Mysteries of Geometric Optics In MTW Chapter 22

In summary: KB approximation and singular perturbation theory are equivalent methods for solving problems in quantum mechanics. They are both expansions in powers of the Heaviside constant ##\hbar##.
  • #1
TerryW
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Homework Statement
I really don't understand what is going on here - can anyone shed any light please?
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At the start of this section §22.5 (Geometric Optics in curved Spacetime), the amplitude of the vector potential is given as:

A = ##\mathfrak R\{Amplitude \ X \ e^{i\theta}\} ##

The Amplitude is then re=expressed a "two-length-scale" expansion (fine!) but it then is modified further to (22.25) by
introducing a "useful" parameter ε...
Introducing a parameter.png
But this amplitude is now varying at a quite different rate (if ε is anything other than unity) so it isn't equivalent to the original vector potential.

When we get into the calculations, (22.7) is readily derived but I have a problem with the gathering terms O(##\frac {1}{ε}##) or O(ε) etc because, if ε reverts to its eventual value unity thereby recovering the original rate of variation, justification for (22.28) and (22.9) looks a bit thin.

Using parameter to derive realtionships.png


Can anyone explain what is going on here and why this is all reasonable?

Regards
TerryW
 
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  • #2
The introduction of the parameter ##\varepsilon## is just a convenient way to "tag" the various terms so that the power of ##\varepsilon## immediately tells you the order of the term in λ/L. A term with ##\varepsilon^2## is second order in λ/L even when ##\varepsilon = 1##. If you want, you could forgo introducing ##\varepsilon##. But, then, you would need to remember that ##a^\mu## is zero order in λ/L, ##b^\mu## is first order, ##c^\mu## is second order, ##\theta## is order -1,etc. When MTW "collect terms of order ##\epsilon^n##", they are just collecting terms of order (λ/L)n, which you could do even if you didn't introduce ##\varepsilon##.
 
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  • #3
It's not different from the usual eikonal expansion in standard Maxwell theory in flat space. Perhaps it helps to read about this first, e.g., in A. Sommerfeld, Lectures on Theoretical Physics, vol. 4 (optics).

In quantum mechanics a mathematically equivalent procedure is known as WKB approximation. There it's an expansion in powers of ##\hbar##.

The mathematical procedure is known as "singular perturbation theory".
 
  • #4
TSny said:
The introduction of the parameter ##\varepsilon## is just a convenient way to "tag" the various terms so that the power of ##\varepsilon## immediately tells you the order of the term in λ/L. A term with ##\varepsilon^2## is second order in λ/L even when ##\varepsilon = 1##. If you want, you could forgo introducing ##\varepsilon##. But, then, you would need to remember that ##a^\mu## is zero order in λ/L, ##b^\mu## is first order, ##c^\mu## is second order, ##\theta## is order -1,etc. When MTW "collect terms of order ##\epsilon^n##", they are just collecting terms of order (λ/L)n, which you could do even if you didn't introduce ##\varepsilon##.
Thanks TSny,

I'd already decided that I didn't need ##\varepsilon## and that I could get to all the results by "remembering" the order of the terms, but your reply is helpful in reassuring me that I wan't missing something crucial.

CheersTerry
 
  • #5
vanhees71 said:
It's not different from the usual eikonal expansion in standard Maxwell theory in flat space. Perhaps it helps to read about this first, e.g., in A. Sommerfeld, Lectures on Theoretical Physics, vol. 4 (optics).

In quantum mechanics a mathematically equivalent procedure is known as WKB approximation. There it's an expansion in powers of ##\hbar##.

The mathematical procedure is known as "singular perturbation theory".
Thanks for replying vanhees71,

I'll see if I can find something about the WKB approximation or singular perturbation theory - it doesn't look like I can get a free download of Sommerfield Vol IV. Maybe I can find something in my ancient copy of the Feynman lectures.RegardsTerry W
 

Related to Mysteries of Geometric Optics In MTW Chapter 22

1. What is geometric optics?

Geometric optics is a branch of optics that deals with the study of light as rays and how it behaves when it interacts with different materials and surfaces.

2. What are the main principles of geometric optics?

The main principles of geometric optics include the law of reflection, the law of refraction, and the principle of least time. These principles explain how light travels in a straight line, reflects off of surfaces, and changes direction when passing through different mediums.

3. What is the difference between ray optics and wave optics?

Ray optics, or geometric optics, describes light as a ray and focuses on the behavior of light at the macroscopic level. Wave optics, on the other hand, describes light as a wave and focuses on the behavior of light at the microscopic level.

4. How is geometric optics used in everyday life?

Geometric optics is used in many everyday applications, such as in the design of lenses for glasses and cameras, the reflection and refraction of light in mirrors and prisms, and in the functioning of optical instruments like microscopes and telescopes.

5. What are some common examples of geometric optics in nature?

Some common examples of geometric optics in nature include the reflection of light off of water surfaces, the refraction of light through raindrops to create a rainbow, and the way light bends and reflects off of objects in a kaleidoscope.

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