Center of Mass Problem: Treating Corrections

[stefano]

I don't know how to treat a center of mass corrections.
Suppose to have a Slater determinant of N single-particle wavefunctions. If I make a traslation to center of mass of system, this is mean to move all particle in the following way:

R_i ---> R_i - (sum_j {R_j}/N)

But how have I to correct the gradient and laplacian terms?

I think that there would be a kinetic energy correction. Is it true?

Thank's!

 

[santoshroy]

Since the gradient and laplacian contain "coordinate" (d/dx, d2/dx2..)terms one has to replace to coordinate with c.m. coordinates, there will ofcourse will be a kinetic energy correction as K.E. contains terms like (d2/dx2).
In another words Physically kinetic energy is not invariant in different coordinate farmes.

 

[R0man]

The center of mass problem in physics involves determining the point in a system where the total mass can be considered to be concentrated. This is an important concept in many areas of physics, including mechanics and quantum mechanics. In the context of treating corrections for the center of mass, it is important to understand the implications of this point for the system's wavefunctions.

In the case of a Slater determinant of N single-particle wavefunctions, the center of mass correction involves moving all particles in the system in a specific way. This is done by subtracting the sum of all the particle positions from each individual particle's position, divided by the total number of particles. This results in a new position for each particle that is centered around the center of mass of the system.

However, this correction also affects the gradient and laplacian terms in the system's wavefunction. The gradient and laplacian operators are mathematical tools used to describe the behavior of particles in a system. When the center of mass correction is applied, these operators must also be adjusted accordingly.

In order to correctly treat these corrections, it is important to consider the kinetic energy term in the system's Hamiltonian. The Hamiltonian is a mathematical operator that describes the total energy of a system. When the center of mass correction is applied, the kinetic energy term must also be adjusted to account for the new positions of the particles.

In conclusion, treating corrections for the center of mass in a system involves adjusting not only the positions of the particles, but also the gradient and laplacian terms and the kinetic energy term in the Hamiltonian. This is an important consideration in accurately describing the behavior of particles in a system and should be carefully addressed in any calculations or simulations involving the center of mass.