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PVB302-Quantum Mechanical Model Of Hydrogen Atom

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  • Course Code: PVB302
  • University: Queensland University Of Technology
  • Country: Australia

Questions:

1. Explain what is fundamentally different between the classical and quantum descriptions of the electron that is bound to the atom’s nucleus and describe the role that wave-particle duality plays.
 
2. Describe what these differences mean in terms of the atom’s stability and the energy of the electron.
 
3. In a general sense, describe and explain what the differences between the classical and quantum models of the electron lead to in terms of our ability to know such things as the electron’s location and its momentum. Use the electron around the hydrogen atom and an electron in an infinite potential well as examples to demonstrate your point.
 
4. Explain why electrons can do things like quantum tunnel through potential barriers yet we cannot. 
 

Answer:

Introduction

After  the  failure  of  the  Rutherford  model  of  an  atom  proposed  in  1911  which  was  not  able  to  explain  the  stability  of  the  atom; Niels  Bohr, a  Danish Physicist,   suggested  in  1913  that   the  electrons  could  only  orbit    in  discrete  orbits  or  shells  with  a  constant  radius  around  the  nucleus. The  electron  could not  exist  in  between  these  shells  and  occupy  only  those  shells  whose  radius  is  given  by  Bohr’s  Formula. According  to  Friedrich (2006), though  the  model successfully  explained  the  Hydrogen  Spectra  precisely, it  was  not  able  to  explain  Zeeman  Effect,  Heisenberg  Uncertainty  Principle  and  atomic  spectra  of heavy  atoms.  Further,  the  model  was  modified  by  the  Introduction  of  new  field  of  Physics  namely  Quantum  Mechanics.

Quantum  Mechanical  Model  of  Atom

1.Classical  and  Quantum  Mechanical  Models: According  to  Verma (1993) , the  Classical  Model  for  an  atom  suggests  that  the  electron  revolving  around  the nucleus  is  a  particle  and  the  force  governing  its  motion  is  Electrostatic  force  of  attraction.  On  the  other  hand,  the  quantum  model  for  electron  defines electron  as  a  wave  with  wavefunction  square  of  which  gives  us  the  probability  of  locating  the  electron  at  the  position  (x)  as  given  in  Gasiorowicz (2007). Before  the  evolution  of  quantum  mechanics,  Neils  Bohr  gave  3  postulates  regarding  motion  of  electron  in  an  atom.  As  stated  in  Saraswati (2017), these were-
 
An  electron  can  revolve  around  the  nucleus  in  certain  fixed  orbits  of  definite  energy  without  emission  of  any  radiant  energy. Such  orbits  are  called stationary  orbits. 
 
 

An  electron  can  make  a  transition  in  the  atom  from  a  stationary  state  of  high  energy  E2 to  a  state  of  low  energy  E1  and  in  doing  so, it  emits  a  single photon  with  frequency,

ν =  E2-E1/H

where  h  is  the  Planck's  constant.

Conversely, on  absorbing  an  energy  (E2−E1)  when  the  electron  is  at  energy  E1, the  electron  can  make  a  transition  from  E1  to  E2.

Only  those  orbits  are  allowed  in  the  atom  corresponding  to  which  the  orbital  angular  momentum (L)  of  the  electron  is  an  integral  multiple of  h/ 2π,

Thus,

 L = where  n = 0, 1, 2,.....

According  to  Griffiths (2004), the  wave  particle  duality  of  electron  tells  us  that  the  electron  exhibit  both  particle  and  wave  nature  but not  simultaneously. This  dual  nature  was  demonstrated  by  the  famous  double  slit  experiment.  The  screen  shows  the  interference  pattern  when  both  the  slits  are  open (even when  single  electron  passes  through  the  slit  at  a  time)  confirming  its  wave  nature.  As  soon  as  we  try  to  make  a  measurement  by  closing  one  slit  so that  we  could  find  through  which  slit  the  electron  passes,  the  wave  nature  vanishes  and  particle  nature  comes  into  picture.

Bohr’s  Postulates  were  based  on  the  fact  that  the  electron  is  a  particle  and  its  energy  was  calculated  which  was  found  to  be  similar  to  the  energy  of  the  electron  calculated  using  Quantum  Mechanics  methods  considering  electron  as  a  wave  i.e.,

E =

The  wave  particle  duality  was  further  backed  up  by  deBroglie hypothesis  who  claimed  that  every  particle  must  be  associated  with  a  wave  called  Matter Waves  whose  wavelength()  is  given  by  the  formula-

Wavelength  of  matter  wave  () =h/mv

Here,    

h - Planck’s  Constant

m - the  mass  of  the  particle

v - velocity  of  the  particle.

2.As  stated  in  Griffiths (2013), according  to  Maxwell’s  Electromagnetic Theory,  an  accelerated  charge  particle  radiate  energy.  Rutherford’s  classical  model  suggested  the  circular  motion  of  the  electron. Due  to  its  circular  motion, the  electron  will  accelerate  toward  the  centre  and  thus  it  will  radiate  energy  which  implies  it  will  loose  energy  and  its  radius  of  circular  path  will  eventually  get  smaller  and  smaller  resulting  in  the  jump  of  electron  into  the  nucleus. Therefore,  the  atom  is  not  stable  and  will  vanish  after  radiating  all  its  energy  of   the  electron. Moreover, this   model  didn’t  explain  the  various  Hydrogen  Spectral  lines  which  were  observed  experimentally.

 

Figure - 1                                                   

Bohr  removed  constraint  of  unstability  of  the  atom  and  also  explained  the  spectra  produced  by  Hydrogen  Atom  by  postulating  that  electron  revolve  in  fixed  orbits  called  Stationary  Orbit  of  fixed  orbits  with  fixed  discrete  energy  levels. According  to  him,  an  atom  can  only  radiate  energy  whenever  there  is  a  transition  from  a  high  energy  level  to  a  low  energy  level. He  calculated  the  formula  for  radius  and  velocity  of  the  electron  in  the  orbit  given  by (Kumar, 2009),

                                              rn  =         and                                                     

                                    v =

 

Figure – 2

But  Bohr’s  Theory  was  not  able  to  explain  the  atomic  spectra  of  non-Hydrogenic  atoms. Also, it  didn’t  explained  the   spectral  lines splitting  on  the application  of  magnetic  field  which  is  nothing  but  Zeeman  Effect. It  was  later  explained  by  the  quantum  mechanical  model  taking  into  consideration  the electron’s  spin  in  the  atom. (Kumar, 2018)

3.The  particle  nature  of  the  electron  gave  simultaneous  measurement  of  the  momentum  and  position  of  the  particle. For   an  electron  moving  in a  circular orbit  of  Hydrogen  atom, we  know  its  radius  exactly  i.e.,  rn = ( Z = 1)   and  thus  ?r = 0. However, since  it  is  moving  in  a  circular  orbit, it  cannot  have  any  radial  velocity, and  thus  pr = 0  and  ?pr = 0. Thus, we  have  simultaneous exact  knowledge  of  both  r  and  pr   which   violates  the  uncertainty  principle.(Ghoshal, 2010)   

Now, considering  the  quantum  mechanical  model  of  Hydrogen  atom  i.e.,  electron  in  an  infinite  potential  well,  we  consider  electron  as  a  wave  whose  wavefuntion  and  energy  is  given  by-

 =  sin ( )     n = 1, 2, 3,...

 =       n = 1, 2, 3,....

According  to  Bransden  and  Joachan (2003), a  single  sinusoidal  wave  has  a  precise  measurable  wavelength (),  so  the  electron  presented  by  a  sine  wave which  is  nothing  but  the  matter  wave  suggested  by  deBroglie  has  a  precise  or  definite  momentum. But  a  single  sine  wave  keeps  going  in  both  the directions  i.e.,  the  wave  is  not  localized  anywhere. So,  the  position  of  the  electron  is  totally  uncertain. When  several  sine  waves  having  unique  wavelength each  are  added  together,  we  get  a  resultant  wave  that  is  localized  to  some  extent. Adding  more  sine  waves  together  gives  us  more  localized  resultant wave , and  it  also  gives  less  uncertainty    about  the  electron's  location. It  is  not  clear  which  wavelength  satisfies  deBroglie's  formula  for  the  calculation  of electron’s  momentum, since, the  resultant  wave  consists  of  a  range  of  wavelengths ( wavelengths  of  the  sinusoidal  waves)  existing  simtaneously. Thus, this gives  us  uncertainty  about  the  electron's  momentum  to  some  extent. When  more  sine  waves  are  added  to  the  wave, the  wave  after  addition  will  give more localization  of  the  electron  but  there  will  also  be  more  uncertainty  in  the  momentum  and  wavelength  of  the  electron  represented  by  the  resultant  wave . Therefore,  the  quantum  mechanical  model  satisfies  the  Heisenberg  uncertainty  principle

4.According  to  Griffiths (2004), the  diagram  of  a  rectangular  potential  barrier  is  represented  in  figure - 3   which  extends  from  x = 0  to  x = a. The  potential  of  the  barrier  is  constant  and  equal  to  U0. On  the  left  and  right  side  of  the  barrier  the  potential  U = 0.

                

Figure – 3

Considering  that  a  stream  of  particles  of  energy  E  be  incident  from  left  on  the  barrier  surface  at  x = 0. The  following  two  cases  arise-

  1. If  E > U0, then  according  to  classical  mechanics  the  particles  will be  wholly  transmitted  and  no  reflection  is  possible. But  quantum  mechanically  there  is  always  some  probability  at  x = 0  and  x = a
  2. If   E < U0, then  classically  the  particles  will  be  wholly  reflected  and  hence  penetration  through  the  barrier  is  impossible. But  quantum  mechanically  there  is  always  some  probability  of  penetration  into  the  barrier  and  appearance  of  the  particles  in  region III. This  finite  probability   of  transmission  through  the  barrier  even  for  E < U0is  called  the  Quantum  Mechanical  Tunnelling  Effect.

After  solving  the  Schrodinger  Equation  for  the  case  E < U0 , we  will  find  that  the  transmission  coefficient  for  the  given  particle  to  pass  through  the  potential  barrier  is  given  by-

                                          T =

Here,  =       ; m - mass  of  the  particle

From  T 0  implies  that  there  is  a  finite  probability  of  transmission  of  the  particle  through  the  potential  barrier  of  height  U0  and  width  a  even  if     E < U0 . This  cannot  be  accounted  for  in  view  of  classical  theory. Transmission  coefficient  depends  on  4  factors  i.e., width  of  barrier (a), mass  of  particle (m), Energy  of  the  particle (E), and  the  barrier  height (U0)   We  know  that  mass  of  human  beings (mhb)  is  much  larger  than  the  mass  of  the  electron (me), i.e., mhb >> me. . So,  in  case  of  humans,  m large  value , so,   

     sinh2() =  =  >> 1

    T =      which  will  be  nearly  zero  for    

While  for  electron,  it  the  wavefunction  will retain  its  amplitude  even  after  penetrating  a  small  barrier.

Hence,  electrons  can  do  quantum  tunnel  through  potential  barriers  yet  we  cannot.

Conclusion

The  spectacular  discovery  of   dual  nature  of  electron  gave  rise to  a  new  field  of  Physics  called  Quantum  Mechanics. Many  complex  problems  of  the  nature could  be  solved  now  using  quantum  techniques. One  thing  that  should  be kept  in  mind  that  the  quantum  mechanics  is  applicable  only  for  very  small particles  like  proton, electron, etc. Quantum  Mechanics gives  us  probabilistic  results  unlike  the  Newtonian  mechanics. Numerous  technology  have  been given using  this  new  field  of  science  like  Scanning  Tunnelling  Microscope  which  uses  tunnelling  effect  of  the  electron, Quantum  Dots  and  Quantum  Computations and  many  more.  In  the  end ,  we  can  conclude  that  the  riddle  of  the  structure  of  atoms  is  successfully  explained  with  the  help  of  Quantum  Mechanics. 

 

References

Bransden  B.H.  and  Joachan  C.J. (2003). Physics  of  Atoms  and  Molecules. 2nd  Edition. London: Pearson

Friedrich  H. (2006). Theoretical  Atomic  Physics. 3rd  Edition. Berlin: Springer

Gasiorowicz  S. (2007). Quantum  Physics. 3rd  Edition. New  Jersey: Wiley

Ghoshal  S.N. (2010). Atomic  Physics. Delhi: S.Chand

Griffiths  D.J. (2013). Introduction to  Electrodynamics. 4th  Edition. London: Pearson  Education

Griffiths  D.J. (2004).  Introduction  to  Quantum  Mechanics. 2nd  Edition. London: Pearson  Education

Kumar  A. (2018). Fundamentals  of  Quantum  Mechanics. Cambridge: Cambridge  University  Press

Kumar  R. (2009). Atomic  and  Molecular  Physics. Meerut: Campus  Books

Saraswati  V. (2017) Quantum  Mechanics  Atomic  and  Molecular  Physics. Delhi: Himanshu  Publications

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