Transmission coefficient and the reflection coefficient

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jby

I tried to work out the transmission coefficient and
the reflection coefficient for a case similar to the
one refered by this website:
http://www.chembio.uoguelph.ca/educmat/chm386/rudiment/models/barrier/barsola.htm [Broken]

but instead this time, I reverse the situation and
now, that particle I is heading from the right, ie
from a higher potential and a possibility that it will
be transmitted to the left, ie to a lower potential,
still with the same E > V.

I've drawn a diagram of the situation which I am
considering in my question using my own notations.
(As this is a bmp file, it may take some time to
load.)

http://www.geocities.com/ace_on_mark9909/reflection.htm


My confusion is regarding to the reflection and its
coefficient, which I've worked in steps here:

I state here the situation I am refering to: Supposing
the particle initially is at the potential V = 0,
heading to the left. At x = 0, there is the sudden
change in the potential to V = -V'.

Using p as the wave number, ie (2pi/lambda) for the
particle when at V = 0, and q as wave number for
particle at V = -V', I obtained the transmission
coefficient, T as 4pq/(p+q)^2.

By the condition of the potential 0 > -V', thus, p <
q, ie the wavelength at V = 0 > wavelength at V = -V'.


Let, q = ap, ie a = ratio of final wavenumber to
initial wave number: q/p. Since, q > p => a > 1.

We simplify the transmission coefficient to from
T = 4pq/(p+q)^2
to
T = 4a/(1+a)^2 ... (1)

From equation 1, it states that T is only dependent on
the ratio of the two wave number and hence dependent
only on the ratio of both wavelengths, and not on any
of the wavelength alone

=> the coefficient T does not discriminate on the size
on any of the wavelength alone but the ratio of the
magnitude of its wavelengths.

=> T does not distinguish between a particle or a
macroscopic object, eg, a ball.

From T = 4a/(1+a)^2, I've drawn a graph of it for a in
the range 0 <= a <= +infinity. I've uploaded to this
website:

http://www.geocities.com/ace_on_mark9909/transmission.htm


From the graph, it looks like there is a turning point
at a = 1, corresponding to T = 1, and slowly goes to
zero, as a -> infinity

By a -> infinity, we can say that the potential height
-V' approaches -infinity.

But, if the potential at x = 0, changes so sharply as
in approaching infinity, the graph shows T = 0, then,
it means that if I were to replace a particle with a
ball/human and is to approach this potential it is
almost likely to be reflected back...

Is there anything wrong with my maths? If not, how do
you interpret this result?
 
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Tom Mattson

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This thread got no love in the Homework forum, so I'm kicking it here to Physics.
 
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The energy of a macroscopic object is going to be so great that the potential isn't going to matter and your going to get complete transmission.

JMD

You should rework the solution, beacuse I doubt you get any reflection, but I haven't worked this out for a few years and can't remember right of the top of my head.
 
Last edited:

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