Question

The Schrodinger equation for a free electron of mass m and energy $E$ written in terms of the wave function $\psi$ is $\frac{d^2\psi }{{dx}^2}+\frac{8{\pi }^{2}mE}{h^2}\psi =0$. The dimensions of the coefficient of $\psi$ in the second term must be

• A. $\left[M^1{L}^1\right]$
• B. $\left[L^{-2}\right]$
• C. $\left[L^{-3/2}\right]$
• D. $\left[M^1{L}^{-1}{T}^1\right]$

Answer 1

The correct option is B $\left[L^{-2}\right]$

The quantity whose dimension is to be determined $Q=\frac{8{\pi }^{2}mE}{h^2}$
We know, dimension of mass $m=\left[M\right]$
Dimension of energy $E=\left[M{L}^2{T}^{-2}\right]$
Dimension ofPlanck's constant $h=\left[M{L}^2{T}^{-1}\right]$
Thus dimensions of $Q=\frac{\left[M\right]\left[M{L}^2{T}^{-2}\right]}{[M{L}^2{T}^{-1}{]}^2}=\left[L^{-2}\right]$