- #1

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- TL;DR Summary
- Made up some unproven(?) statements about simple one particle QM.

Does anyone know if a proof exists for these statements about 1d quantum mechanics?

1. If the potential energy where a particle moves is of the form

##V(x) = c_2 x^2 + c_4 x^4 + c_6 x^6 + \dots##

or

##V(x) = c_2 x^2 + c_3 |x|^3 + c_4 x^4 + c_5 |x|^5 + c_6 x^6 + \dots##

with ##c_j \geq 0## for all ##j\in\mathbb{N}##, then the standard deviation of the particle's position in the eigenstates of ##\hat{H}## increases monotonically with increasing quantum number.

2. If the ##V(x)## is like that above, then the energy level spacing ##E_{n+1} - E_n## can only increase or stay constant with increasing quantum number ##n##.

3. A system having any spectrum like ##E_n = a + bn^c## with ##a,b,c## constants, ##1\leq c \leq 2## and ##b>0## can be produced with a ##V(x)## of the form given above.

The claim number 2 seems to be correct, because if you begin with the harmonic oscillator potential ##V(x) = \frac{1}{2}kx^2## and add a small perturbation of form ##V'(x) = \lambda x^{2n}## with ##n\in\mathbb{N}## and ##n\geq 2##, then the first order correction to the ##m##:th energy eigenvalue is ##E_{m}^{'} = \lambda\int\limits_{x=-\infty}^{x=\infty}\psi_{m}^{*}(x)x^{2n}\psi_m (x)dx##. This clearly gets larger with increasing ##m## (basically because of what was said in claim 1, i.e. the particle is more likely to be far from the origin ##x=0## for higher excited states), so this perturbation will only increase energy level spacings. But it's not rigorously clear that one can assume claim 1 is true and the first-order change can be used to predict the result of finite perturbations.

I invented claim 3 because of the observation that a spectrum like ##E_n = a + bn^c## usually fits very accurately to the first 10 - 20 eigenenergies of a system with a convex potential energy function ##V(x)##. Only with larger quantum numbers it's easy to see the difference. So the claim 3 is the converse of that statement. However, the potential energy ##V(x) = c_2 x^2 + c_4 x^4 + c_6 x^6 + \dots## is not necessarily the only one that produces the same spectrum - a system with constant energy level spacings can also be created with the singular "isotonic oscillator potential" ( https://www.sciencedirect.com/science/article/abs/pii/037596017990197X ) instead of a harmonic oscillator. One way to reconstruct a ##V(x)## corresponding to a given spectrum ##E_n## is the inverse scattering theorem, but it's not necessarily easy to calculate the limit of Eqn. (5a) in the link https://arxiv.org/pdf/0811.1389.pdf when all eigenvalues are included and the determinant becomes "infinitely large".

1. If the potential energy where a particle moves is of the form

##V(x) = c_2 x^2 + c_4 x^4 + c_6 x^6 + \dots##

or

##V(x) = c_2 x^2 + c_3 |x|^3 + c_4 x^4 + c_5 |x|^5 + c_6 x^6 + \dots##

with ##c_j \geq 0## for all ##j\in\mathbb{N}##, then the standard deviation of the particle's position in the eigenstates of ##\hat{H}## increases monotonically with increasing quantum number.

2. If the ##V(x)## is like that above, then the energy level spacing ##E_{n+1} - E_n## can only increase or stay constant with increasing quantum number ##n##.

3. A system having any spectrum like ##E_n = a + bn^c## with ##a,b,c## constants, ##1\leq c \leq 2## and ##b>0## can be produced with a ##V(x)## of the form given above.

The claim number 2 seems to be correct, because if you begin with the harmonic oscillator potential ##V(x) = \frac{1}{2}kx^2## and add a small perturbation of form ##V'(x) = \lambda x^{2n}## with ##n\in\mathbb{N}## and ##n\geq 2##, then the first order correction to the ##m##:th energy eigenvalue is ##E_{m}^{'} = \lambda\int\limits_{x=-\infty}^{x=\infty}\psi_{m}^{*}(x)x^{2n}\psi_m (x)dx##. This clearly gets larger with increasing ##m## (basically because of what was said in claim 1, i.e. the particle is more likely to be far from the origin ##x=0## for higher excited states), so this perturbation will only increase energy level spacings. But it's not rigorously clear that one can assume claim 1 is true and the first-order change can be used to predict the result of finite perturbations.

I invented claim 3 because of the observation that a spectrum like ##E_n = a + bn^c## usually fits very accurately to the first 10 - 20 eigenenergies of a system with a convex potential energy function ##V(x)##. Only with larger quantum numbers it's easy to see the difference. So the claim 3 is the converse of that statement. However, the potential energy ##V(x) = c_2 x^2 + c_4 x^4 + c_6 x^6 + \dots## is not necessarily the only one that produces the same spectrum - a system with constant energy level spacings can also be created with the singular "isotonic oscillator potential" ( https://www.sciencedirect.com/science/article/abs/pii/037596017990197X ) instead of a harmonic oscillator. One way to reconstruct a ##V(x)## corresponding to a given spectrum ##E_n## is the inverse scattering theorem, but it's not necessarily easy to calculate the limit of Eqn. (5a) in the link https://arxiv.org/pdf/0811.1389.pdf when all eigenvalues are included and the determinant becomes "infinitely large".

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