- #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.
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: