# Ground state of harmonic oscillator

## Homework Statement

Verify that the ground state (n=0) wavefunction is an eigenstate of the harmonic
oscillator Hamiltonian. Using the explicit wavefunction of the ground state to calculate
the average potential energy <0|$$\hat{v}$$|0> and average kinetic energy <0|$$\hat{T}$$| 0>

## Homework Equations

$$\int^{\infty}_{0}$$(x$$^{2n}$$ e$$^{-ax^{2}}$$)dx=$$\frac{1x3x5x...x(2n-1)}{2^(n+1)a^n}$$$$\sqrt{\frac{\pi}{a}}$$

## The Attempt at a Solution

I did the ground state harmonic oscillation standard alteration with "a" and got $$\\Psi_{0}$$(x)=1/($$\pi^{1/4}$$$$\sqrt{x_{0}}$$)*e^-x^2/2x$$^{2}_{0}$$