Number of bound states and index theorems in quantum mechanics?

In summary, the conversation discusses the possibility of an index theorem for a n-dimensional Hamiltonian that counts the bound states in the discrete spectrum. It is mentioned that this has been studied extensively in the past and there are some useful references available. However, due to the complexity of the potential in n variables, it is unlikely that such an index theorem exists. The conversation then suggests looking into analytical indices as an alternative approach.
  • #1
tom.stoer
Science Advisor
5,779
172
Just an idea: is there an index theorem for an n-dimensional Hamiltonian

[tex] H = -\triangle^{(n)} + V(x)[/tex]

which "counts" the bound states

[tex] (H - E) \,u_E(x) = 0[/tex]

i.e. eigenfunctions and eigenvalues in the discrete spectrum of H?
 
Physics news on Phys.org
  • #2
Due to the huge dosis of arbitrary in your setting (the potential in n variables needn't be seprarable), I'm quite sure the answer is <no>.
 
  • #3
and what would be the conditions for an index theorem?
 
  • #5
yes, something like that
 
  • #6
I probably should not ask for an index theorem but simply for an analytical index; the idea is to have a formula to calculate the number of bound states instead of solving the SE and counting them
 
Last edited:
  • #7
tom.stoer said:
Just an idea: is there an index theorem for an n-dimensional Hamiltonian

[tex] H = -\triangle^{(n)} + V(x)[/tex]

which "counts" the bound states

[tex] (H - E) \,u_E(x) = 0[/tex]

i.e. eigenfunctions and eigenvalues in the discrete spectrum of H?

This was studies a lot about 40 years ago. Volume 3 of Thirring's ''Course in mathematical physics'' has some results. scholar.google for >>bounds on the number of "bound states"<<
or variations turns up useful references. For related recent results see, e.g.,
http://arxiv.org/pdf/1108.1002 or
http://www2.imperial.ac.uk/~alaptev/Papers/Lapt_Beijing.pdf
 
  • #8

1. What is the index theorem in quantum mechanics?

The index theorem, also known as the Atiyah-Singer index theorem, is a mathematical theorem that relates the number of bound states in a quantum mechanical system to the topology of the underlying space. It provides a way to calculate the number of bound states without explicitly solving the Schrödinger equation.

2. What is the significance of the index theorem in quantum mechanics?

The index theorem is significant because it provides a powerful tool for understanding the behavior of quantum mechanical systems. It allows for the calculation of the number of bound states in a given system, which can provide insight into its stability and properties.

3. How does the index theorem relate to the number of bound states?

The index theorem states that the number of bound states in a quantum mechanical system is equal to the difference between the number of positive and negative energy eigenvalues of the Hamiltonian operator. This is known as the Atiyah-Singer index, and it is a topological invariant that remains constant even if the system is perturbed or deformed.

4. Can the index theorem be applied to all quantum mechanical systems?

The index theorem is a general mathematical theorem that can be applied to a wide range of quantum mechanical systems, including those with multiple dimensions, different potentials, and even relativistic systems. However, it may not be applicable to highly complex systems with continuously varying potentials.

5. How does the number of bound states affect the physical properties of a system?

The number of bound states in a quantum mechanical system can have a significant impact on its physical properties. For example, a system with a large number of bound states may be more stable and have a longer lifetime compared to a system with fewer bound states. Additionally, the energy levels of a system are determined by the number of bound states, which can affect its spectral and transport properties.

Similar threads

  • Quantum Physics
Replies
11
Views
2K
Replies
2
Views
1K
Replies
2
Views
640
Replies
1
Views
687
Replies
5
Views
860
  • Quantum Physics
Replies
2
Views
907
Replies
12
Views
672
Replies
2
Views
600
Replies
18
Views
2K
Replies
16
Views
1K
Back
Top