- #1
jby
I tried to work out the transmission coefficient and
the reflection coefficient for a case similar to the
one referred by this website:
http://www.chembio.uoguelph.ca/educmat/chm386/rudiment/models/barrier/barsola.htm
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 referring 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?
the reflection coefficient for a case similar to the
one referred by this website:
http://www.chembio.uoguelph.ca/educmat/chm386/rudiment/models/barrier/barsola.htm
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 referring 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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