9: Quantum Mechanics

Example number 9.1, Page number 202

In [1]:
#importing modules
import math
from __future__ import division

#Variable declaration
V = 100;     #Accelerating potential for electron(volt)

#Calculation
lamda = math.sqrt(150/V)*10**-10;     #de-Broglie wavelength of electron(m)

#Result
print "The De-Broglie wavelength of electron is",lamda, "m"

The De-Broglie wavelength of electron is 1.22474487139e-10 m

Example number 9.2, Page number 203

In [2]:
#importing modules
import math
from __future__ import division

#Variable declaration
e = 1.6*10**-19;     #Energy equivalent of 1 eV(J/eV)
h = 6.626*10**-34;    #Planck's constant(Js)
m = 9.11*10**-31;     #Mass of the electron(kg)
Ek = 10;     #Kinetic energy of electron(eV)

#Calculation
p = math.sqrt(2*m*Ek*e);     #Momentum of the electron(kg-m/s)
lamda = h/p ;     #de-Broglie wavelength of electron from De-Broglie relation(m)
lamda = lamda*10**9;     #de-Broglie wavelength of electron from De-Broglie relation(nm)
lamda = math.ceil(lamda*10**2)/10**2;     #rounding off the value of lamda to 2 decimals

#Result
print "The de-Broglie wavelength of electron is",lamda, "nm"

The de-Broglie wavelength of electron is 0.39 nm

Example number 9.3, Page number 203. theoritical proof

Example number 9.4, Page number 203

In [3]:
#importing modules
import math
from __future__ import division

#Variable declaration
h = 6.626*10**-34;      #Planck's constant(Js)
m = 9.11*10**-31;       #Mass of the electron(kg)
v = 1.1*10**6;     #Speed of the electron(m/s)
pr = 0.1;        #precision in percent

#Calculation
p = m*v;     #Momentum of the electron(kg-m/s)
dp = pr/100*p;    #Uncertainty in momentum(kg-m/s)
h_bar = h/(2*math.pi);     #Reduced Planck's constant(Js)
dx = h_bar/(2*dp);         #Uncertainty in position(m)

#Result
print "The uncertainty in position of electron is",dx, "m"

The uncertainty in position of electron is 5.26175358211e-08 m

Example number 9.5, Page number 203

In [4]:
#importing modules
import math
from __future__ import division

#Variable declaration
e = 1.6*10**-19;     #Energy equivalent of 1 eV(J/eV)
h = 6.626*10**-34;    #Planck's constant(Js)
dt = 10**-8;      #Uncertainty in time(s)

#Calculation
h_bar = h/(2*math.pi);    #Reduced Planck's constant(Js)
dE = h_bar/(2*dt*e);       #Uncertainty in energy of the excited state(m)

#Result
print "The uncertainty in energy of the excited state is",dE, "eV"

#answer given in the book is wrong

The uncertainty in energy of the excited state is 3.2955020404e-08 eV

Example number 9.6, Page number 204

In [5]:
#importing modules
import math
from __future__ import division

#Variable declaration
c = 3*10**8;      #Speed of light(m/s)
dt = 10**-8;      #Average lifetime(s)
lamda = 400;    #Wavelength of spectral line(nm)

#Calculation
lamda = lamda*10**-9;      #Wavelength of spectral line(m)
#From Heisenberg uncertainty principle,
#dE = h_bar/(2*dt) and also dE = h*c/lambda^2*d_lambda, which give
#h_bar/(2*dt) = h*c/lambda^2*d_lambda, solving for d_lambda
d_lamda = (lamda**2)/(4*math.pi*c*dt);     #Width of spectral line(m)

#Result
print "The width of spectral line is",d_lamda, "m"

The width of spectral line is 4.24413181578e-15 m

Example number 9.7, Page number 204. theoritical proof

Example number 9.8, Page number 204. theoritical proof

Example number 9.9, Page number 205. theoritical proof

Example number 9.10, Page number 205. theoritical proof

Example number 9.11, Page number 205. theoritical proof

Example number 9.12, Page number 206. theoritical proof

Example number 9.13, Page number 206. theoritical proof

Example number 9.14, Page number 207

In [7]:
#importing modules
import math
from __future__ import division
from scipy.integrate import quad

#Variable declaration
a = 2*10**-10;    # Width of 1D box(m)
x1=0;    # Position of first extreme of the box(m)
x2=1*10**-10;   # Position of second extreme of the box(m)

#Calculation
def intg(x):
    return ((2/a)*(math.sin(2*math.pi*x/a))**2)
S=quad(intg,x1,x2)[0]

#Result
print "The probability of finding the electron between x = 0 and x = 10**-10 is",S

The probability of finding the electron between x = 0 and x = 10**-10 is 0.5
In [ ]: