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					                                            Exercises



                          Part 1: The origins of quantum theory

1. Describe briefly the Davisson-Germer and Thomson experiments and the use of the Wilson

cloud chamber for the observation of particle tracks.


2. A perfectly elastic ping-pong ball is dropped in vacuum from a height equal to ten times its

radius onto a perfectly elastic fixed sphere of the same radius. Neglecting effects due to earth

motion, estimate the largest number of bounces against the fixed sphere that the ball can be

expected to make under optimum condition of release.


3. A beam of monoenergetic electrons in used to excite a particular level of an atom in a

Franck-Hertz experiment. If this level is of short duration, owing to radiation back to the ground

state, show that the inelastically scattered electrons that have lost energy to produce the excited

level will not all be expected to have the same final energy. If the excited level lasts about 10-10 sec,

what is the order of magnitude of the electron energy spread, measured in electron-volts?


4. Find the de Brogile wavelength of each of the following:


     a) a 70kg man traveling at 6km/h;


     b) a 1kg stone traveling at 10m/s;


     c) a 10-6g particle of dust moving at 1m/s;


5. If the electron in a hydrogen atom is assumed to follow a circular orbit, show that the

requirement that its de Brogile wavelength be equal to the wavelength of a standing wave around

the orbit is equivalent to Bohr’s assumption that mvr  n (n=1,2,3,…).


6. Find the frequency, wavelength and momentum associated with a photon of energy 13.6eV.

What would be the momentum of a free electron with this kinetic energy? Compare the frequency

and de Brogile wavelength of the electron with that of the photon.


7. A neodymium (Nd) laser operates at a wavelength   1.06 10 m . If the laser is operated in
                                                                       6
                                                   11
a pulsed mode, emitting pulses of duration 3 10         s , what is the minimum spread in

  (a) frequency and (b) wavelength of the laser beam?


8. Explain what was learned about quantization of radiation or mechanical system from two of the

following experiments:


          a) Photoelectric effect.


          b) Black body radiation spectrum.


          c) Franck-Hertz experiment.


          d) Davisson-Germer experiment.


          e) Compton scattering.


9. Consider an experiment in which a beam of electrons is directed at a plate containing two slits,

labeled A and B. Beyond the plate is a screen equipped with an array of detectors which enables

one to determine where the electrons hit the screen. For each of the following cases draw a rough

graph of the relative number of incident electrons as a function of position along the screen and

give a brief explanation.


          f)   Slit A open, slit B closed.


          g) Slit B open, slit A closed.


          h) Both slits open.


 What is the effect of making the beam intensity so low that only one electron is passing through

the apparatus at any time?




                  Part 2:     Wave functions and Schrodinger Equations

  1. Use the arguments of this Chapter to set up a differential equation for        that involves a
  second time derivative of      , in the case of a free particle. Discuss any solutions that this
  equation has that are not shared by the free-particle Schrodinger equation.
     2. Show that if the potential energy V (r ) is changed everywhere by a constant, the

     time-independent wave functions are unchanged. What is the effect on the energy eigenvalues?


     3. Suppose


                                0if 0  x  a
                       V ( x)  
                                otherwise

A particle in this potential is completely free, except at the two ends (x=0 and x=a), where an

infinite force prevents it from escaping. What is the energy states of the particle and the

corresponding functions?


4.     A particle in the infinite square well has the initial wave function
 ( x, 0)  Ax(a  x),  x  a  for some constant A. Outside the well, of course,   0 .
Find  ( x, t ) .


5. A particle in the infinite square well has the initial wave function


                                        Ax0  x  a / 2
                             ( x, 0)  
                                        A a  x a / 2  x  a

            i)    Sketch  ( x, 0) , and determine the constant A.


            j) Find  ( x, t ) .

            k) What is the probability that a measurement of the energy would yield the value E1 .

            l) Find the expectation value of the energy.

6. A particle of mass m in the infinite square well (of width a) starts out in the left half of the well,

and is (at t=0) equally likely to be found at any point in that region.


           a) What is its initial wave function? (Assume it is real. Don’t forget to normalize it)


           b) What is the probability that a measurement of the energy would yield the value

                2
                      / 2ma 2 .

7. Find the first excited state of and the expectation value of the potential energy in the nth state
of the harmonic oscillator.


8. A particle in the harmonic oscillator potential starts out in the state


                               ( x, t )  A[3 0 ( x)  4 1 ( x)]

                                                          2
     a) Find A and construct  ( x, t ) and  ( x, t ) .



     b) Find    x   and     p . If you measured the energy of this particle, what values might you
    get, and with what probability?


9. A free particle, which is initially localized in the range: a  x  a , is released at time t=0:


                                           Aif 0  x  a
                               ( x, 0)  
                                          0otherwise

    Where A and a are positive real constants. Find  ( x, t ) .


10. A free particle has the initial wave function  ( x, 0)  Ae
                                                                       a | x|
                                                                                 , where A and a are positive real

constants.


     a) Normalize  ( x, 0) .


     b) Construct  ( x, t ) , in the form of an integral.


     c) Discuss the limiting cases (a very large, and a very small).


11. Consider the “step” potential:


                                           0if x  a
                                  V ( x)  
                                           V0 if x  a

     a) Calculate the reflection coefficient, for the case E<V0, and comment on the answer.


     b) Calculate the reflection coefficient for the case E>V0.


     c) For a potential such as this, which does not go back to zero to the right of the barrier, the

                                                                
          transmission coefficient is not simply F  A (with A the incident amplitude and F
         the transmission amplitude), because the transmitted wave travels at a different

         speed.

     d) For E>V0, calculate the transmission coefficient for the step potential, and check that

          T  R 1.

12. Solve the time-independent Schrodinger equation with appropriated boundary conditions for

the “centered” square well: V(x)=0 (for –a<x<a), V ( x)   (otherwise).


13. Let u1 ( r ) and u2 (r ) be two eigenfunctions of the same Hamiltonian that correspond to the

same energy eigenvalue; they may be the same function, or they may be degenerate. Show that


                                 u (r )( xp        px x)u2 (r )d 3r  0
                                   *
                                   1           x



Where the momentum operator px  i ( / x) operates on everything to its right.


13. Apply the Bohr-Sommerfeld quantization rules to the determination of the energy levels of a

harmonic oscillator and of the circular orbits in a hydrogen atom. Compare with the results

obtained in this Chapter.


14. A particle in the infinite square well has the initial wave function


                                 ( x, 0)  A sin 3 ( x / a)         ( 0 x  a )
                                                                                 .


  Determine A, find  ( x, t ) , and calculate  x , as a function of time. What is the expectation

  value of the energy?


15. A particle of mass m is in the ground state of the infinite square well. Suddenly the well

  expands to twice its original size----the right wall moving from a to 2a-----leaving the wave

  function undisturbed. The energy of the particle is now measured.


     a) what is the most probable result? What is the probability of getting that result?


     b) what is the next most probable result, and what is its probability?


     c) what is the expectation value of the energy?


16. A particle of mass m is in the potential
                          x  0
                         
                V ( x)    / ma 2 0  x  a
                          x  a
                         

    a) How many bound states are there?


    b) In the highest-energy bound state, what is the probability that the particle would be found

          outside the well ( x  a )?


17. What is the order of magnitude of the spread of quantum numbers and energies of the states

that contribute significantly to the oscillating-wave-packet solution for the harmonic oscillator?


18. Show that if the potential energy V (r ) is changed everywhere by a constant, the

time-dependent wave functions are unchanged. What is the effect on the energy eigenvalues?


19. Consider the one-dimensional time-independent Schrodinger equation for some arbitrary

potential V(x). Prove that if a solution  ( x ) has the property that  ( x)  0 as x   ,

then the solution must be nondegenerate and therefore real, appear from a possible overall phase

factor.


20. Consider a one-dimensional bound particle


    a) Show that


                      d  *
                      dt 
                             ( x, t ) ( x, t )dx  0       ( need not be a stationary state)



    b) Show that, if the particle is in a stationary state at a given time, then it will always remain in

      a stationary state.


    c) If at t  0 the wave function is constant in the region a  x  a and zero elsewhere,

      express the complete wave function at a subsequent time in terms of the eigenstates of the

      system.


21. Obtain the binding energy of a particle of mass m in one dimension due to the following

short-range potential: V ( x)  V0 ( x) .
22. A approximate model for the problem of an atom near a wall is to consider a particle moving

under the influence of the one-dimensional potential given by


                                                  V  ( x)x  d
                                         V ( x)   0
                                                    x   d

Where  ( x ) is the so-called “delta function”.


     a) Find the modification of the bound-state energy caused by the wall when it is far away.

     Explain also how far is “far away”.


     b) What is the exact condition on V0 and d for the existence of at least one bound state?


23. Consider a particle beam approximated by a plane wave directed along the x-axis from the left

and incident upon a potential V ( x)   ( x) ,   0 ,  ( x ) is the Dirac delta function.


     a) Give the form of the wave function for x  0 .


     b) Give the form of the wave function for x  0 .


     c) Give the conditions on the wave function at the boundary between the regions.


     d) Calculate the probability of transmission.


24. Obtain an approximate analytic expression for the energy level in a square well potential

( l  0 ) when V0 a 2 is slightly greater than 
                                                   2   2
                                                           / 8m .

25. Consider a one-dimensional bound particle.


     a) Show that


                      d  *
                      dt 
                             ( x, t ) ( x, t )dx  0 ( need not be a stationary state).
                                     


     b) Show that, if the particle is in a stationary state at a given time, then it will always remain

    in a stationary state.


      c) If at t  0 the wave function is constant in the region a  x  a and zero elsewhere,

    express the complete wave function at a subsequent time in terms of the eigenstates of the
   system.


26. An electron is confined in the ground state in a one-dimensional box of width 10-10m. Its

energy is 38eV. Calculate:


      a) The energy of the electron in its first excited state.


      b) The average force on the walls of the box when the electron is in the ground state.




                   Part 3           Dynamical variables in quantum mechanics

1. Prove the commutation relations:


           ^ ^           V
      a)
            H , Px   i x ;
                   


           ^ ^            ^
                                            ^ ^            ^
                                                                         ^ ^            ^
      b)
            Lx , Ly   i L z ;            Ly , Lz   i L x ;        Lz , Lx   i L y
                                                                              


          ^2 ^2 ^2 ^ 
      c)  Lx + Ly + Lz ,L i  0(i  x, y, z )
                            
                            
         


           ^ ^           ^
                                 ^ ^            ^
                                                        ^ ^
      d)
            Lz , Px   i P y ,  Lz , Py   i P x ,  Lz , Pz   0
                                                             


                                ^       ^              ^
2. Using the forms for L x , L y and L z given in Chapter 3, show that each component of the

angular momentum operator commutes with the parity operator.

                                                           ^         ^
3. Prove that for any pair of linear operators A and B


                               ^ 2  ^ ^  ^ ^  ^ ^ 
                                    ^

                               A ,B  =  A ,B  B+ B  A ,B 
                                                         


                      ^                 ^      ^               ^
   Show that [ x, Px ]  2i Px , [ x, Px ]  3i Px .
                          2                        3
4. Prove the following commutator identity:


                                AB,   A  B, C    A, C  B
                                   C                

        a) Show that  x , p   i nx .
                        n            n 1
                     

                                                         df
        b) show more generally that    f ( x), p  i      , for any function.
                                                         dx

5. Show that the radial momentum operator


                                               1
                                     Pr  i   
                                              r r 

 satisfies the commutation relation [r , Pr ]  i . Explain the physical significance of this.


                                                            ^
6. If    is an eigenfunction of the linear operator A , prove that it can simultaneously be an
                                            ^                           ^
eigenfunction of another linear operator B that commutes with A .

7. Consider the operator


                           ^     d
                         Qi
                                d

 Where  is the usual polar coordinate in two dimensions. (This operator might arise in a

                                                                                  ^
physical context if we were studying the bead-on-a-rising), Is Q hermitian? Find its

eigenfunctions and eigenvalues.


                                                                                      ^
8. Suppose that f ( x ) and g ( x) are two eigenfunctions of an operator Q , with the same


                                                                                                 ^
eigenvalue q . Show that any linear combination of f and g is itself an eigenfunction of Q ,

with eigenvalue q .


9. a) Find the eigenfunctions and eigenvalues of the momentum operator.
   b) Find the eigenfunctions and eigenvalues of the position operator.


10. Is the ground state of the infinite square well an eigenfunction of momentum? If so, what is its

momentum? If not, why not?


11. The Hamiltonian for a certain two-level system is


                                  H  E 1 1  2 2  1 2  2 1 ,

     a) Where 1        and 2          is an orthonormal basis and E is a number with the dimensions of

          energy.


     b) Find its eigenvalues and (normalized) eigenvectors (as linear combinations of 1           and

           2 ).


                                   A
12. Suppose          ( x,0)           ,for constants A and a.
                                 x  a2
                                  2




    a) Determine A, by normalized  ( x, 0) .


    b) Find the expectations of x and x2, and  x (at time t=0).


    c) Find the momentum space wave function ( p, 0) , and check that it is normalized.


    d) Use ( p, 0) to calculate the expectations of p and p , and
                                                                     2
                                                                             p (at time t=0).

    e) Check the Heisenberg uncertainty principle for this state.


13. A harmonic oscillator is in a state such that a measurement of the energy yield either
(1/ 2)  or (3 / 2)  , with equal probability. What is the largest possible value of the
expectation of p in such a state? If it assumes this maximal value at time t  0 , what is

 ( x, t ) ?

14. Given three degenerate eigenfunctions that are linearly independent although not necessarily

orthogonal, find three linear combinations of them that are orthogonal to each other and are

normalized. Are the three new combinations eigenfunctions? If so, are they degenerate?


15. Find expressions for the eigenfunctions and energy levels of a particle in a two-dimensional
circular box that has perfectly rigid walls.


16. The Schrodinger equation for a rigid body that is constrained to rotate about a fixed axis and

that has a moment of inertia I about this axis is


                                          2
                                               2
                               i                 ,
                                    t    2 I  2

     a) Where  ( , t ) is a function of the time of t and of the angle of rotation,  , about the

         axis.


     b) What boundary conditions must be applied to the solutions of this equation? Find the

         normalized energy eigenfunctions and eigenvalues. Is there any degeneracy?


17. Show that the integral in Equation


                                                 b
                                        f g   f ( x)* g ( x)dx
                                                 a




satisfies the conditions for an inner product, and show that the set of all square-integrable

functions is a vector space.


18. Show that if         h Q h  Qh h           for all functions   h (in Hilbert space), then
  f Q g  Qf g         for all f and g .


19. Show that the sum of two hermitian operations is hermitian, and the position operator and the

Hamiltonian operator are hermitian.


20. Suppose Q is hermitian, and          is a complex number. Under what condition (on  ) is
 Q hermitian? And when is the product of two herimitian operators herimitian?

                                                                                
21. The hermitian conjugate (or adjoint) of an operator Q is the operator Q         such that


                                    f Q g  Q f g         (for all f and g )


     a) Find the hermitian conjugates of x , i , and d / dx .


     b) Construct the hermitian conjugate of the harmonic oscillator raising operator.
                                   
     c) Show that (QR)  R Q .


22. At the time t = 0 the wave function for hydrogen atom is


                             1
                (r , 0)       (2 100  210  2 211  3 211 )
                             10

 a) What is the expectation value for the energy of this system?


 b) What is the probability of finding the system with l  1 , m  1 as a function of time?


 c) What is the probability of finding the electron within 10-10 cm of the proton (at time t =0)? (A

good approximate result is acceptable here.)


 d) How does this wave function evolve in time; i.e., what is    (r , t ) ?

 e) Suppose a measurement is made which shows that L = 1 and Lz =+1, Describe the wave

function immediately after such a measurement in terms of the  nlm used above.




    Part 4    Formalism - The representation of states and dynamical variables

1. Assume that any hermitian matrix can be diagonalized by a unitary matrix. From this, show that

the necessary and sufficient condition that two hermitian matrices can be diagonalized by the same

unitary transformation is that they commute.


2. Show that a nonsingular matrix of finite rank must be square. Also show that in this case the

equation AA1  1 implies the equation A1 A  1.

3. Find two matrices A and B that satisfy the following equations:

                                                                  
                   A2  0                   
                                     AA  A A1                  B A A

where 0 is the null matrix and 1 is the unit matrix. Show that B2  B . Obtain explicit
expressions for A and B in a representation in which B is diagonal, assuming that it is

nondegenerate. Can A be diagonalized in any representation?


5. An anti-hermitian (or skew-hermitian) operator is equal to minus its hermitian conjugate:
Q   Q , show that the expectation value of an anti-hermitian operator is imaginary. Also show
that the commutator of two hermitian operators is anti-hermitian. How about the commutator of

two anti-hermitian operators?


6. The Hamiltonian for a certain three-level system is represented by the matrix


                                   a0 b 
                                               
                              H   0c 0    where a, b, and c are real numbers.
                                   b0 a 
                                               

 a) If the system starts out in the state


                             0
                      (0)  1 
                                         what is  (t ) ?
                             0
                              

 b) If the system starts out in the state


                             0
                      (0)   0 
                                         what is  (t ) ?
                             1 
                              

7. Consider a free-particle wave packet in one dimension. At t=0 it satisfies the minimum

uncertainty relation


                                                     2
                             ( x) ( p) 
                                     2        2

                                                    4

In addition, we know  x  p  0                      ( t  0 ). Using the Heisenberg picture, obtain

 (x)2  t as a function of t (t  0) when  (x)2  t 0 is given.

           '             "                                                                    '       "
8. Let a       and a          be eigenstates of a Hermitian operator A with eigenvalues a and a ,

respectively ( a  a ). The Hamiltonian operator is given by
                '        "




                                   H  a '  a"  a"  a ' , where  is just a real number.

  a) Clearly,       a'       and    a"   are not eigenstates of the Hamiltonian. Write down the
       eigenstates of the Hamiltonian. What are their energy eigenvalues?


                                                                at t  0 . Write down the state vector in
                                                            '
  b) Suppose the system is known to be in state a

       the Schrodinger picture for t  0 .


  c) What is the probability for finding the system in a "             for t  0 if the system is known

                              at t  0 .
                          '
       to be in state a


9. Use the one-dimensional simple harmonic oscillator as an example illustrate the difference

between the Heisenberg picture and the Schrodinger picture. Discuss in particular how (a) the

dynamic variables x and p and (b) the most general state vector evolve with time in each of the

two picture.


10. Consider a particle of mass m subject to a one-dimensional potential of the following form:


                                          1 2
                                           kx  forx  0
                                 V ( x)   2
                                           forx  0
                                          


 a) What is the ground-state energy?

                                           2
 b) What is the expectation value x            for the ground state?


