Inverse spectral problem in QM

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In summary: Eigenfunctions.In summary, Ilja is arguing that it is possible to identify the V(x) for a particle using only the spectrum and not the Eigenfunctions. However, he notes that this is not always the case, and sometimes you only have information about the spectrum.
  • #1
zetafunction
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I have taken QM , and i find it very interesting but my question is , we have the Hamiltonian eigenvalue problem

[tex] i \partial _t \Psi (x,t) = \lambda _n \Psi (x,t)=(p^2+V(x))\Psi(x,t) [/tex]

of course in general, we know the potential V(x) but my question is the inverse, if we knew how the spectrum is or for example the (approximate) value of [tex] \sum_n e^{it \lambda _n} [/tex]

for example for Harmonic oscillator since all energies are lineal we know that

[tex] \sum_n e^{it \lambda _n}=(1-exp(it/2))^{-1} [/tex] from this could we deduce that V(x) is a quadratic potential x*x
 
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  • #2
I don't think you can (although I can't be bothered to try and prove it mathematically.. might be related to the v-representability problem though.).

If you knew the eigenfunctions, then you could expand the potential on some basis and then treat it as a minimization problem. (an 'inverse Ritz variational method' if you like)
 
  • #3
Knowing energy eigenvalues is not sufficient to restore the Hamiltonian. As alxm said, you need to know all eigenfunctions too.

Two different Hamiltonians [tex]H[/tex] and [tex]H' = UHU^{-1}[/tex] have exactly the same energy spectra if [tex]U[/tex] is any unitary operator.
 
  • #4
meopemuk said:
Knowing energy eigenvalues is not sufficient to restore the Hamiltonian. As alxm said, you need to know all eigenfunctions too.

Two different Hamiltonians [tex]H[/tex] and [tex]H' = UHU^{-1}[/tex] have exactly the same energy spectra if [tex]U[/tex] is any unitary operator.

But H' would be most probable not of the form [tex]H' = \Delta + V(x)[/tex], so that this simple consideration is not sufficient to answer the question if it is possible to identify V(x).

It is not possible, and I have used this as an argument against many worlds, see http://arxiv.or/abs/arXiv:0901.3262" [Broken] and the references therein.
 
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  • #5
Ilja said:
But H' would be most probable not of the form [tex]H' = \Delta + V(x)[/tex], so that this simple consideration is not sufficient to answer the question if it is possible to identify V(x).
QUOTE]

If you choose a unitary operator [tex] U [/tex] commuting with momentum, then the transformed Hamiltonian is of the form [tex]H' = \Delta + V(x,p)[/tex]. I don't see anything wrong with having potential V that depends on both position and momentum of the particle.
 
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  • #6
a potential [tex] V(x,p) [/tex] that depend on momentum is not common in physics since you should modifie Newton law or E-L equation

the idea is that sometimes you only known the spectrum but not the Eigenfunctions , perhaps using WKB approximation , you would know that eigenfunctions depend on the potential [tex] \Psi (x) = exp (i \oint (E-V(x))^{1/2}) [/tex]

but in general you only have information about the espectrum
 

What is the inverse spectral problem in QM?

The inverse spectral problem in quantum mechanics is a mathematical problem that involves determining the potential or Hamiltonian of a quantum system based on the known energy eigenvalues and wavefunctions of the system. It is an important problem in quantum mechanics as it allows us to understand the underlying dynamics and properties of a system.

Why is the inverse spectral problem important in QM?

The inverse spectral problem is important in quantum mechanics because it allows us to extract information about a quantum system that is not directly observable. By solving this problem, we can determine the potential or Hamiltonian of a system, which can provide insights into the behavior and properties of the system.

What are the challenges in solving the inverse spectral problem?

One of the main challenges in solving the inverse spectral problem is the non-uniqueness of solutions. This means that multiple potentials can produce the same energy eigenvalues and wavefunctions, making it difficult to determine the exact potential of a system. Additionally, the inverse spectral problem often involves solving complex mathematical equations, which can be challenging even for experienced mathematicians.

What techniques are used to solve the inverse spectral problem in QM?

There are several techniques that are used to solve the inverse spectral problem in quantum mechanics. These include the Gel'fand–Levitan–Marchenko equation, the factorization method, and the scattering transform method. Each of these techniques has its own strengths and limitations, and the choice of method often depends on the specific problem at hand.

What are the applications of the inverse spectral problem in QM?

The inverse spectral problem has various applications in quantum mechanics, including in the study of quantum systems with complex potentials, such as the Schrödinger equation with a delta potential or a periodic potential. It is also used in the design and optimization of quantum systems, such as in quantum computing and quantum sensors. Additionally, the inverse spectral problem has connections to other fields of mathematics, such as inverse scattering theory and integrable systems.

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