Hamiltonian for homogeneous system

In summary, the conversation is about simplifying the Hamiltonian for a homogeneous system using scaled coordinates. The first two terms can be converted easily, but there is some difficulty with the last term, which involves integrating over n squared divided by the distance between two points. The expression for the scaled coordinates is given, and there is confusion about how to reduce the two integration variables to just one. The issue seems to be related to some missing exponents in the expression, which can be rewritten in scaled coordinates as a simpler integral.
  • #1
greisen
76
0
I have to show that the hamiltonian for a homogeneous system can be simplified in scaled coordinates.
The first two terms I can convert to scaled coordinates <T>+<V> whereas I have some trouble for the last term

-½*[tex] \int d³r d³r' \frac{n²}{|r-r'|} [/tex]

where n is the density. The scaled coordinates can be expressed as \tilde{r}=\frac{a_0}{r_s} - r_s is a average distance between electrons and the expression can be written as

[tex] -\frac{3}{4\pi} \int d³\tilde{r} \frac{1}{\tilde{r}} [/tex]

I have some troubles getting the last part - how can the two d³r d³r' be reduced to d³\tilde{r} - any hints or advise appreciated
thanks in advance
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
I can see that some of the exponents have vanished so the integral which gives me problems is

- \int dr^{3} dr'^{3} \frac{n^{2}}{|r-r'|}

which in scaled coordinates can be written as

- \frac{3}{4*pi}\int d\tilde{r}^{3} \frac{1}{\tilde{r}}
 

1. What is a Hamiltonian for a homogeneous system?

A Hamiltonian for a homogeneous system is a mathematical equation that describes the total energy of a system that is invariant under translations in space. It takes into account the kinetic and potential energies of all the particles in the system.

2. How is the Hamiltonian for a homogeneous system different from other Hamiltonians?

The Hamiltonian for a homogeneous system differs from other Hamiltonians in that it only considers the total energy of the system, rather than individual particle energies. This is because the system is homogeneous, meaning that it is uniform and has no preferred direction or location.

3. What is the significance of the Hamiltonian for a homogeneous system?

The Hamiltonian for a homogeneous system is significant because it allows us to understand and predict the behavior of the system. It can be used to calculate the equations of motion and determine the stability and energy conservation of the system.

4. How is the Hamiltonian for a homogeneous system used in physics?

The Hamiltonian for a homogeneous system is used in classical mechanics and quantum mechanics to describe the dynamics of physical systems. It is also used in statistical mechanics to study the behavior of large systems and in thermodynamics to calculate the thermodynamic properties of a system.

5. Can the Hamiltonian for a homogeneous system be applied to non-uniform systems?

No, the Hamiltonian for a homogeneous system is only applicable to systems that are uniform and have no preferred direction or location. For non-uniform systems, different Hamiltonians must be used to accurately describe the system's energy and dynamics.

Similar threads

  • Advanced Physics Homework Help
Replies
5
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
302
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
5
Views
2K
  • Advanced Physics Homework Help
Replies
3
Views
628
  • Advanced Physics Homework Help
Replies
5
Views
831
  • Advanced Physics Homework Help
Replies
19
Views
702
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
1K
Back
Top