# 6: Schrodinger Wave Mechanics¶

## Example number 8, Page number 228¶

In [10]:
#importing modules
import math
from __future__ import division

#Variable declaration
m=9.1*10**-31;        #mass of electron(kg)
e=1.6*10**-19;        #charge(coulomb)
a=10**-10;            #width(m)
h=6.62*10**-34;       #planck's constant
n1=1;
n2=2;
n3=3;

#Calculation
Ex=h**2/(8*e*m*a**2);  #energy(eV)
E1=Ex*n1**2;         #energy at 1st level(eV)
E2=Ex*n2**2;         #energy at 2nd level(eV)
E3=Ex*n3**2;         #energy at 3rd level(eV)

#Result
print "energy levels are",int(round(E1)),"eV",int(round(E2)),"eV",int(round(E3)),"eV"
print "answer given in the book is wrong"

energy levels are 38 eV 150 eV 339 eV
answer given in the book is wrong


## Example number 9, Page number 229¶

In [13]:
#importing modules
import math
from __future__ import division

#Variable declaration
deltax=1*10**-10;        #width
a=15*10**-10;            #width(m)

#Calculation
W=2*deltax/a;            #probability of finding the particle

#Result
print "probability of finding the particle is",round(W,3)

probability of finding the particle is 0.133


## Example number 10, Page number 229¶

In [28]:
#importing modules
import math
from __future__ import division

#Variable declaration
E=1;          #energy(eV)
V0=2;         #voltage(eV)
m=9.1*10**-31;        #mass of electron(kg)
e=1.6*10**-19;        #charge(coulomb)
chi=1.05*10**-34;
a=2*10**-10;          #potential barrier

#Calculation
x=math.sqrt(2*m*(V0-E)*e);
y=16*E*(1-(E/V0))/V0;
T=y*math.exp(-2*a*x/chi);     #probability of transmission of electron

#Result
print "probability of transmission of electron is",round(T,1)
print "answer given in the book is wrong"

probability of transmission of electron is 0.5
answer given in the book is wrong


## Example number 11, Page number 230¶

In [32]:
#importing modules
import math
from __future__ import division

#Variable declaration
E=0.080*10**-19;          #energy(eV)
E_V0=0.016*10**-19;       #voltage(eV)

#Calculation
x=math.sqrt(E);
y=math.sqrt(E_V0);
R=(x-y)/(x+y);       #fraction of electrons reflected
T=1-R;                    #fraction of electrons transmitted

#Result
print "fraction of electrons reflected is",round(R,2)
print "fraction of electrons transmitted is",round(T,2)

fraction of electrons reflected is 0.38
fraction of electrons transmitted is 0.62


## Example number 12, Page number 231¶

In [37]:
#importing modules
import math
from __future__ import division

#Variable declaration
E=0.34;          #energy(eV)
E_V0=0.01;       #voltage(eV)

#Calculation
x=math.sqrt(E);
y=math.sqrt(E_V0);
T=4*x*y/(x+y)**2;       #fraction of electrons transmitted
R=1-T;                  #fraction of electrons reflected

#Result
print "fraction of electrons transmitted is",round(T,4)
print "fraction of electrons reflected is",round(R,4)

fraction of electrons transmitted is 0.4998
fraction of electrons reflected is 0.5002


## Example number 13, Page number 232¶

In [50]:
#importing modules
import math
from __future__ import division

#Variable declaration
e=1.6*10**-19;  #charge(coulomb)
E1=1*e;         #energy(J)
E2=2*e;         #energy(J)
V0=5*e;         #voltage(J)
m=9.1*10**-31;        #mass of electron(kg)
chi=1.054*10**-34;
a1=10*10**-10;         #potential barrier(m)
a2=20*10**-10;         #potential barrier(m)

#Calculation
beta1=math.sqrt(2*m*(V0-E1)/(chi**2));
y1=16*E1*((V0-E1)/(V0**2));
T1=y1*math.exp(-2*a1*beta1);     #transmission coefficient
beta2=math.sqrt(2*m*(V0-E2)/(chi**2));
y2=16*E2*((V0-E2)/(V0**2));
T2=y2*math.exp(-2*a1*beta2);     #transmission coefficient in 1st case
T3=y2*math.exp(-2*a2*beta2);     #transmission coefficient in 2nd case

#Result
print "transmission coefficient is",round(T1*10**9,2),"*10**-9"
print "transmission coefficient in 1st case is",round(T2*10**8,2),"*10**-8"
print "transmission coefficient in 2nd case is",round(T3*10**15,2),"*10**-15"

transmission coefficient is 3.27 *10**-9
transmission coefficient in 1st case is 7.62 *10**-8
transmission coefficient in 2nd case is 1.51 *10**-15