Friday, September 5, 2014

Selected Solutions - Intro. to Quantum Mechanics (3)


3) The wave function for a particle confined to moving in a 1D box is given by:

 
y(x)  =   A [sin (npx/L)]  

 
Use the normalization condition on y(x)   to show the constant A is given by:  A =  Ö2/ ÖL  

(Refer to the normalization condition applied to the system in 5(b), i.e. solving for A)


4) From the information and the Heisenberg Uncertainty Principle:
 
DP x (D x)  > h/ 2p  = 1.05 x 10-34 J-s  = 1.05 x 10-27 erg-s   (cgs units)



The uncertainty in the x-component is 0.5 angstrom where 1 A = 1.0 x 10-8 cm


 
Then the uncertainty in the x-component is: Dx =  0.5A = 5.0 x 10-9 cm



The uncertainty in the x-component of the  momentum of the electron is:


 
DP x»    (h/ 2p)  / D x  =    1.05 x 10-27 erg-s/  5.0 x 10-9  cm


 
Or, 2 x 10-19 g cm-s



5 (a)) In classical theory the probability that the particle would be in the region between x and x + L/3 is 0.33.  Since no other information is given then it is equally probable that it’s in any place in the region – so one third of the time will be spent over any given line segment, i.e. which is 0ne-third the total length.


 
b) For a quantum mechanical interpretation, define the wave function (See e.g. http://brane-space.blogspot.com/2013/05/solutions-to-quantum-box-problems.html:


y = A sin 2kx


 
sin 2kL = 0  and k (wave number ) =   n p/L


 
y = A sin (n px/L)


 P(x) =  y 2  = A2 sin 2 (n px/L)


 
For normalization:



1 =  =     ò L 0   P(x)   dx  =   A2  ò L 0    sin 2 (n px/L) dx


 
A2  (L/ n p) [ u/2 – sin 2u/ 4] 0 n p           = 1


 A2  (L/ n p) (n p/2) = 1


 Or:  A2  =  2/L   so that:   A =  Ö2/ ÖL  


 
Then the probability it will lie between 0 and L/3 is:


 
òL/3 0    sin2 (npx/L) dx


 
=  2/ n p [n p/ 6  -  ¼ (sin 2 p/ 3 )]  = 


2/ p [p/ 6  -  ¼ (sin 2 p/ 3 )]      = 0.19


 
(Since n = 1 is the lowest energy state)


 
c) Since n = 2 is the second lowest state the probability the particle is between 0 and L/3 is:


 
  (2/L) L/ 2 p  [p/ 3  -  ¼ (sin 4 p/ 3 )]  =  0.40 

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