[Q]Hamiltonian of many body problem.

[good_phy]

Hi

I thought Hamiltonian for two particle system is $H = \frac{-\hbar^2}{2m}\frac{\partial}{\partial x_{1}} + \frac{-\hbar^2}{2m}\frac{\partial}{\partial x_{2}} + v(x)$

But is it right? I just want to know This way is right to construct every hamiltonian of many

body system.

What i want to know more is How can we solve this problem?

I have heard that computational methode can be used after transforming wave equation to

matrix form. Is it right?


Do you have any recommandation practice to improve integral and solve difficult differential

equation ?

 

[Manchot]

First, I should say that the derivatives in your equation should be second derivatives, not first derivatives. Secondly, while you are correct that a general quantum mechanical system has the above form (derivative error aside), there are additional constraints imposed by the types of particles you are considering. If they are indistinguishable, the state of the system must be either symmetric or antisymmetric under exchange of the two particles (depending on whether they are bosons or fermions).

Obviously, if you want to solve the problem computationally, how you go about doing so will vary depending on the form of the potential. For example, if the two particles are weakly coupled, then you can get away with solving the individual particle wavefunctions and perturbatively adding back the coupling term. Or, if one particle is much heavier than the other (e.g., the hydrogen atom), the Born-Oppenheimer approximation would work nicely. In most cases, you won't find a nice closed-form solution, and some approximation is necessary.

 

[Entropia]

Hi there,

The Hamiltonian for a many body system is slightly different than what you have written for a two particle system. For a many body system, the Hamiltonian is given by H = \sum_{i=1}^{N}\frac{-\hbar^2}{2m}\frac{\partial}{\partial x_{i}} + \sum_{i=1}^{N}v(x_{i}), where N is the number of particles in the system. This takes into account the kinetic energy of each particle as well as their interactions through the potential energy term, v(x).

To solve the many body problem, there are various methods that can be used depending on the specific system and its properties. One approach is to use computational methods, as you mentioned, where the wave equation is transformed into a matrix form and then solved numerically. Another approach is to use perturbation theory or variational methods, which involve approximating the solution and then improving upon it iteratively.

To improve your skills in solving integrals and differential equations, I would recommend practicing regularly and using resources such as textbooks, online tutorials, and problem sets. It is also helpful to break down complex problems into smaller, more manageable parts and to seek guidance from a mentor or professor if needed. With practice and determination, you will become more comfortable and skilled in solving these types of problems.