Chapter 2: Quantum Mechanics

Example 2.1, Page 79

In [1]:
from math import *

#Variable Declaration
V = 50;   # Given potential difference, V

#Calculations
lamda = 12.24/sqrt(V);   # Wavelength of the light, angstrom

#Result
print "The de-broglie wavelength of electron = %4.2f angstrom"%lamda

The de-broglie wavelength of electron = 1.73 angstrom

Example 2.2, Page 79

In [14]:
from math import *

#Variable Declaration
h = 6.62e-34;   # Planck's constant, J-s
m0 = 1.6e-27;   # Rest mass of proton, kg
c = 3.0e+8;   # Speed of light, in m/s
v = c/20;    # Velocity of the proton, in m/s

#Calculations
lamda = (h*sqrt(1-v**2/c**2))/(m0*v);

#Result
print "The de broglie wavelength associated with the proton = %4.2e m"%lamda
#answer differs due to rounding-off errors

The de broglie wavelength associated with the proton = 2.75e-14 m

Example 2.3, Page 79

In [16]:
#Variable Declaration
c = 3.0e+8;   # Speed of light, m/s
v = 2.0e+8;   # Velocity of the proton, m/s
m0 = 1.6e-27;   # Rest mass of proton, kg
h = 6.62e-34;   # Plancks constant,J-s

#Calculations
lamda = (h*sqrt(1-(v**2/c**2)))/(m0*v);

#Result
print "The wavelength of matter wave associated with the proton = %5.3e m"%lamda

#answer differs due to rounding-off errors

The wavelength of matter wave associated with the proton = 1.542e-15 m

Example 2.5, Page 80

In [4]:
#Variable Declaration
a = 0.003;    # Accuracy of the electron,in percent
s = 5e+03;    # Speed of the electron,in m/s
del_v = (a/100)*s;    # Change in velocity,in m/s
m0 = 9.1e-31;    # Rest mass of the electron,in kg
hcut = 1.054e-34;    # Plancks constant,J-s

#Calculations
del_x = hcut/(2*del_v*m0);

#Result
print "The uncertainity in the position of the electron = %4.2e m"%del_x

The uncertainity in the position of the electron = 3.86e-04 m

Example 2.6, Page 81

In [5]:
#Variable Declaration
del_t = 2.5e-14;    # Lifetime of the hydrogen atom in excited state
hcut = 1.054e-34;    # Planck's constant,in J-s
e = 1.6e-19;    # Charge on electron,in C

#Calculations
del_E = hcut/(2*del_t*e);    # Energy of the state, in eV

#Result
print "The minimum error in measurement of lifetime of excited state of hydrogen atom = %6.4f eV"%del_E

The minimum error in measurement of lifetime of excited state of hydrogen atom = 0.0132 eV

Example 2.7, Page 81

In [6]:
#Variable Declaration
del_x = 1e-09;      # Uncertainty in position of the electron, m
m0 = 9.1e-031;    # Rest mass of an electron, kg
hcut = 1.054e-034;    # Planck's constant,in J-s

#Calculations
del_v = hcut/(2*del_x*m0);    # Uncertainity in velocity of the electron

#Result
print "The uncertainity in the velocity of an electron = %4.2e m/s"%del_v
#Incorrect answer in the textbook

The uncertainity in the velocity of an electron = 5.79e+04 m/s

Example 2.8, Page 81

In [7]:
#Variable Declaration
hcut = 1.054e-34;    # Reduced Planck's constant, Js
v = 3e+07;    # Velocity of the electron, m/s
c = 3e+08;    # Speed of light in vacuum, m/s
m0 = 9.1e-31;    # Rest mass of an electron, kg
del_v = 3e+08;    # Uncertainty in velocity of the electron, m/s

#Calculations
del_x = (hcut*sqrt(1-v**2/c**2))/(2*m0*del_v);

#Result
print "The smallest possible uncertainity in position of the electron = %6.4f angstrom"%(del_x/1e-010)

The smallest possible uncertainity in position of the electron = 0.0019 angstrom

Example 2.9, Page 82

In [8]:
#Variable Declaration
n = 1;
m0 = 9.1e-031;    # Mass of the electron, kg
a = 1e-10;    # Width of the box, m
h = 6.63e-034;    # Planck's constant, J-s

#Calculations
E = n**2*h**2/(8*m0*a**2);

#Result
print "The energy of the electron moving in 1D infinetly high potential box = %5.2e J"%E

The energy of the electron moving in 1D infinetly high potential box = 6.04e-18 J

Example 2.10, Page 83

In [25]:
#Variable Declaration
#n = [1,2];    # Shell numbers for two lowest permitted energy of the electron 
m0 = 9.1e-31;    # Mass of the electron, kg
a = 2.5e-10;    # Width of the box, m
h = 6.63e-34;    # Planck's constant, J-s
e = 1.6e-19;    # Charge on electron, C

#Calculations&Results
E = round((h**2)/(8*m0*a**2*e));
for n in range(1,3):
    print "The lowest two permitted energy values of an electron are"
    print E*n**2,"eV"

The lowest two permitted energy values of an electron are
6.0 eV
The lowest two permitted energy values of an electron are
24.0 eV

Example 2.11, Page 83

In [10]:
#Variable Declaration
m0 = 1.67e-27;    # Rest mass,in kg
a = 1e-14;    #  Size of the box
h = 6.63e-34;    # Planck's constant,in J-s
n = 1;   # Quantum number for lowest energy state

#Calculations
E_n = n**2*h**2/(8*m0*a**2);

#Result
print "The lowest energy of the neutron confined to the nucleus = %4.2e J"%E_n

The lowest energy of the neutron confined to the nucleus = 3.29e-13 J

Example 2.12, Page 83

In [11]:
#Variable Declaration
m0 = 9.1e-31;    # Rest mass, kg
a = 1e-10;    # Length of the box, m
h = 6.62e-34;    # Planck's constat, J-s
n1 = 1;    # Ground state
n2 = 2;    # First excited state
e = 1.6e-19;    # Charge on electron, C

#Calculations
E1 = (n1**2*h**2)/(8*m0*a**2*e);
E2 = (n2**2*h**2)/(8*m0*a**2*e);
del_E = E2-E1; 

#Result
print "The energy difference between the ground state and the first excited state = %5.1f eV"%del_E

The energy difference between the ground state and the first excited state = 112.9 eV