[jennyberries]

I'm supposed to write the ground state of the harmonic oscillator as a linear combination of the free particle eigenstates but I'm a bit confused as to how to go about it.

So, I know that using the free particle eigenstates, phi(x,t)=integral a(k,t)*exp(i*k*t) dk, that a(k,t)=a(k)exp(-i*h*k**2*t/2*m) and that a(k)=(1/2*pi)*integral exp(-i*k*x)*phi(x,0) dx but I'm a bit confused about what to use for phi(x,0), is it the ground state for the harmonic oscillator, (m*omega/pi*h)**(1/4)*exp(-m*omega*x**2/2*h)? Or something else? Because using that one, won't I just get back that function times some sort of time dependency?

Any insight would be much appreciated. Thanks!