Help with this semiclassical sum please.

[mhill]

I know how to obtain a semiclassical expression for the sum:

$$\sum_{n=0}^{\infty} e^{-a E_{n}}\approx \iint _{\Gamma}e^{-aH(x,p)} dx, dp$$

where the integral is extended over the whole phase space of particle (x,p) , my question is if E-->oo is a big (but finite) energy , then what would be the asymptotic expression to evaluate:

$$\sum_{E_n \le E}e^{-aE_{n}}$$

this is just the sum of the function exp(-x) taken over all the Energies of Hamiltonian operator that are less or equal to a finite given Energy 'E' then what would be the expression for this last sum ?? , thanks in advance.

EDIT: i am not pretty sure, however i would say that if A is a constant then .. the last sum could be approximated by the integral:

$$\int_{0}^{N}exp(-E)dE$$ with E_n=E so 'N' is the N-th quantum

level whose energy is precisely 'E'

 

[eljose79]

Thank you for your question. To answer your question, I will first provide a brief explanation of the semiclassical approximation and then address the sum in question.

The semiclassical approximation is a mathematical technique used in quantum mechanics to approximate the behavior of a quantum system in terms of classical mechanics. This is achieved by replacing the quantum Hamiltonian with its classical counterpart and then solving the resulting classical equations of motion. The resulting solution is then used to approximate the quantum behavior of the system.

Now, to address your question, let us consider the sum in question:

\sum_{E_n \le E}e^{-aE_{n}}

As you correctly pointed out, this sum can be approximated by the integral:

\int_{0}^{N}exp(-E)dE

However, in the semiclassical approximation, we can further simplify this expression by replacing the integral with an integral over the phase space, similar to the original expression:

\int_{\Gamma}exp(-aH(x,p))dx dp

This integral is extended over the phase space of the system, where x and p represent the position and momentum variables, respectively. This integral is essentially the sum over all possible states of the system with energies less than or equal to E.

Therefore, the asymptotic expression for the sum can be written as:

\sum_{E_n \le E}e^{-aE_{n}} \approx \int_{\Gamma}exp(-aH(x,p))dx dp

This expression provides a more accurate approximation for the sum in question, taking into account the semiclassical behavior of the system.

I hope this helps answer your question. If you have any further doubts or queries, please feel free to ask.

 

[denian]

I would first like to commend you on your understanding of the semiclassical expression for the sum and your attempt to find an asymptotic expression for the sum in question. Your intuition about approximating the sum with an integral is correct, but let's delve a little deeper into the concept.

In the semiclassical expression, we are considering the sum of the function exp(-aE_n) over all possible energy levels, which is equivalent to integrating over the entire phase space. However, in the sum in question, we are only considering energies lower than or equal to a finite given energy E. This means that we can restrict our integral to the region of phase space where the energy is less than or equal to E.

So, the asymptotic expression for the sum can be written as:

\sum_{E_n \le E}e^{-aE_{n}} \approx \iint _{\Gamma_E}e^{-aH(x,p)} dx, dp

where \Gamma_E represents the phase space region where the energy is less than or equal to E.

Now, to find an explicit expression for this integral, we need to determine the boundaries of \Gamma_E. This can be done by solving the Hamiltonian equation H(x,p) = E for both x and p. This will give us the boundaries in phase space that correspond to the energy level E. Let's say these boundaries are x_min(E), x_max(E), p_min(E), and p_max(E).

Then, the asymptotic expression for the sum can be written as:

\sum_{E_n \le E}e^{-aE_{n}} \approx \int_{p_min(E)}^{p_max(E)}\int_{x_min(E)}^{x_max(E)}e^{-aH(x,p)} dx dp

This integral can be further simplified by substituting the Hamiltonian equation into the expression for H(x,p). This will give us an integral in terms of E, which can be evaluated using standard techniques.

In summary, the asymptotic expression for the sum in question can be approximated by an integral over the region of phase space where the energy is less than or equal to E. The boundaries of this region can be found by solving the Hamiltonian equation H(x,p) = E. I hope this helps in your further exploration of this problem.