Questions about the Dimensions Analysts changing the equation

[Guaicai]

The first equation of the left hand side is the Schrodinger Equation of the Harmonic Oscillator.The second equation of the left hand side is the new equation doing by the Dimensions Analysts. But I don't understand how the new equation come from according with Dimension Analysts of the right hand side...
How the new equation have the extra term ,why not just E=ħw ?[emoji53]

 

[blue_leaf77]

$P_o$ and $X_o$ have the dimension of momentum and displacement respectively. Thus, the terms inside the bracket in the 2nd equation in the LHS is dimensionless and the dimension of $\hat{E}$ is determined by $\hbar \omega$ alone.

How the new equation have the extra term ,why not just E=ħw ?

The new equation is obtained by factoring out $\hbar \omega$ from the first equation.