/sci/ - Science & Math

>>9769567
Not the original anon, but I do mathematical physics.
>Schrodinger equation
>linear
Only in mean field approximation my child, and this approximation can often miss phase transitions in strongly-correlated systems.
>so how exactly hilber spacea come to play?
In general the Hamiltonian [math]\hat{H}[/math] is defined as the generator of time evolution along the physical trajectories on a symplectic Hilbert space [math]\mathcal{H}[/math]. The Schrodinger's equation [math]i\dot{\psi} = \hat{H}\psi[/math] is then literally just the usual flow equation due to time evolution. The fact that it's the total energy comes from Noether's theorem (time translation symmetry = conservation of energy).
Now what exactly that Hilbert space is comes from Born's postulate: i.e. that wavefunctions are sections of a Hilbert ([math]L^2[/math] Hermitian) line bundle [math]L\rightarrow M[/math] with a structure group [math]U(1)[/math], which implies probability conservation (again, by Noether's). If we denote [math]\rho: U(1) \rightarrow S^1[/math] the representation and [math]L^2[/math] the representation space, the associated vector bundle is [math]L\times_{U(1)}L^2 \rightarrow M[/math], where the first factor (Hermitian line bundle) gives the [math]\mathbb{C}[/math]-valued-ness of the wavefunction while the second factor gives the square integrability over your position space [math]M[/math].
The fact that PDE's arise from, for instance, Schrodinger's equation is precisely due to this representation. The Banach algebra of bounded operators [math]\mathcal{B}(\mathcal{H})[/math] on the Hilbert space admits a representation as differential operators on [math]L^2[/math]. The Hamiltonian [math]\hat{H}\in\mathcal{B}(\mathcal{H})[/math] is hence also represented as a differential operator, so Schrodinger equation in general is a differential equation, which is not necessarily linear.
There, you now know the entirety of QM.