# Chapter 15: Quantum Mechanics¶

## Example 15.1, Page 15.24¶

In [1]:
from math import *

# Given
E = 1000 # energy of electron in eV
delta_x = 1e-10 # error in position in m
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
h = 6.62e-34 # Planck constant in J-sec

#Calculations
p = sqrt(2 * m * E * e)
delta_p = h / (4 * pi * delta_x)
P = (delta_p / p) * 100

#Result
print "Percentage of uncertainty in momentum is %.1f%%"%P

Percentage of uncertainty in momentum is 3.1%


## Example 15.3, Page 15.25¶

In [2]:
from math import *

# Given
E = 500 # energy of electron in eV
delta_x = 2e-10 # error in position in m
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
h = 6.62e-34 # Planck constant in J-sec

#Calculations
p = sqrt(2 * m * E * e)
delta_p = h / (4 * pi * delta_x)
P = (delta_p / p) * 100

#Result
print "Percentage of uncertainty in momentum is %.2f%%"%P

Percentage of uncertainty in momentum is 2.18%


## Example 15.4, Page 15.25¶

In [3]:
from math import *

# Given
delta_lambda = 1e-6 # accuracy in wavelength of its one part
lamda = 1e-10 # wavelength of x-ray in m
h = 6.62e-34 # Planck constant in J-sec

#Calculations
delta_x = lamda / (4 * pi * delta_lambda)

#Result
print "Uncertainty in position is %.2f micrometer"%(delta_x*10**6)

Uncertainty in position is 7.96 micrometer


## Example 15.5, Page 15.26¶

In [4]:
from math import *

# Given
delta_x = 1e-10 # error in position in m
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
h = 6.62e-34 # Planck constant in J-sec

#Calculations
delta_p = h / (4 * pi * delta_x)

#Result
print "Uncertainty in momentum is %.2e kg m/sec"%delta_p

Uncertainty in momentum is 5.27e-25 kg m/sec


## Example 15.6, Page 15.26¶

In [5]:
from math import *

# Given
M = 5.4e-26 # momentum of electron in kg-m/sec
p = 0.05 # percentage accuracy in momentum
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
h = 6.62e-34 # Planck constant in J-sec

#Calculations
delta_m = p * M / 100
delta_x = h / (4 * pi * delta_m)

#Result
print "Uncertainty in position is %.3f micrometer"%(delta_x * 10**6)

Uncertainty in position is 1.951 micrometer


## Example 15.7, Page 15.27¶

In [6]:
from math import *

# Given
r = 0.53e-10 # radius of hydrogen atom in m
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
h = 6.62e-34 # Planck constant in J-sec

#Calculations
delta_M = h / (4 * pi * r)
delta_k = delta_M**2 / (2 * m)

#Result
print "Minimum energy of electron is %.3e J"%delta_k

Minimum energy of electron is 5.428e-19 J


## Example 15.8, Page 15.27¶

In [7]:
from math import *

# Given
v = 5e3 # speed of electron in m/sec
a = 0.003 # percentage accuracy in measurement of speed
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
h = 6.62e-34 # Planck constant in J-sec

#Calculations
delta_v = v * a / 100
delta_p = m * delta_v
delta_x = h / (4 * pi * delta_p)

#Result
print "Uncertainty in determining the position of electron is %.3e m"%delta_x

Uncertainty in determining the position of electron is 3.859e-04 m


## Example 15.9, Page 15.27¶

In [8]:
from math import *

# Given
v = 6.6e4 # speed of electron in m/sec
a = 0.01 # percentage accuracy in measurement of speed
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
h = 6.6e-34 # Planck constant in J-sec

#Calculations
delta_v = v * a / 100
delta_p = m * delta_v
delta_x = h / (4 * pi * delta_p)

#Result
print "Uncertainty in determining the position is %.2e m"%delta_x

Uncertainty in determining the position is 8.74e-06 m


## Example 15.10, Page 15.28¶

In [9]:
from math import *

# Given
v = 3e7 # speed of electron in m/sec
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
h = 6.62e-34 # Planck constant in J-sec
c = 3e8 # speed of light in m/sec

#Calculations
delta_p = m * v / sqrt(1 - (v/c)**2)
delta_x = h / (4 * pi * delta_p)

#Result
print "Uncertainty in determining the position is %.2e m"%delta_x

Uncertainty in determining the position is 1.92e-12 m


## Example 15.11, Page 15.28¶

In [10]:
from math import pi

# Given
t = 2.5e-14 # life time of hydrogen atom in exited state in sec
h = 6.62e-34 # Planck constant in J-sec

#Calculations
delta_E = h / (4 * pi * t)

#Result
print "Minimum error in measurement of the energy is %.2e J"%delta_E

Minimum error in measurement of the energy is 2.11e-21 J


## Example 15.12, Page 15.28¶

In [12]:
from math import *

# Given
t = 10**-8 # life time of atom in exited state in sec
h = 6.62e-34 # Planck constant in J-sec

#Calculations
delta_f = 1 / (4 * pi * t)

#Result
print "Minimum uncertainty in frequency is %.2e sec"%delta_f

Minimum uncertainty in frequency is 7.96e+06 sec


## Example 15.13, Page 15.29¶

In [13]:
from math import pi

# Given
delta_x = 20e-10 # uncertainty in position in m
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
m_ = 1.67e-27 # mass of proton in kg
c = 3e8 # speed of light in m/sec
h = 6.62e-34 # Planck constant in J-sec

#Calculations
delta_v1 = h / (4 * pi * m * delta_x)
delta_v2 = h / (4 * pi * m_ * delta_x)
r = delta_v2 / delta_v1

#Result
print "Ratio of uncertainty in velocity of a proton and an electron is %.2e"%r

Ratio of uncertainty in velocity of a proton and an electron is 5.45e-04


## Example 15.14, Page 15.29¶

In [14]:

# Given
delta_x = 1e-10 # width of box in m
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
c = 3e8 # speed of light in m/sec
h = 6.62e-34 # Planck constant in J-sec
n = 1 # for n=1

#Calculations
E = (n**2 * h**2) / (8 * m * delta_x**2)
n = 2 # for n=2
E_ = (n**2 * h**2) / (8 * m * delta_x**2)

#Result
print "Energy of electron - \nFor (n=1) energy is %.2e J\nFor (n=2) energy is %.2e J"%(E,E_)

Energy of electron -
For (n=1) energy is 6.02e-18 J
For (n=2) energy is 2.41e-17 J


## Example 15.15, Page 15.30¶

In [15]:
# Given
l = 1e-10 # width of box in m
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
c = 3e8 # speed of light in m/sec
h = 6.62e-34 # Planck constant in J-sec
n = 1 # for n=1

#Calculations
E = (n**2 * h**2) / (8 * m * l**2)
n = 2 # for n=2
E_ = (n**2 * h**2) / (8 * m * l**2)
d = E_ - E

#Result
print "Energy difference is %.2e J"%d

Energy difference is 1.81e-17 J


## Example 15.16, Page 15.30¶

In [16]:

# Given
l = 3e-10 # width of box in m
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
c = 3e8 # speed of light in m/sec
h = 6.62e-34 # Planck constant in J-sec
n = 1 # For n=1

#Calculations
E = (n**2 * h**2) / (8 * m * l**2)
n = 2 # For n=2
E_ = (n**2 * h**2) / (8 * m * l**2)
n = 3 # For n=3
E__ = (n**2 * h**2) / (8 * m * l**2)

#Result
print "Energy of electron -\nFor (n=1) is %.1e J\nFor (n=2) is %.2e J\nFor (n=3) is %.2e J"%(E,E_,E__)

Energy of electron -
For (n=1) is 6.7e-19 J
For (n=2) is 2.68e-18 J
For (n=3) is 6.02e-18 J


## Example 15.17, Page 15.30¶

In [17]:

# Given
l = 2.5e-10 # width of box in m
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
c = 3e8 # speed of light in m/sec
h = 6.62e-34 # Planck constant in J-sec

#Calculations
n = 1 # for n=1
E = (n**2 * h**2) / (8 * m * l**2)
n = 2 # for n=2
E_ = (n**2 * h**2) / (8 * m * l**2)

#Result
print "Energy of electron -\nFor (n=1) is %.2e J\nFor (n=2) is %.3e J"%(E,E_)

Energy of electron -
For (n=1) is 9.63e-19 J
For (n=2) is 3.853e-18 J


## Example 15.18, Page 15.31¶

In [20]:

# Given
l = 1e-14 # width of box in m
e = 1.6e-19 # charge on an electron in C
m = 1.67e-27 # mass of neutron in kg
c = 3e8 # speed of light in m/sec
h = 6.62e-34 # Planck constant in J-sec

#Calculations
n = 1 # for n=1
E = (h**2) / (8 * m * l**2)

#Result
print "Lowest energy of neutron confined in the nucleus is %.2e J"%E

Lowest energy of neutron confined in the nucleus is 3.28e-13 J


## Example 15.19, Page 15.31¶

In [21]:

# Given
l = 1e-10 # width of box in m
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
c = 3e8 # speed of light in m/sec
h = 6.63e-34 # Planck constant in J-sec

#Calculations
n = 1 # for n=1
p1 = (n * h) / (2 * l)
E = (n**2 * h**2) / (8 * m * l**2)
n = 2 # for n=2
p2 = (n * h) / (2 * l)
E_ = (n**2 * h**2) / (8 * m * l**2)

