- #1

- 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: