Can the Energy for Non-Integer Values be Modeled in a Quantum System?

[eljose]

Let,s suppose we have a qunatum system so the energies are the roots of the function f(x)=0..then my question is that we could calculate the roots to obtain E(0),E(1),E(2),...but the problem comes when we have the integral..

$$\int_0^{\infty}E(n)dn$$ my question is if for this case we could modelize E(n) for non-integer n in the form:

$$E(n)=\sum_{k=0}^{\infty}E(n)\delta(n-k)$$

so for this case the sum becomse the series: $$\sum_{k}E(k)$$ for every positive integer... thanx.

 

[solidspin]

We use density matrices all the time w/r/t our pulse sequences in solid-state NMR. They're extremely powerful from a practical pov, b/z the off-diagonal terms represent coherences w/ very useful physical meaning.

A reasonable book w/ some very good practical application is called Spin Dynamics, by Malcolm Levitt. Of course, this is about NMR, but you will quickly see that manipulation of large nxn matrices, after some manipulation of sandwich operators, yields some great results w/o too much headscratching.

 

[denian]

I would say that there is ongoing research in the field of quantum mechanics to understand and model the behavior of energy for non-integer values. While it is true that the energies in a quantum system are typically represented by the roots of a function, there are other ways to represent and calculate these energies, such as using the Hamiltonian operator.

In regards to the integral \int_0^{\infty}E(n)dn, it is important to note that in quantum mechanics, the energy levels are discrete and not continuous. This means that there are only a finite number of energy levels, and the integral would not be applicable in this case.

As for your proposed model of E(n), it is important to consider the physical interpretation and implications of using a delta function. While it may be mathematically convenient, it may not accurately represent the physical behavior of the system. Additionally, the use of a delta function can lead to divergences and other mathematical issues.

In summary, the modeling of energy for non-integer values in a quantum system is an active area of research, and there is no definitive answer at this time. However, it is important to carefully consider the physical implications and limitations of any proposed models.