Chapter 5:Quantum Mechanics

Example 5.3.1, Page number 5-5

In [57]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 10**-2          #mass of object(kg)
v = 1               #velocity(m/s)
l = 1*10**-10       #wavelength(m)
d = 10**-3          #distance travelled(m)

#Calculations
lamda = h/(m*v)

v = h/(m*l)

t1 = d/v
t = t1/(365*24*60*60)

#Results
print "de Brogile wavelength =",round(lamda/1e-32,2),"*10^-32 m"
print "Velocity =",round(v/1e-22,2),"*10^-22 m/s(Calculation mistake in the textbook)"
print "Distance travelled =",round(t/1e+10,2),"*10^10 years(Calculation mistake in the textbook)"

de Brogile wavelength = 6.63 *10^-32 m
Velocity = 6.63 *10^-22 m/s(Calculation mistake in the textbook)
Distance travelled = 4.78 *10^10 years(Calculation mistake in the textbook)

Example 5.3.2, Page number 5-6

In [9]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 9.1*10**-31     #mass of electron(kg)
lamda = 1*10**-10   #wavelength(m)

#calculation
v = h/(m*lamda)

#Result
print "Velocity =",round(v/1e+6,2),"*10^6 m/s"

Velocity = 7.29 *10^6 m/s

Example 5.3.3, Page number 5-6

In [13]:
#Variable declaration
h = 6.63*10**-34     #Planck's constant(J-s)
m = 9.1*10**-31      #mass of electron(kg)
lamda = 5000*10**-10 #wavelength(m)

#calculation
E = (h**2/(2*m*lamda**2))/(1.6*10**-19)

#Result
print "Kinetic energy =",round(E/1e-6,5),"*10^-6 eV"

Kinetic energy = 6.03804 *10^-6 eV

Example 5.3.4, Page number 5-7

In [17]:
import math

#Variable declaration
h = 6.63*10**-34      #Planck's constant(J-s)
m = 1.676*10**-27     #mass of object(kg)
E = 0.025*1.6*10**-19 #energy(J)

#Calculation
lamda = h/math.sqrt(2*m*E)

#Result
print "Wavelength =",round(lamda/1e-10,2),"A"

Wavelength = 1.81 A

Example 5.3.5, Page number 5-7

In [18]:
import math

#Variable declaration
h = 6.63*10**-34      #Planck's constant(J-s)
m = 9.1*10**-31       #mass of electron(kg)
E = 120*1.6*10**-19   #energy(J)

#Calculation
lamda = h/math.sqrt(2*m*E)

#Result
print "Wavelength =",round(lamda/1e-10,2),"A"

Wavelength = 1.12 A

Example 5.3.6, Page number 5-7

In [23]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 1.67*10**-27    #mass of neutron(kg)
lamda = 1*10**-10   #wavelength(m)

#calculation
v = h/(m*lamda)

E = (h**2/(2*m*lamda**2))/(1.6*10**-19)

#Result
print "Velocity =",round(v,2),"m/s"
print "Kinetic energy =",round(E/1e-2,3),"*10^-2 eV"

Velocity = 3970.06 m/s
Kinetic energy = 8.225 *10^-2 eV

Example 5.3.7, Page number 5-8

In [26]:
import math

#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 1               #mass of object(kg)
v = 1               #velocity(m/s)
V = 182             #potential differnce(V)

#Calculation
#Case i
lamda = 12.27/math.sqrt(V)

#Case ii
l = h/(m*v)

#Results
print "de Brogile wavelength for accelerated electron=",round(lamda,2),"A"
print "de Brogile wavelength for object=",l,"A"

de Brogile wavelength for accelerated electron= 0.91 A
de Brogile wavelength for object= 6.63e-34 A

Example 5.3.8, Page number 5-9

In [28]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 9.1*10**-31     #mass of electron(kg)
lamda = 10**-14     #wavelength(m)

#Calculations
p = h/lamda

E = (p**2/(2*m))/(1.6*10**-13)

#Results
print "Momentum =",p,"kg-m/s"
print "Energy =",round(E,2),"MeV"

Momentum = 6.63e-20 kg-m/s
Energy = 15095.09 MeV

Example 5.3.9, Page number 5-10

In [55]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
V = 3000            #potential differnce(V)

#Calculation
#Case i
lamda = 12.27/math.sqrt(V)

#Case ii
p = h/(lamda*10**-10)

#Case iii
lamda_b = 1/(lamda*10**-10)

#Case iv
d = 2.04*10**-10  #m
n = 1             #for first order
theta = math.degrees(math.asin((n*lamda*10**-10)/(2*d)))

#Results
print "Momentum =",round(p/1e-23,2),"*10^-23 kg-m/s"
print "de Brogile wavelength =",round(lamda,3),"A"
print "Wave number =",round(lamda_b/1e+10,3),"*10^10 /m"
print "Bragg angle =",round(theta,3),"degrees"

Momentum = 2.96 *10^-23 kg-m/s
de Brogile wavelength = 0.224 A
Wave number = 4.464 *10^10 /m
Bragg angle = 3.147 degrees

Example 5.3.10, Page number 5-11

In [49]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
V = 10*10**3        #potential differnce(V)

#Calculation
lamda = 12.27/math.sqrt(V)

p = h/(lamda*10**-10)

#result
print "Wavelength =",lamda,"A"
print "Momentum =",round(p/1e-23,3),"*10^-23 kg-m/s"

Wavelength = 0.1227 A
Momentum = 5.403 *10^-23 kg-m/s

Example 5.3.12, Page number 5-12

In [4]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 6.68*10**-27    #mass of particle(kg)
E = 1.6*10**-16     #energy(J)

#Calculations
lamda = h/math.sqrt(2*m*E)

v = h/(m*lamda)

#Results
print "Wavelength =",round(lamda/1e-13,3),"*10^-13 A"
print "Velocity =",round(v/1e+5,2),"*10^5 m/s"

Wavelength = 4.535 *10^-13 A
Velocity = 2.19 *10^5 m/s

Example 5.3.13, Page number 5-12

In [18]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
c = 3*10**8         #velocity of light(m/s)
m = 9.1*10**-31     #mass of electron(kg)
E = 1.6*10**-19     #energy(J)

#Calculations
lamda_ph = (h*c)/E            #wavelength of photon
 
lamda_e = h/math.sqrt(2*m*E)  #wavelength of electron

#Result
print "Wavelength of proton =",round(lamda_ph/1e-6,3),"*10^-6 m"
print "Wavelength of electron =",round(lamda_e/1e-10,3),"A"

Wavelength of proton = 1.243 *10^-6 m
Wavelength of electron = 12.286 A

Example 5.3.14, Page number 5-13

In [23]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
c = 3*10**8         #velocity of light(m/s)
m = 9.1*10**-31     #mass of electron(kg)

#Calculations
E = m*c**2
lamda = h/math.sqrt(2*m*E)

#Result
print "de Brogile wavelength =",round(lamda/1e-12,3),"*10^-12 m"

de Brogile wavelength = 1.717 *10^-12 m

Example 5.7.1, Page number 5-26

In [26]:
import math

#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 9.1*10**-31     #mass of electron(kg)
del_v = (0.01*400)/100

#Calculation
del_x = h/(4*math.pi*m*del_v)

#Result
print "Accuracy =",round(del_x/1e-3,2),"mm"

Accuracy = 1.45 mm

Example 5.7.2, Page number 5-27

In [32]:
import math

#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 9.1*10**-31     #mass of electron(kg)
del_x = 10*10**-9   #m
E = 1.6*10**-16     #energy(J)

#Calculation
del_px = h/(4*math.pi*del_x)
p = math.sqrt(2*m*E)
per = (del_px/p)*100

#Result
print "Percentage of uncertainity =",round(per,4),"%"

Percentage of uncertainity = 0.0309 %

Example 5.7.3, Page number 5-27

In [1]:
import math

#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 9.1*10**-31     #mass of electron(kg)
del_v = (0.01*4*10**5)/100

#Calculation
del_x = h/(4*math.pi*m*del_v)

#Result
print "Accuracy =",round(del_x/1e-6,2),"*10^-6 m"

Accuracy = 1.45 *10^-6 m

Example 5.7.4, Page number 5-27

In [39]:
import math

#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 9.1*10**-31     #mass of electron(kg)
del_v = (1.88*10**6)/100

#Calculation
del_x = h/(4*math.pi*m*del_v)

#Result
print "Precision =",round(del_x/1e-9,3),"*10^-9 m"

Precision = 3.084 *10^-9 m

Example 5.7.5, Page number 5-28

In [44]:
import math

#Variable declaration
lamda = 4*10**-7    #wavelength(m)
c = 3*10**8         #velocity of light(m/s)
del_l = 8*10**-15   #spectral width(m)

#calculation
del_t = lamda**2/(4*math.pi*c*del_l)

#Result
print "Time spent by the elctrons =",round(del_t/1e-9,3),"*10^-9 s"

Time spent by the elctrons = 5.305 *10^-9 s

Example 5.7.6, Page number 5-29

In [50]:
import math

#Variable declaration
h = 6.63*10**-34      #Planck's constant(J-s)
del_t = 1.4*10**-10   #time spent(s)

#calculation
E = (h/(4*math.pi*del_t))/(1.6*10**-19)

#Result
print "Uncertainity =",round(E/1e-6,2),"*10^-6 eV"

Uncertainity = 2.36 *10^-6 eV

Example 5.7.7, Page number 5-29

In [2]:
import math

#Variable declaration
lamda = 546*10**-9  #wavelength(m)
c = 3*10**8         #velocity of light(m/s)
del_l = 10**-14     #spectral width(m)

#calculation
del_t = lamda**2/(4*math.pi*c*del_l)

#Result
print "Time spent by the elctrons =",round(del_t/1e-9,2),"*10^-9 s"

Time spent by the elctrons = 7.91 *10^-9 s

Example 5.15.1, Page number 5-41

In [1]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 9.1*10**-31     #mass of electron(kg)
L = 2*10**-10       #width(m)

#Calculation
E1 = (h**2/(8*m*L**2))/(1.6*10**-19)
E2 = 4*E1
E3 = 9*E1

#Result
print "Energy of electron in ground state =",round(E1,3),"eV"
print "Energy of electron in first state =",round(E2,3),"eV"
print "Energy of electron in second state =",round(E3,3),"eV"

Energy of electron in ground state = 9.434 eV
Energy of electron in first state = 37.738 eV
Energy of electron in second state = 84.91 eV

Example 5.15.2, Page number 5-42

In [2]:
#Variable declaration
L = 5.6*10**-3      #width(m)

#Calculation
E = L/4

#Result
print "Ground state energy =",E,"eV"

Ground state energy = 0.0014 eV

Example 5.15.4, Page number 5-43

In [9]:
import scipy
from scipy import integrate

#Variable declaration
#Intervals
x1 = 0
x2 = 1./2

#Calculation
x = lambda x: 3*x**2
P = integrate.quad(x, x1, x2)

#Result
print "Probability =",P[0]

 Probability = 0.125

Example 5.15.5, Page number 5-44

In [41]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m1 = 9.1*10**-31    #mass of electron(kg)
m2 = 10**-9         #mass of grain dust(kg)
L1 = 10**-9         #width(m)
L2 = 10**-4         #width(m)

#Calculation
#For electron
print "For an electron, the lowest thre energy states obtained for n=1,2 and 3 are"
for n in range(1,4):
    En1 = ((n**2*h**2)/(8*m*L1**2))/(1.6*10**-19)
    print round(En1,4),"eV"
    
#For the grain of dust
print "\nFor a grain of dust, the lowest thre energy states obtained for n=1,2 and 3 are"
for n in range(1,4):
    En2 = ((n**2*h**2)/(8*m2*L2**2))/(1.6*10**-19)
    print round(En2/1e-32,3),"eV"

For an electron, the lowest thre energy states obtained for n=1,2 and 3 are
0.3774 eV
1.5095 eV
3.3964 eV

For a grain of dust, the lowest thre energy states obtained for n=1,2 and 3 are
3.434 eV
13.737 eV
30.907 eV

Example 5.15.6, Page number 5-45

In [48]:
import math

#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 9.1*10**-31     #mass of electron(kg)
E1 = 38*1.6*10**-19 #energy(J)

#Calculation
L = math.sqrt(h**2/(8*m*E1))
              
#Result
print "Width of well =",round(L/1e-10,4),"A"

Width of well = 0.9965 A

Example 5.15.7, Page number 5-45

In [55]:
#Variable declaration
h = 6.63*10**-34    #Planck's constant(J-s)
m = 9.1*10**-31     #mass of electron(kg)
L = 5*10**-10       #width(m)
n1 = 1
n2 = 2
c = 3*10**8         #velocity of light(m/s)

#Calculation
E = ((3*h**2)/(8*m*L**2))   #E2-E1
Ev = E/(1.6-10**-19)        #J
lamda = (h*c)/E

#Result
print "Energy =",round(Ev/1e-19,2),"eV"
print "Wavelength =",round(lamda/1e-7,3),"*10^-7 m"

Energy = 4.53 eV
Wavelength = 2.745 *10^-7 m