#Result
print "Energy of electron -\nFor (n=1) is %.2e J\nFor (n=2) is %.2e J"%(E,E_)
print "\nMomentum of electron -\nFor (n=1) is %.3e kg-m/sec\nFor (n=2) is %.2e kg-m/sec"%(p1,p2)

Energy of electron -
For (n=1) is 6.04e-18 J
For (n=2) is 2.42e-17 J

Momentum of electron -
For (n=1) is 3.315e-24 kg-m/sec
For (n=2) is 6.63e-24 kg-m/sec


## Example 15.20, Page 15.32¶

In [22]:

# Given
l = 1e-10 # length of box in m
m = 9.1e-31 # mass of electron in kg
c = 3e8 # speed of light in m/sec
h = 6.62e-34 # Planck constant in J-sec

#Calculations
n = 1 # for n=1
E1 = (n**2 * h**2) / (8 * m * l**2)
lambda1 =2*l
n = 2 # for n=2
E2 = (n**2 * h**2) / (8 * m * l**2)
lambda2 =2*l/2
n = 3 # for n=3
E3 = (n**2 * h**2) / (8 * m * l**2)
lambda3 =2*l/3

#Results
print "Energy Eigen value of electron -\nFor (n=1) is %.2e J\nFor (n=2) is %.2e J\nFor (n=3) is %.2e J"%(E1,E2,E3)
print "\nde-Broglie wavelength of electron -\nFor (n=1) is %.f A\nFor (n=2) is %.f A \nFor (n=3) is %.3f A"%(lambda1*1e10,lambda2*1e10,lambda3*1e10)

Energy Eigen value of electron -
For (n=1) is 6.02e-18 J
For (n=2) is 2.41e-17 J
For (n=3) is 5.42e-17 J

de-Broglie wavelength of electron -
For (n=1) is 2 A
For (n=2) is 1 A
For (n=3) is 0.667 A


## Example 15.21, Page 15.32¶

In [23]:

# Given
E1 = 3.2e-18 # minimum energy possible for a particle entrapped in a one dimensional box in J
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
c = 3e8 # speed of light in m/sec
h = 6.62e-34 # Planck constant in J-sec

#Calculations
E1 = E1 / e # in eV
n = 2 # for n=2
E2 = n**2 * E1
n = 3 # for n=3
E3 = n**2 * E1
n = 4 # for n=4
E4 = n**2 * E1

#Result
print "Energy Eigen values -\nFor (n=2) for %.f eV\nFor (n=3) is %.f eV\nFor (n=4) is %.f eV"%(E2,E3,E4)

Energy Eigen values -
For (n=2) for 80 eV
For (n=3) is 180 eV
For (n=4) is 320 eV


## Example 15.22, Page 15.33¶

In [24]:
from math import *

# Given
l = 4e-10 # width of box in m
E = 9.664e-17 # energy of electron in J
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
c = 3e8 # speed of light in m/sec
h = 6.62e-34 # Planck constant in J-sec

#Calculations
n = 1 # for n=1
E1 = (n**2 * h**2) / (8 * m * l**2)
N = sqrt(E / E1)
p = ((N) * h) / (2 * l)

#Result
print "Order of exited state is %d\nMomentum of electron is %.2e kg-m/sec"%(N,p)

Order of exited state is 16
Momentum of electron is 1.33e-23 kg-m/sec


## Example 15.23, Page 15.33¶

In [25]:

# Given
l = 10e-10 # width of box containing electron in m
E = 9.664e-17 # energy of electron in J
M = 0.001 # mass of glass marble in kg
l_ = 0.2 # width of box containing marble in m
e = 1.6e-19 # charge on an electron in C
m = 9.1e-31 # mass of electron in kg
c = 3e8 # speed of light in m/sec
h = 6.62e-34 # Planck constant in J-sec

#Calculations
# For electron
n = 1 # for n=1
E1 = (n**2 * h**2) / (8 * m * l**2)
E2 = 2**2* E1
E3 = 3**2 * E1
# For glass marble
E1_ = h**2/(8*M*l_**2)
E2_ = 2**2 * E1_
E3_ = 3**2 *E1_

#Result
print "\nEnergy levels of electron- \nFor (n=1) is %.2e J\nFor (n=2) is %.2e J\n For (n=3) is %.2e J"%(E1,E2,E3)
print "\nEnergy levels of marble- \nFor (n=1) is %.2e J\nFor (n=2) is %.2e J\nFor (n=3) is %.2e J"%(E1_,E2_,E3_)

Energy levels of electron-
For (n=1) is 6.02e-20 J
For (n=2) is 2.41e-19 J
For (n=3) is 5.42e-19 J

Energy levels of marble-
For (n=1) is 1.37e-63 J
For (n=2) is 5.48e-63 J
For (n=3) is 1.23e-62 J