# This worksheet is used to calculate the transition probability through a finite by VjLbQ2V2

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```									Equation 2.23: Transition Probability through finite rectangular barrier
for high energy

Goals

This worksheet is used to calculate the transition probability through a finite
rectangular barrier of length L and height V
as a function of incident energy E . The energy E is greater than the barrier height V .
The mass of the particle is m .

Excel skills required for this workbook

1. Writing formulas
2. Plotting functions

Click on …                                      To…

Transition probability                          Calculate and plot the transition probability
for various values of m , L , and V
Data                                            Display data used for the graph

Colour Code:

The value within a yellow cell can be set as part of the
exploration process. The value is typically used to calculate data points in
a plot or for a "one-point" calculation of a given property. Changing this value, then,
results in a change in the appearance of a displayed plot or in a calculated value.
The value within a violet cell is the result of a "one-point" calculation using values
in the yellow cells.
Equation 2.23: The transition probability for E > V as function of energy E

The transition probability is given by eqn 2.23. We specify values of the mass m , barrier
height V and barrier length L .
Use atomic units for which hbar = 1; m is a multiple of m e, V is a multiple of the
hartree (E h), L is a multiple of the Bohr radius a 0.

T = [1 + (sin2(k ' L )/( 4(E /V )(E /V – 1))]–1                 k ' = [2mV (E /V –1)]1/2

mVL                       1                0.5             10                       Choose values of m /m e V /E h   L /a 0
mVL                       1                0.5              4
mVL                       1                0.5              2

Transition probability

1.2

1

0.8
Series2
0.6                                                                                            Series3
T

Series4
0.4

0.2

0
1.3

1.8

2.3

2.8

3.3
1.05

1.55

2.05

2.55

3.05

E/V

Exploration

Explore the conditions under which the effects of scattering resonances increase
(i.e. there are more peaks in T ).
Row 13 contains values of E/V (unitless)

1.05            1.1           1.15      1.2     1.25       1.3    1.35      1.4     1.45
0.253325    0.999028652    0.607342246 0.504366 0.57616     0.7498 0.93619 0.999237 0.938841
0.256671     0.32609109    0.408396784 0.501777 0.601881 0.701972 0.794197 0.871649 0.930233
0.528944    0.557363278    0.585142794 0.61218 0.638383 0.663673 0.687985 0.711266 0.733476
1.5     1.55      1.6     1.65      1.7     1.75      1.8     1.85      1.9
0.856535 0.806093 0.795298 0.817572 0.862405 0.916369 0.964234 0.99344 0.999438
0.969334 0.991178 0.999514 0.998388 0.991365 0.981225 0.969952 0.958864 0.948777
0.754588 0.774586 0.793465 0.81123 0.827894 0.843479 0.858011 0.871524 0.884053
1.95       2      2.05      2.1     2.15      2.2     2.25      2.3     2.35
0.986663 0.964325 0.941319 0.923647 0.914222 0.913628 0.920905 0.934091 0.950619
0.940147 0.933189 0.927959 0.92441 0.922438 0.921899 0.922632 0.924468 0.927238
0.895641 0.906328 0.916161 0.925184 0.933443 0.940983 0.947849 0.954086 0.959736
2.4     2.45      2.5     2.55      2.6     2.65      2.7     2.75     2.8
0.967679 0.982642 0.993489 0.999147 0.99959 0.995687 0.988864 0.980731 0.972764
0.930775 0.934919 0.939521 0.944439 0.949545 0.954722 0.959867 0.964889 0.96971
0.964841 0.969438 0.973568 0.977265 0.980563 0.983495 0.98609 0.988378 0.990384
2.85      2.9     2.95       3      3.05      3.1     3.15      3.2     3.25
0.966126 0.961589 0.959534 0.960001 0.962748 0.967321 0.973124 0.979489 0.985749
0.974266 0.978504 0.982387 0.985885 0.988983 0.991671 0.993953 0.995837 0.997338
0.992133 0.993648 0.994951 0.996061 0.996997 0.997776 0.998413 0.998923 0.99932
3.3     3.35      3.4     3.45      3.5
0.991305 0.995688 0.998594 0.999909 0.999696
0.998477 0.999279 0.999772 0.999985 0.999951
0.999614 0.999819 0.999943 0.999996 0.999988

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