- #1

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?

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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