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Transcript
(Quantum Mechanics)
1. State basic concepts (or postulates) of Bohr's model and derive the energies
of the electron-proton system in a hydrogen atom.
2.
Calculate the de Broglie wavelengths of i) an electron with kinetic energy 1
eV and ii) a proton with kinetic energy 1 keV.
3. A mass  oscillates in one dimension with a natural frequency  . The total
energy  is given as,




      ,



Estimate the ground state energy using the uncertainty principle.
4. One of the uncertainty relations is expressed as  =  , where ∆ is the
uncertainty in the position  and ∆ , that in the momentum  .
(a) Using the uncertainty relation, estimate the ground state energy and radius
of the hydrogen atom.
(b) Similary, estimate the ground state energy of one-dimensional harmonic
oscillator.
5. Let  be the angle variable in plane polar coordinates. Show that the

≡  is hermitean. Find the eigenfunctions and eigenvectors for this
operator 

operator.
6. Find out the eigenstates and eigenvalues of a point mass of  in an infinite
well of a width of . Draw the wave functions of the lowest three states.
7. Draw the (schematic) wavefunctions of the lowest three states in a finite well
of width  .
8. A particle with mass  and energy  moves from  =-
∞
toward the
following potential barrier    .
       





 

  
  
If the barrier width  is not too large compared with the de Broglie wavelength
if the particle, the particle has a finite probability to pass through the barrier
even if     . This phenomena is called as the quantum tunneling.
(a) Why the tunneling is possible? Give a quantum mechanical explanation.
(b) Write down the Schrödinger equations for three regions  <0(region I),
     (region II), and    (region III)?
(c) The solutions in three regions are related because the overall wavefunction
should
be
physical.
What
are
the
boundary
conditions
satisfied
by
the
wavefunction for this purpose?
(d) What are the factors determining the tunneling probability?
9. The Hamiltonian  , the ground state wafefunction   , and the 1st excited
state wavefunction  are given below for a one-dimentional harmonic oscillator
with mass  and natural frequency  .




          ,


  
   exp      ,



  
   ′  exp    



(a) Using the relation       , calculate the ground state and 1st excited
state energies.
(b) What is the value  , if  is normalized.
(c) Show that

∞
∞
     . Why this integral is zero?.
(d) Calculate the expectation value    for a state given as


 
  
 .






10. The position operator in the momentum representation is     .
Using this, prove the following relation.
       
11.
Show
that
two
eigenstates
of
a
hermitean
operator
with
different
eigenvalues are mutually orthogonal.
12. Show that the time derivative of the expectation value of an operator 
can be written as
〈 〉  〈 〉 


〈〉  





where  is the Hamiltonian of the system.
13. If a particle is in the state of the angular momentum quantum l=2, state all
the possible z-components of the angular momentum.
14. Consider the raising and lowering operators
a+
a
for the one+
dimensional harmonics oscillator. The commutator of two operators
,




and the Hamiltonian is    ×     . Show that
In other word, if
eigenvalue
|E >
a
and
[ a , a ]=1
is the lowering operator.
is an eigenstate of  ,    is also an eigenstate with
E -ℏω. (You must know or obtain
   .)
15. Consider a mass  in a one-dimensional box (or infinite potential well) of
width  . Find the energy eigenvalues of the mass. If the potential in the


left-half of the well is raised by an amount  =  , what is the first order
energy shift of the system?
16. A hydrogen atom is in a uniform electric field. Show that the first order
energy shift for the ground state is zero.
17. The Hamiltonian for the one-dimensional harmonics oscillator is
            .
If
we
define
two
operators
a = mω/2ℏ x+ ip/ 2mℏω
and
†  
     
 , the Hamiltonian can be rewritten as




   †     .
The eigenvalue equation for this Hamiltonian is




          .
(a) Obtain the matrix element
    .
(b) Obtain the matrix element
       .
Use the relations    
     †   
  
18. The Hamiltonian of a nonlinear oscillator is given as
              .
Considering the last term as a perturbation interaction, explain that the
first-order correction to the ground state energy is zero.
19. Describe the method of obtaining the first order shift of two degenerate
energy levels under the perturbation   .