What is the Partition Function for a Singular Exponential Hamiltonian?

  • Thread starter jarvGrad
  • Start date
In summary, the problem is that the integral of the exponential of the Hamiltonian is singular. There is a way to solve for the integral, but it is more difficult than just changing the variable and integrating over the hypersphere.
  • #1
jarvGrad
3
0
I need to find the integral as follows:

I am given a Hamiltonian of the form:

[tex]\

H=\Sigma {(x_n+d y_n)^2}< 2 m E

[/tex]

(This should be a sum over n, but its not showing in the preview)


we integrate the exponential in n-space as
[tex]\
\begin{equation}
\int \exp{H} d^{3n}x d^{3n}y
\end{equation}
[/tex]
so that
[tex]\
\begin{equation}
\int \exp{(x+dy)^2} d^{3n}x d^{3n}y
\end{equation}
[/tex]
where (x+dy)^2 < E

I found a solution that tells me
[tex]\
\begin{equation}
\int \exp{(ax^2+bxy+cx^2)^2} d^{3n}x d^{3n}y
\end{equation}
[/tex]
which equals

[tex]\
\begin{equation}
\pi^{m/2}/{det[A]}
\end{equation}
[/tex]
where A is the 2-D matrix
A=[a b
b c]

However, the determinant is zero as I am given

[tex]\

x^2+2mwxy+(mwy)^2

[\tex]

so this doesn't work. I found this solution at http://srikant.org/thesis/node13.html .
There is a bit more work shown on the website. My professor assured me that the solution is closed form.
 
Last edited:
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  • #2
Do you mean
[tex]
\
\begin{equation}
\int \exp{(ax^2+bxy+cy^2)^2} d^{3n}x d^{3n}y
\end{equation}

[/tex]

Why not diagonalise A by doing a change of variables in you integral, or possibly a proof by induction?
 
  • #3
Oops sorry I miss wrote the integral in one of the lines above. It should read

[tex]\
Z=\int exp[ax^2+bxy+cy^2] d^{3n}x d^{3n}y
[/tex]

my problem happens to be specified that the exponential is of the form

\\x^2+(m w y)^2 + 2mwxyThis makes the matrix in the above solution singular and the integral cannot be performed.

**************************************
Note the problem that I was working on should not be solved like this. All I needed to do for that problem was do a simple variable change and integrate over a hypersphere described by the Hamiltonian. Not the exponential of the Hamiltonian. It is the usual integration one performs for multiplicity calculations. But another way to solve this problem is to calculate the partition function instead of the multiplicity. This calculation involves the partition function, which is denoted Z above.
********************************************************************
 

Related to What is the Partition Function for a Singular Exponential Hamiltonian?

1. What is n-dim Gaussian?

N-dim Gaussian, also known as n-dimensional Gaussian, is a probability distribution function that describes the random movement of particles in an n-dimensional space. It is often used in scientific and statistical analysis to model natural phenomena.

2. How is n-dim Gaussian evaluated?

N-dim Gaussian is evaluated by calculating its probability density function, which involves determining the mean and standard deviation of the distribution. This can be done using mathematical formulas or computational methods.

3. What is the significance of evaluating n-dim Gaussian?

Evaluating n-dim Gaussian allows scientists to understand and make predictions about complex systems and processes in various fields such as physics, chemistry, and biology. It also plays a crucial role in machine learning and data analysis.

4. What are some real-life applications of n-dim Gaussian?

N-dim Gaussian is used in a wide range of scientific and engineering applications, including image processing, signal processing, finance, and weather forecasting. It is also used in the study of Brownian motion and diffusion processes.

5. Are there any limitations to the evaluation of n-dim Gaussian?

One limitation of evaluating n-dim Gaussian is that it assumes a normal distribution, which may not always be the case in real-world scenarios. Additionally, it may be challenging to accurately estimate the parameters of the distribution, especially in high-dimensional spaces.

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