Analogues between QM- and CM N-body problem

[olgerm]

In CM general formulation of N-body problem is:
[itex]x(N;D;T) = \iint \sum_{n=0}^{N_{max}} (\frac {(x(N;D;t)-x(n;D;t))*(m_N*m_n*G+q_N*q_n/(4*π*ε_0))}{(\sum_{d=0}^{D_{max}}((x(N;d;t)-x(n;d;t))^2))^{3/2}*m_N}) \, dt^2[/itex]

Where x(N;D;T) is D´th coordinate of N´th body at time T.
But to get equation of motion you need more information for example: speed and velocity of all bodies at given time.

Is it analogues in QM where general formulation of N-body problem is:
[itex]U_{System Potential Energy}(r_1,r_2,r_3,...,r_n,t)-\sum_{n=1}^{n_{max}}(\sum_{d=0}^{d_{max}}(\frac{d^2Ψ(r_1,r_2,r_3,...,r_n,t)}{dx_n^2})*\frac{ħ^2}{m_n})=i*ħ \frac{dΨ(r_1,r_2,r_3,...,r_n,t)}{dt}[/itex]
And to get wave function ψ we also need more information? What information could it be?
Could tihis information be function [itex]f(r_1,r_2,r_3,...,r_n)=Ψ(r_1,r_2,r_3,...,r_n,t_{given}[/itex])

If we knew [itex]f(r_1,r_2,r_3,...,r_n)[/itex] then it were possible to solve QM N-body problem or I also had to use condition that wave function has to be continuous function?

 

[eljose79]

I would like to clarify that the general formulation of the N-body problem in classical mechanics (CM) is different from that in quantum mechanics (QM). In CM, the equation of motion can be derived from the positions and velocities of all bodies at a given time. In QM, the equation of motion is described by the Schrödinger equation, which includes the potential energy of the system and the second derivative of the wave function with respect to the position of each particle.

To solve the QM N-body problem, we need to know the wave function ψ, which describes the quantum state of the system. This wave function is a continuous function of the positions of all particles, and it is determined by the initial conditions and the system potential energy. Therefore, we do not need any additional information to solve the QM N-body problem, as long as we know the wave function and the potential energy.

In some cases, it may be helpful to have additional information, such as the symmetry of the system or the boundary conditions. However, these are not necessary to solve the problem. The wave function ψ is the key to solving the QM N-body problem, and it is determined by the initial conditions and the potential energy of the system.