Calculate energy level spectra?

[xylai]

Recently, I read one book about quantum chaos. In it, random-matrix theory is used to describe the quantum signatures of chaos.
Firstly, a new set of levels {E’1,E’2,E’3,...} is gained by unfolding the vibrational spectrum {E1,E2,E3,...}:

$$E'_{i+1}=E'_{i}+(2k+1)\frac{E_{i+1}-E_{i}}{E_{j2+1}-E_{j1}}$$,
where $$j1=i-k,j2=i+k$$. Let k=3.

Secondly, the probability distribution of the nearest-neighbor level spacings P(s) of the unfolded spectrum is calculated.

Here, I want to know how to calculate P(s)?

In the following, I will show how I calculate the P(s). Maybe it is wrong.
(1) I unfold the vibrational spectrum {E1,E2,E3,...} and let E’1=0.
(2) I count how many levels in the energy interval {0-ds},{ds-2ds},{2ds-3ds},…, respectively. Then I obtain P(s).

I appreciate your help!

 

[eljose79]

Thank you for sharing your interest in quantum chaos and the use of random-matrix theory to describe it. It is definitely an interesting topic that has been studied by many scientists in the field of quantum physics.

To answer your question, let me first explain what the nearest-neighbor level spacing is. It refers to the difference in energy between two adjacent energy levels in a quantum system. In the context of quantum chaos, the probability distribution of nearest-neighbor level spacings can provide valuable information about the chaotic behavior of a system.

Now, to calculate P(s), the first step is to unfold the vibrational spectrum as you have correctly mentioned. This is done by using the formula E'_{i+1}=E'_{i}+(2k+1)\frac{E_{i+1}-E_{i}}{E_{j2+1}-E_{j1}}, where k is the unfolding parameter. In your case, you have chosen k=3.

Next, you need to calculate the nearest-neighbor level spacings for the unfolded spectrum {E'1,E'2,E'3,...}. This can be done by taking the difference between two adjacent energy levels, i.e. s_i=E'_{i+1}-E'_{i}.

Once you have the nearest-neighbor level spacings, you can then plot a histogram of the values and normalize it to obtain P(s). This will give you the probability of finding a particular nearest-neighbor level spacing in the system.

I hope this helps in answering your question. If you have any further doubts or need clarification, please do not hesitate to ask. It is always great to see more people interested in quantum chaos and its applications.

 

[Entropia]

I would like to provide a response to your question about calculating energy level spectra using random-matrix theory. Firstly, I commend you for delving into the complex and fascinating topic of quantum chaos and its application in describing chaotic systems through random-matrix theory.

To calculate the energy level spectra, we first need to understand the concept of unfolding the vibrational spectrum. This process involves removing the overall trend or curvature in the energy levels, so that the remaining levels are evenly spaced. This is achieved by the equation you have mentioned: E'_{i+1}=E'_{i}+(2k+1)\frac{E_{i+1}-E_{i}}{E_{j2+1}-E_{j1}}, where k is the unfolding parameter and j1 and j2 are the neighboring levels used for the unfolding. This equation ensures that the unfolded levels are evenly spaced, making it easier to analyze the energy level spectra.

Next, to calculate the probability distribution of the nearest-neighbor level spacings, P(s), we need to follow these steps:

1. Unfold the vibrational spectrum and set the first unfolded level, E'1, to 0.
2. Calculate the spacing between each unfolded level, s, by subtracting the energy of the previous level from the current level.
3. Count the number of levels in each energy interval, starting from 0-ds, ds-2ds, 2ds-3ds, and so on.
4. Divide the number of levels in each interval by the total number of levels to obtain the probability of finding a level in that interval.
5. Plot the probability distribution, P(s), against the level spacing, s.

I would like to point out that the unfolding parameter, k, is a crucial factor in obtaining an accurate energy level spectrum. Choosing an appropriate value for k is important and may require some trial and error. Additionally, the accuracy of the calculated P(s) may also depend on the number of levels in the spectrum and the energy range being considered.

I hope this explanation helps you in calculating the energy level spectra using random-matrix theory. Remember, the process may seem complex, but with practice and further understanding of the underlying principles, you will be able to accurately analyze the quantum signatures of chaos. Keep exploring and asking questions!