Why the reflection and transmission?

In summary, when solving the time-dependent Schrödinger's equation for the delta potential, non-normalizable solutions are obtained. By using the boundary condition and comparing coefficients, the probability of transmission and reflection can be determined. However, it may be questioned how one can prove this from a mathematical standpoint, especially for a wave traveling from -infinity to +infinity. One possible approach is to use the probability current as a hand-waving argument, but a more rigorous method, such as using the general form of a wave packet and an existence proof involving epsilon and delta, may be needed for mathematicians.
  • #1
tim_lou
682
1
in solving the time-dependent Schrödinger's equation for the delta potential, one obtain a set of non-normalizable solutions.

form the boundary condition and comparing the coefficients of the solution, one obtains the probability of transmission and reflection.

However, how can one be sure that such events occur in a mathematical standpoint? suppose one has a wave traveling from -infinity (a time-dependent localized wave packet that solves the time-dependent Schrodinger's equation), how does one prove that after a very long time (as time approaches infinity),

the integral:
[tex]\lim_{t\rightarrow +\infty}\int_{-\infty}^0\left|\Psi(x,t)\right|^2dx[/tex]

is the reflection coefficient?
 
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  • #2
hmmm, I think I see it now, the probability current can give me some hand waving argument to this...

But is there a more rigorous way (mathematically), perhaps something involving the general form of a wave packet?
 
  • #3
For a physicist, the probability that a particle moving to the right has not passed go in an infinite time defines R. A mathemetician would need an existence proof using epsilon and delta.
 

Why do objects reflect light?

Objects reflect light because light is made up of tiny particles called photons that bounce off the surface of an object when they hit it. This bouncing of photons is what allows us to see objects that are not directly in the path of a light source.

Why do some objects reflect more light than others?

The amount of light an object reflects depends on its surface properties, such as texture, color, and material. Smooth and shiny surfaces tend to reflect more light than rough and dull surfaces. The color of an object also plays a role, as darker colors absorb more light while lighter colors reflect more light.

What factors affect the transmission of light?

The transmission of light through an object depends on various factors, such as the material of the object, its thickness, and any impurities or imperfections in the material. Materials that are transparent, like glass, allow most of the light to pass through, while opaque materials, like wood, absorb or reflect most of the light.

Why do some materials allow light to pass through while others don't?

The ability of a material to allow light to pass through it is determined by its atomic and molecular structure. Materials with tightly packed and regular atomic arrangements, such as glass, allow light to pass through, while materials with more disordered and irregular atomic arrangements, like wood, prevent light from passing through.

How is reflection and transmission used in everyday life?

Reflection and transmission play a crucial role in everyday life, from the functioning of mirrors and windows to the design of solar panels and optical instruments. Understanding the principles of reflection and transmission also helps in creating efficient lighting and communication systems, as well as in developing new materials and technologies.

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