# Analogues between QM- and CM N-body problem

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1. Jun 25, 2015

### olgerm

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

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:
$U_{System Potential Energy}(r_1,r_2,r_3,...,r_n,t)-\sum_{n=1}^{n_{max}}(\sum_{d=0}^{d_{max}}(\frac{d^2Ψ(r_1,r_2,r_3,...,r_n,t)}{dx_n^2})*\frac{ħ^2}{m_n})=i*ħ \frac{dΨ(r_1,r_2,r_3,...,r_n,t)}{dt}$
And to get wave function ψ we also need more information? What information could it be?
Could tihis information be function $f(r_1,r_2,r_3,...,r_n)=Ψ(r_1,r_2,r_3,...,r_n,t_{given}$)

If we knew $f(r_1,r_2,r_3,...,r_n)$ 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?

2. Jun 30, 2015