and RHS: [ dp dx ] = [ ] [ ] [ ] = [ ] [ dp dx ] = [ MT ⁻¹] [ MT ⁻¹] = [ M ^2 L ^0 T ⁻²] Answer: [ M ^2 L ^0 T ⁻² ]