D. Tong in his lecture note "String Theory" has mentioned on pages 11 and 12 on the contest of relativistic point particle that:
"We introduce a wavefunction $\Psi (X)$ which satisfies the usual Schrödinger equation $$i\frac{\partial \Psi}{\partial \tau}=H\Psi.\tag{p.11}$$ But, computing the Hamiltonian $H=\dot{X}^{\mu}p_{\mu}-L$ we find that it vanishes $H=0$. It is simply telling that the wavefunction doesn't depend on $\tau$ .... "
Here $$S=m\int d\tau \sqrt{-\dot{X}^{\mu}\dot{X}^{\nu}\eta_{\mu\nu}}.\tag{1.2}$$
In another paragraph he mentioned that $\Psi (X)$ satisfies in an operator equation (the Klein-Gordon equation) $$\Big( -\frac{\partial}{\partial X^{\mu}}\frac{\partial}{\partial X^{\nu}}\eta^{\mu\nu}+m^2\Big) \Psi (X)=0\tag{1.7}$$ (which comes from mass-shell condition). My main question is that: Doesn't the Schrödinger equation matter anymore in string theory? Do we have to use the Klein-Gordon equation instead of Schrödinger equation in string theory? I'm a little confused here. I was wondering if someone could explain me a little bit more.
