# Semiclassical exact expression ?

• zetafunction
In summary, the question asks if two functions, Z(t) and U(t), are equal given certain conditions on the eigenvalue function N(x) and the potential function V(x). The answer is yes, and to prove this, we would need to use the fact that the potential function is even and its Fourier transform is real and symmetric.
zetafunction
let be $$N(x)= \sum_{n} H(x-E_{n})$$ the eingenvalue 'staircase' function

and let be a system so $$V(x)=V(-x) [tex]and [tex] V^{-1}(x)=\sqrt \pi \frac{d^{1/2}}{dx^{1/2}} N(x)$$

then would it be true that the two function

$$\sum_{n}exp(-tE_{n})=Z(t)= \int_{0}^{\infty}dN(x)exp(-tx)$$

and $$\int_{0}^{\infty}dx\int_{0}^{\infty}dpexp(-tp^{2}-tV(x))=U(t)$$

would be equal ?? i have just compared the two results $$\int_{0}^{\infty}dnN(x)exp(-tx)=\int_{0}^{\infty}dx\int_{0}^{\infty}dpexp(-tp^{2}-tV(x))$$ i have taken the Laplace transform inside and get the desired result assuming that the potential V(x) is EVEN

. Yes, they would be equal. To prove this, you would need to use the fact that for an even function, its Fourier transform is real and symmetric. This means that the Laplace transform of the even function can be written as a sum of two terms: one term containing the real part of the Fourier transform, and the other term containing the imaginary part. Using this, you can then show that the two functions are indeed equal.

## What is the semiclassical exact expression?

The semiclassical exact expression is a mathematical equation used to describe the behavior of quantum mechanical systems in the classical limit. It takes into account both quantum mechanical and classical effects.

## How is the semiclassical exact expression derived?

The semiclassical exact expression is derived using the semiclassical approximation, which involves expanding the quantum mechanical wave function in terms of classical trajectories.

## What are the limitations of the semiclassical exact expression?

The semiclassical exact expression is only valid in the classical limit, where the quantum effects are small. It also does not take into account quantum fluctuations or interactions between particles.

## What types of systems can the semiclassical exact expression be applied to?

The semiclassical exact expression can be applied to a wide range of systems, including atoms, molecules, and condensed matter systems.

## How is the semiclassical exact expression used in scientific research?

The semiclassical exact expression is used to study the behavior of quantum systems in the classical limit, and can provide insights into the behavior of complex systems. It is also used in theoretical calculations and simulations to understand and predict the behavior of physical systems.

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