Reconstruction of potential V(x)

[zetafunction]

the idea is, let us suppose we know the trace

$$Tr(h(\hat H ))= \sum_{n=0}^{\infty}h(E_n )$$

here 'h' can be a real or complex exponential of the form exp(-ax) and 'H' is the usual Hamiltonian operator

$$H=p^2 + V(x)$$

what information about the spectrum of Hamiltonian would i need in order to obtain V(x) ??

for example: for the Harmonic Oscillator in Planck's unit so h=1 and w=1 i have that

$$\sum _{n=0}^{\infty} exp(-s(n+1/2))= \frac{exp(-s/2)}{1-exp(-s)}$$

then from the expression above could i conclude that potential goes like $$V(x)=ax^{2}$$

 

[azntoon]

??The answer is no. In general, it is not possible to determine the potential V(x) from the trace Tr(h(\hat H )). The trace includes information about the spectrum of the Hamiltonian, but does not contain enough information to uniquely determine the potential. To do this, one would need additional information, such as the eigenfunctions of the Hamiltonian or knowledge of the exact form of the Hamiltonian.

 

[Entropia]

I would like to clarify that this is a theoretical concept and not a real-world experiment. In order to obtain V(x), we would need information about the spectrum of the Hamiltonian, which includes the energy levels and corresponding eigenfunctions. This information can be obtained through theoretical calculations or experimental measurements.

In the example given, the potential V(x) can be reconstructed using the information about the energy levels and eigenfunctions of the Harmonic Oscillator. However, this may not be the case for other systems. The reconstruction of potential V(x) depends on the specific system and the corresponding Hamiltonian.

Additionally, the reconstruction of potential V(x) is not a straightforward process and may require advanced mathematical techniques such as inverse scattering or spectral analysis. It also depends on the accuracy and completeness of the information about the Hamiltonian.

In conclusion, the reconstruction of potential V(x) from the given expression requires knowledge about the spectrum of the Hamiltonian, which can be obtained through theoretical calculations or experimental measurements. However, the specific method of reconstruction may vary depending on the system and the accuracy of the information.