# Analogues between QM- and CM N-body problem

• olgerm
In summary, the general formulation of the N-body problem in QM is different from that in CM, and it can be solved by knowing the wave function ψ and the potential energy of the system. Additional information may be useful, but it is not necessary to solve the problem.
olgerm
Gold Member
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?

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.

## 1. What is the difference between the quantum mechanical (QM) and classical mechanical (CM) N-body problem?

The QM N-body problem describes the behavior of a system of particles at the quantum level, where particles can exist in multiple states simultaneously. The CM N-body problem, on the other hand, describes the behavior of a system of particles at the classical level, where particles are treated as point-like objects with well-defined positions and momenta.

## 2. How are the equations of motion different for the QM and CM N-body problems?

The equations of motion for the QM N-body problem are described by the Schrödinger equation, which is a differential equation in terms of the wave function. The equations of motion for the CM N-body problem are described by the laws of classical mechanics, such as Newton's laws of motion.

## 3. Can the QM and CM N-body problems be solved analytically?

While there are some special cases where the QM and CM N-body problems can be solved analytically, such as the Hydrogen atom in QM and the two-body problem in CM, in general, both problems require numerical methods for solution.

## 4. Are there any similarities between the QM and CM N-body problems?

Yes, there are some similarities between the two problems. Both describe the behavior of a system of particles, and both involve solving for the evolution of the system over time. Additionally, in certain limits, the equations of motion for both problems can converge to the same classical equations of motion.

## 5. How do the results of the QM and CM N-body problems differ?

The results of the QM N-body problem are probabilistic, meaning that they give the probability of finding a particle in a certain state. The results of the CM N-body problem, on the other hand, are deterministic, meaning that they give the exact position and momentum of each particle at any given time. Additionally, the QM N-body problem can exhibit phenomena such as quantum entanglement, which do not occur in the CM N-body problem.

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