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