# Chapter 7 Particle in a potential well¶

## Example 7.4 Page no 134¶

In [14]:
#Given
m=9.1*10**-31                               #Kg
h=1.05*10**-34                              #Js
ev=1.6*10**-19
n1=1
n2=2
n3=3
a=10**-10                                   #m

#Calculation
import math
E1=((n1**2*math.pi**2*h**2)/(8.0*m*a**2))/(1.6*10**-19)                    #ev
E2=n2**2*E1
E3=n3**2*E1

#Result
print"three lowest energy levels are ",round(E1,1),"ev,", round(E2,1),"ev and ",round(E3,2),"ev"
print "their eigenfunctions are 1/10**-5*cos(pie*x/2*10**-10),1/10**-5*sin(pie*x/10**-10) and 1/10**-5*cos(3*pie*x/2*10**-10)"

three lowest energy levels are  9.3 ev, 37.4 ev and  84.08 ev
their eigenfunctions are 1/10**-5*cos(pie*x/2*10**-10),1/10**-5*sin(pie*x/10**-10) and 1/10**-5*cos(3*pie*x/2*10**-10)


## Example 7.5 Page no 135¶

In [84]:
#Given
m=10.0*10**-3                                            #kgram
l= 10.0*10**-2                                           #Length in m
h=1.054*10**-34
n1=1
n2=2
n3=3

#Calculation
E1=(((3.14*h*n1)**2)/(2.0*m*(l**2)))/(1.6*10**-19)
E2=(((3.14*h*n2)**2)/(2.0*m*(l**2)))/(1.6*10**-19)
E3=(((3.14*h*n3)**2)/(2.0*m*(l**2)))/(1.6*10**-19)

#Result
print"energies are ", round(E1,46),"ev,",round(E2,46),"ev,",round(E3,45),"ev"
print"these energies are extremely small and close together and hence can't be measured"

energies are  3.4e-45 ev, 1.37e-44 ev, 3.1e-44 ev
these energies are extremely small and close together and hence can't be measured


## Example 7.7 Page no 136¶

In [15]:
#Given
L=10**-9                                         #Width in m
v=9.0*10**-9
h=1.054*10**-34                                  #Js
c=3*10**8                                        #m/s
m=9.1*10**-31
v1=(9.0+1)*10**-9
v2=(9.0-1)*10**-9

#Calculation
import math
n=math.sqrt((4*c*m*(L**2))/(v*math.pi*h))
n1=math.sqrt((4*c*m*(L**2))/(v1*math.pi*h))
n2=math.sqrt((4*c*m*(L**2))/(v2*math.pi*h))

#Result
print"value of n is ",round(n,0),", When + sign is taken ",round(n2,0),", when -ve sign is taken ",round(n1,0)

value of n is  19.0 , When + sign is taken  20.0 , when -ve sign is taken  18.0


## Example 7.8 Page no 136¶

In [16]:
#Given
L1=0.4
L2=0.6
L=1                                                #Say

#Calculation
import math
dx=(L2-L1)*L
#for ground state
P1=2/L*(math.sin(math.pi*L/2.0*L))**2*dx
#for first excited state
P2=2/L*(math.sin(2*math.pi*L/2.0*L))**2*dx
#for second excited state
P3=2/L*(math.sin(3*math.pi*L/2.0*L))**2*dx

#Result
print"(a) probability for ground state ", P1
print"(b) probability for first excited state ",round(P2,1)
print"(c) Probability for second excited state ", P3

(a) probability for ground state  0.4
(b) probability for first excited state  0.0
(c) Probability for second excited state  0.4


## Example 7.9 Page no 140¶

In [17]:
#Given
a=10.0**-14                             #m
m=1.6*10**-27                         #mass of a nucleon in kg
h=1.054*10**-34                       #Js

#Calculation
import math
Emin=((3*(math.pi**2)*(h**2))/(2.0*m*(a**2)))/(1.6*10**-19)

#Result
print"minimum energy of a nucleon is ", round(Emin*10**-6,1),"Mev"

minimum energy of a nucleon is  6.4 Mev