# chapter 03 : Matter Waves, Wave particle duality & Uncertainty principle¶

## Ex3.1 : : Pg: 77¶

In [1]:
#Kinetic energy of an electron
h = 6.6e-034     # Planck's constant, J-s
m = 9.1e-031     # mass of an electron, kg
L = 9e-010     # wavelength of an electron, m
# since E = (m*v**2)/2, Energy of an electron, joule
# thus v = sqrt(2*E/m), solving for L in terms of E, we have
# L = h/sqrt(2*m*E), wavelength of an electron, m
# On solving for E
E = h**2/(2*m*L**2)
print "The kinetic energy of an electron = %6.4f eV"% (E/1.6e-019)

The kinetic energy of an electron = 1.8468 eV


## Ex3.2 : : Pg: 78¶

In [2]:
from math import sqrt
#Wavelength of electrons
h = 6.6e-034     # Planck's constant, J-s
m = 9.1e-031     # mass of an electron, kg
e = 1.6e-019     # Charge on an electron, coulomb
E = 100*e     # Energy of beam of electrons, joule
# since E = (m*v**2)/2     # Energy of beam of electron, joule
p = sqrt(2*m*E)     # Momentum of beam of electrons, kg-m/s
L = h/p     # wavelength of a beam of electron, m
print "The wavelength of electrons = %4.2f angstorm"% (L/1e-010)

The wavelength of electrons = 1.22 angstorm


## Ex3.3 : : Pg: 78¶

In [3]:
#Momentum of photon
h = 6.624e-034     # Planck's constant, J-s
L = 6e-07     # wavelength of photon, m
M = h/L     # Momentum of photon, kg-m/s
print "The momentum of photon = %5.3e kg-m/s"% (M)

The momentum of photon = 1.104e-27 kg-m/s


## Ex3.4 : : Pg: 78¶

In [4]:
from math import sqrt
#Momentum of an electron
m = 9.1e-031     # Mass of an electron, kg
E = 1.6e-010     # Kinetic energy of an electron, joule
# Since E = p**2/2*m     # Kinetic energy of an electron, joule
p = sqrt(2*m*E)     # Momentum of an electron, kg-m/s
print "The momentum of an electron = %3.1e kg-m/s"% p

The momentum of an electron = 1.7e-20 kg-m/s


## Ex3.5 : : Pg: 79¶

In [5]:
from math import sqrt
#wavelength of a particle
h = 6.624e-034     # Planck's constant, J-s
m = 9e-031     # Mass of an electron, kg
U = 1.6e-017     # Kinetic energy of an particle, joule
# Since U = (m*v**2)/2     # Kinetic energy of a particle, joule
# such that v = sqrt(2*U/m)     # Velocity of the particle, m/s
L = h/sqrt(2*m*U)     # wavelength of a particle, m
print "The wavelength of a particle = %5.3f angstorm"% (L/1e-010)

The wavelength of a particle = 1.234 angstorm


## Ex3.6 : : Pg: 79¶

In [6]:
#Comparison of energy of photon and neutron
m = 1.67e-027     # Mass of neutron, kg
L = 1e-010     # Wavelength of neutron and photon, m
c = 3e+08     # Velocity of light, m/s
h = 6.624e-034     # Plancks constant, joule second
U_1 = h*c/L     # Energy of photon, joule
# Since  U_2 = (m*v**2)/2, Energy of neutron, joule
# Thus v = h/m*L_2, Velocity of the particle, m/s
# on solving for U_2
U_2 = h**2/(2*m*L**2)     # Energy of photon, joule
print "The ratio of energy of photon and neutron = %4.2e "% (U_1/U_2)

The ratio of energy of photon and neutron = 1.51e+05


## Ex3.7: : Pg: 80¶

In [7]:
from math import sqrt
#de-Broglie wavelength of electrons
L_1 = 3e-07     # Wavelength of ultraviolet light, m
L_0 = 4e-07     # Threshold wavelength of ultraviolet light, m
m = 9.1e-031     # Mass of an electron, kg
c = 3e+08     # Velocity of light, m/s
h = 6.624e-034     # Plancks constant, joule-second
U = h*c*(1/L_1-1/L_0)     # Maximum Kinetic energy of emitted electrons, joule
# since U = m*v**2/2, Kinetic energy of electrons, joule
# Thus v = sqrt(2*U/m), so that L_2 becomes
L_2 = h/sqrt(2*m*U)     # wavelength of electrons, m
print "The wavelength of the electrons = %3.1f angstorm"% (L_2/1e-010)

The wavelength of the electrons = 12.1 angstorm


## Ex3.8 : :Pg: 80¶

In [8]:
from math import sqrt
#de-Broglie wavelength of accelerated electrons
m = 9.1e-031     # Mass of an electron, kg
e = 1.6e-019     # Charge on an electron, Coulamb
h = 6.624e-034     # Plancks constant, joule second
V = 1     # For simplicity, we assume retarding potential to be unity, volt
# Since  e*V = (m*v**2)/2     # Energy of electron, joule
v = sqrt(2*e*V/m)     # Velocity of electrons, m/s
L = h/(m*v)     # Wavelength of electrons, m
print "The de-Broglie wavelength of accelerated electrons = %5.2f/sqrt(V) "% (L/1e-010)

The de-Broglie wavelength of accelerated electrons = 12.28/sqrt(V)


## Ex3.9 : : Pg: 81¶

In [9]:
from math import sqrt
#Wavelength of matter waves
E = 2e-016     # Energy of electrons, joule
h = 6.624e-034     # Planck's constant, J-s
m = 9.1e-031     # mass of the electron, kg
# since E = (m*v**2)/2, the energy of an electron, joule
# such that v = sqrt(2*E/m)     # Velocity of electron, m/s
# As L = h/m*v, wavelength of the electron, m
# on solving for L in terms of E
L = h/sqrt(2*m*E)     # wavelength of the electron, m
print "The wavelength of the electron = %5.3f angstorm"% (L/1e-010)

The wavelength of the electron = 0.347 angstorm


## Ex3.10 : : Pg: 81¶

In [10]:
from math import sqrt
#Momentum of proton
U = 1.6e-010     # Kinetic energy of proton, joule
h = 6.624e-034     # Planck's constant, J-s
m = 1.67e-027     # mass of proton, kg
v = sqrt(2*U/m)     # Velocity of proton, m/s
p = m*v     # Momentum of proton, kg m/s
print "The momentum of proton = %4.2e kgm/s"% p

The momentum of proton = 7.31e-19 kgm/s


## Ex3.11 : : Pg: 82¶

In [11]:
from math import sqrt
#Wavelength of an electron
U = 1.6e-013     # Kinetic energy of the electron, joule
h = 6.624e-034     # Planck's constant, J-s
m = 9.1e-031     # Mass of the electron, kg
v = sqrt(2*U/m)     # Velocity of the electron, m/s
L = h/(m*v)     # Wavelength of the electron, m
print "The wavelength of an electron = %5.3e angstorm" % (L/1e-010)

The wavelength of an electron = 1.228e-02 angstorm


## Ex3.12: :Pg: 82¶

In [12]:
from math import sqrt
#De-Broglie wavelength of thermal neutrons
m = 1.6749e-027     # Mass of neutron, kg
h = 6.624e-034     # Plancks constant, joule second
k = 1.38e-021     # Boltzmann constant, joule per kelvin
T = 300     # Temperature of thermal neutrons, kelvin
# Since m*v**2/2 = (3/2)*k*T     # Energy of neutron, joule
v = sqrt(3*k*T/m)     # Velocity of neutrons, m/s
L = h/(m*v)     # Wavelength of neutrons, m
print "The de-Broglie wavelength of thermal neutrons = %5.3f angstorm "% (L/1e-010)

The de-Broglie wavelength of thermal neutrons = 0.145 angstorm


## Ex3.13 : :Pg: 82¶

In [13]:
#Kinetic energy of a proton
L = 1e-010     # wavelength of proton, m
m = 1.67e-027     # Mass of proton, kg
h = 6.624e-034     # Plancks constant, joule second
# Since L = h/(m*v)     # wavelength of proton, m
v = h/m*L     # Velocity of proton, m/s
v_k = h**2/(2*L**2*m)     # Kinetic energy of proton, joule
print "The kinetic energy of proton = %3.1e eV "% (v_k/1.6e-019)

The kinetic energy of proton = 8.2e-02 eV


## Ex3.14: : Pg: 85¶

In [14]:
#Energy of electrons in a one dimensional box
n1 = 1; l = 0; ml = 0; ms = 1.0/2     # Quantum numbers of first electron
n2 = 1; l = 0; ml = 0; ms = -1.0/2     # Quantum numbers of second electron
# The lowest energy corresponds to the ground state of electrons
n = n1     # n1 = n2 = n
m = 9.1e-031     # Mass of electron, kg
h = 6.626e-034     # Planck's constant, Js
a = 1     # For convenience, length of the box is assumed to be unity
E = 2*n**2*h**2/(8*m*a**2)     # Lowest energy of electron, joule
print "The lowest energy of electron = %6.4e/a**2"% E

The lowest energy of electron = 1.2062e-37/a**2


## Ex3.15: : Pg:85¶

In [15]:
#Lowest energy of three electrons in box
n1 = 1; l = 0; ml = 0; ms = 1.0/2     # Quantum numbers of first electron
n2 = 1; l = 0; ml = 0; ms = -1.0/2     # Quantum numbers of second electron
n3 = 2; l = 0; ml = 0; ms = +1.0/2     # Quantum numbers of third electron
# The lowest energy corresponds to the ground state of electrons
m = 9.1e-031     # Mass of electron, kg
h = 6.626e-034     # Planck's constant, Js
a = 1.0     # For convenience, length of the box is assumed to be unity
E = (n1**2*h**2/(8*m*a**2)+n2**2*h**2/(8*m*a**2))+n3**2*h**2/(8*m*a**2)     # Lowest energy of electron, joule
print "The lowest energy of electron = %6.4e/a**2"% E

The lowest energy of electron = 3.6185e-37/a**2


## Ex3.16 : :Pg: 86¶

In [16]:
#Zero point energy of system
m = 9.1e-031     # Mass of an electron, kg
a = 1e-010     # Length of box, m
h = 6.624e-034     # Plancks constant, joule second
n = 1     # Principal quantum number for the lowest energy level
E1 = 2*h**2/(8*m*a**2)     # Energy for the two electron system in the n =1 energy level, joule
E2 = 8*(2**2*h**2)/(8*m*a**2)     # Energy for the eight electron system in the n = 2 energy level, joule
E = E1 +E2     # Total lowest energy of system, joule
print "The zero point energy of system = %4.2e J "% E

The zero point energy of system = 2.05e-16 J


## Ex3.17 : :Pg: 86¶

In [17]:
#Mean energy per electron at 0K
m = 9.1e-031     # Mass of an electron, kg
a = 50e-010     # Length of molecule, m
h = 6.624e-034     # Plancks constant, joule second
E = h**2/(8*m*a**2)     # Energy per electron, joule
print "The mean energy per electron at 0K = %3.1e eV "% (E/1.6e-019)

The mean energy per electron at 0K = 1.5e-02 eV


## Ex3.18 : :Pg: 87¶

In [18]:
#Lowest energy of two electron system
m = 9.1e-031     # Mass of an electron, kg
a = 1e-010     # Length of box, m
h = 6.624e-034     # Plancks constant, joule second
E = 2*h**2/(8*m*a**2)     # Energy of two electron system, joule
print "The lowest energy of two electron system = %4.1f, eV"% (E/1.6e-019)

The lowest energy of two electron system = 75.3, eV


## Ex3.19 : :Pg: 87¶

In [19]:
#Total energy of the three electron system
m = 9.1e-031     # Mass of an electron, kg
h = 6.624e-034     # Plancks constant, joule second
a = 1e-010     # Length of the molecule, m
E = 6*h**2/(8*m*a**2)     # Energy of three electron system, joule
print "The total energy of three electron system = %6.2f, eV "%(E/1.6e-019)

The total energy of three electron system = 226.02, eV


## Ex3.20 : :Pg: 92¶

In [20]:
#Minimum uncertainity in the velocity of an electron
m = 9.1e-031     # Mass of an electron, kg
del_x = 1e-010     # Length of the box, m
h_bar = 1.054e-034     # Reduced Plancks constant, joule second
del_v = h_bar/(m*del_x)     # Minimum uncertainity in velocity, m/s
print "The minimum uncertainity in the velocity of electron = %4.2e m/s "% del_v

The minimum uncertainity in the velocity of electron = 1.16e+06 m/s


## Ex3.21 : :Pg: 92¶

In [21]:
#Uncertainity in momentum and kinetic energy of the proton
m = 1.67e-027     # Mass of a proton, kg
del_x = 1e-014     # Uncertainity in position, m
h_bar = 1.054e-034     # Reduced Plancks constant, joule second
del_p = h_bar/del_x     # Minimum uncertainity in momentum, kgm/s
del_E = del_p**2/(2*m)     # Minimum uncertainity in kinetic energy, joule
print "The minimum uncertainity in momentum of the proton = %5.3e kgm/s"%del_p
print "The minimum uncertainity in kinetic energy of the proton = %5.3e eV"% (del_E/1.6e-019)

The minimum uncertainity in momentum of the proton = 1.054e-20 kgm/s
The minimum uncertainity in kinetic energy of the proton = 2.079e+05 eV


## Ex3.22 : :Pg: 93¶

In [22]:
from math import pi
#Uncertainity in the position of an electron
m = 9.1e-031     # Mass of an electron, kg
v = 600     # Speed of electron, m/s
h_bar = 6.6e-034     # Reduced Plancks constant, joule second
p = m*v     # Momentum of electron, kgm/s
del_p = 5e-05*m*v     # Minimum uncertainity in momentum, kgm/s
del_x = h_bar/(4*pi*del_p)     # Uncertainity in position, m
print "The uncertainity in the position of the electron = %5.3f mm"% (del_x/1e-03)

The uncertainity in the position of the electron = 1.924 mm


## Ex3.23 : :Pg: 93¶

In [23]:
from math import pi
#Uncertainity in the position of a bullet
m = 0.025     # Mass of an bullet, kg
v = 400     # Speed of bullet, m/s
h_bar = 6.6e-034     # Reduced Plancks constant, joule second
p = m*v     # Momentum of bullet, kgm/s
del_p = 2e-04*p     # Minimum uncertainity in momentum, kgm/s
del_x = h_bar/(4*pi*del_p)     # Uncertainity in position, m
print "The uncertainity in the position of the bullet = %5.3e m"% del_x

The uncertainity in the position of the bullet = 2.626e-32 m


## Ex3.24 : :Pg: 94¶

In [24]:
from math import pi
#Unertainity in the position of an electron
m = 9.1e-31     # Mass of an electron, kg
v = 300     # Speed of electron, m/s
h_bar = 6.6e-034     # Reduced Plancks constant, joule second
p = m*v     # Momentum of electron, kgm/s
del_p = 1e-04*p     # Minimum uncertainity in momentum, kgm/s
del_x = h_bar/(4*pi*del_p)     # Uncertainity in position, m
print "The uncertainity in the position of the electron = %5.3f mm"% (del_x/1e-03)

The uncertainity in the position of the electron = 1.924 mm


## Ex3.25 : :Pg: 94¶

In [25]:
from math import pi
#Unertainity in the velocity of an electron
m = 9.1e-31     # Mass of an electron, kg
del_x = 1e-10     # Length of box, m
h_bar = 6.6e-034     # Reduced Plancks constant, joule second
del_p = m*del_v     # Uncertainity in Momentum of electron, kgm/s
del_v = h_bar/(2*pi*del_x*m)     # Minimum uncertainity in velocity of an electron, m/s
print "The uncertainity in the velocity of the electron = %3.2e m/s"% del_v

The uncertainity in the velocity of the electron = 1.15e+06 m/s


## Ex3.26 : :Pg: 94¶

In [26]:
#Minimum uncertainity in the energy of the excited state of an atom
del_t = 1e-08     # Life time of an excited state of an atom, seconds
h_bar = 1.054e-034     # Reduced Plancks constant, joule second
del_E = h_bar/del_t     # Minimum uncertainity in the energy of excited state, joule
print "The minimum uncertainity in the energy of the excited state = %5.3e joule"% del_E

The minimum uncertainity in the energy of the excited state = 1.054e-26 joule