# Inner products harmonic oscillator

How do I compute these inner products;

$$(\Delta x)^2 = \frac{\int_R \!x^2\psi_m\psi_mdx}{\int_R \!\psi_m\psi_mdx}$$

$$(\Delta p)^2 = \frac{\int_R \!\psi_m\frac{d^2}{dx^2}\psi_mdx}{\int_R \!\psi_m\psi_mdx}$$

Where $$\psi(x) = h(x)e^{-x^2/2}$$

$$h_0 = 1$$

$$h_1 = 2x$$

$$h_2 = 2x^2-1$$

$$h_3 = \frac{4x^3}{3}-2x$$

and how do I compute:

$$\Delta x\Delta p$$

• It is far easier to compute these things using ladder operators. – jobe Feb 24 at 13:11