Error of the WKB approximation

In summary, someone is looking for a measure of the error of the WKB approximation and is wondering if there is a book that has this information.
  • #1
phyQu
2
1
TL;DR Summary
How to calculate the error made using the WKB approximation?
hello everyone

I tell you a little about my situation.
I already found the approximate wavefunctions for the schrodinger equation with the potential ##V(x) = x^2##, likewise, energy, etc.
I have the approximate WKB solution and also the exact numeric solution.

What I need to do is to calculate the error of the approximate solution with respect to the exact solution,
but I need help to start, because I am lost with this topic.

Will it be possible to make an error analysis?
Will it be possible apply that is defined as ##e = |y_{exact} - y_{approx}|##?

Do you know if someone has already worked out the error for the WKB approximation?
Do you know of any recent article dealing with this?
I've searched the internet and I can't find anything that tells me specifically about the error of this method, please, can someone help me.

Thanks in advance
 
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  • #2
I am not aware of any previous results concerning the error of the WKB approximation. Since the quality of the approximation depends on the variation of the potential with respect to the oscillation (wave number) of the wave function, I would be surprised if there was a general answer to this.

I would calculate the relative error on the eigenenergy as well as the L2 norm of the difference in wave functions,
$$
\int_{-\infty}^\infty \left| \psi_\textrm{exact}(x) - \psi_\textrm{approx}(x) \right|^2 \, dx
$$
However, this might be too much influenced by the complex phase, which is not physically relevant, so I would also calculate the difference in probability
$$
\int_{-\infty}^\infty \left| \psi_\textrm{exact}(x)\right|^2 - \left|\psi_\textrm{approx}(x) \right|^2 \, dx
$$
 
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  • #3
Thank you very much, by the way, do you know a book that has this topic you are talking about? I would help me to reference my work.
 
  • #4
The latter integral is not a good measure for the "distance of states", because the overall factor is just determined by the normalization condition, i.e., that integral should be 0 when normalizing both wave functions properly.
 
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  • #5
DrClaude said:
However, this might be too much influenced by the complex phase, which is not physically relevant, so I would also calculate the difference in probability
$$
\int_{-\infty}^\infty \left| \psi_\textrm{exact}(x)\right|^2 - \left|\psi_\textrm{approx}(x) \right|^2 \, dx
$$
vanhees71 said:
The latter integral is not a good measure for the "distance of states", because the overall factor is just determined by the normalization condition, i.e., that integral should be 0 when normalizing both wave functions properly.
Well, you could integrate over the absolute value of the differences:
$$
\int_{-\infty}^\infty \left|\left| \psi_\textrm{exact}(x)\right|^2 - \left|\psi_\textrm{approx}(x) \right|^2\right| \, dx
$$
 
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  • #6
vanhees71 said:
The latter integral is not a good measure for the "distance of states", because the overall factor is just determined by the normalization condition, i.e., that integral should be 0 when normalizing both wave functions properly.
Obviously :confused:. I should've thought a bit more about it. I was after a measure of the difference in local probability distribution. Maybe something like
$$
\max_x \left( \left| \psi_\textrm{exact}(x)\right|^2 - \left|\psi_\textrm{approx}(x) \right|^2 \right)
$$
 
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Related to Error of the WKB approximation

What is the WKB approximation?

The WKB approximation, also known as the Wentzel-Kramers-Brillouin approximation, is a method commonly used in quantum mechanics to approximate the wave function of a particle in a potential well. It is based on the assumption that the wavelength of the particle is much smaller than the dimensions of the potential well.

What is the error of the WKB approximation?

The error of the WKB approximation refers to the difference between the exact solution of the Schrödinger equation and the approximate solution obtained using the WKB method. This error can be significant for certain potential wells and is a topic of interest in quantum mechanics research.

How is the error of the WKB approximation calculated?

The error of the WKB approximation can be calculated by comparing the approximate solution obtained using the WKB method with the exact solution of the Schrödinger equation. This can be done by evaluating the difference between the two solutions at various points within the potential well.

What factors can affect the error of the WKB approximation?

The error of the WKB approximation can be affected by various factors, such as the shape and depth of the potential well, the energy of the particle, and the accuracy of the WKB method used. In general, the error tends to decrease as the energy of the particle increases.

How can the error of the WKB approximation be minimized?

The error of the WKB approximation can be minimized by using a more accurate version of the WKB method, such as the higher-order WKB approximation, or by using other methods such as perturbation theory. Additionally, choosing an appropriate energy level and potential well shape can also help reduce the error.

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