11. A particle in one dimension is trapped between two rigid walls:


                                          for  x  L
                                V ( x)  
                                          forx  0, x  L

At t  0 it is known to be exactly at x  L / 2 with certainty. What are the relative

probabilities for the particle to be found in various energy eigenstates?


12. Consider a particle in one dimension bound to a fixed center by a   function potential of
the form       V ( x)  v0 ( x) , ( v0 real and positive). Find the wave function and the binding
energy of the ground state. Are there excited bound states?


13. A particle of mass m in one dimension is bound to a fixed center by an attractive
  function potential of the form             V ( x)   ( x) , (   0 ). At t  0 , the potential is
suddenly switched off (that is, V  0 for t  0 ). Find the wave function for t  0 . (Be

quantitative! But you need not attempt to evaluate an integral that may appear. )


14. You are given a real operator A satisfying the quadratic equation A2-3A+2=0. This is the

lowest-order equation that A obeys.


  a) What are the eigenvalues of A?


  b) What are the eigenstates of A?


  c) Prove that A is an observable.


15. For a simple harmonic oscillator with H = (p2/m + kx 2)/2, show that the energy of the ground

state has the lowest value compatible with the uncertainty principle.


16. Making use of raising or lowering operators, but without solving any differential equation,

write down the (non-normalized) wave function of the first excited state of the harmonic oscillator.


17. The wave function of the state where the uncertainty principle minimum is realized is a

Gaussian function exp( x ) . Making use of this fact, but without solving any differential
                                 2


equation, find the value of   .

18. Assume that the eigenstates of a hydrogen atom isolated in space are all known and designated

as usual by


                       nlm (r , ,  )  Rnl (r )Ylm ( ,  ) .

Suppose the nucleus of a hydrogen atom is located at a distance d from an infinite potential wall

which, of course, tends to distort the hydrogen atom.


  a) Find the explicit form of the ground state wave function of this hydrogen atom as d

  approaches zero.


  b) Find all other eigenstates of this hydrogen atom in half-space, i.e. d  0 , in terms of the
   Rnl and Ylm .

19. A certain state      is an eigenstate of L2 and Lz:
                   L2   l (l  1)        2
                                                ,     Lz   m  .

  For this state calculate        Lx     and     L2 .
                                                  x



20. Suppose an electron is in a state described by the wave function


                               1                                                    
                    
                               4
                                  (ei sin   cos  ) g (r )         where     0
                                                                                        | g (r ) |2 r 2 dr  1 and  , 

are the azimuth and polar angles respectively.


  a) What are the possible results of a measurement of the z-component Lz of the angular moment

  of the electron in this state?


  b) What is the probability of obtaining each of the possible results in part (a)?


  c) What is the expectation value of Lz ?




                                  Part 5 Spin and identical particle

1. Prove


   a) [S x , S y ]  i S z ;      [S y , Sz ]  i Sx ;     [Sz , Sx ]  i S y

   b) [ x ,  y ]  2i z ;      [ y ,  z ]  2i x ; [ z ,  x ]  2i y ;  x y z  i

   c) [ L , J ]  0 ;             [S 2 , J 2 ]  0 ;       [ L2 , J z ]  0 ;                 
           2   2
                                                                                    [S 2 ,J z ] 0

   d) [ J x , J y ]  iJ z  [ Lx , S y ]  [Sx , Ly ] ;   [ Li , S j ]  0     ( i, j  x, y, z )


2. What would be the unperturbed ground-state wave function of helium if each electron had spin

angular momentum               and obeyed Einstein-Bose statistics?


3. Write down the unperturbed ground-state wave function for a neutral lithium atom.


4. Find a set of six 2  2 unitary matrices that represent the 3 6 permutations of three
                                                                !=
objects.


5. Show that, if two pure states of a spin 1/2 particle are orthogonal, the polarization vectors for
these states are equal and opposite.


6. Show that the trace of the product of any two perpendicular components of             for a spin s
particle is zero.


7. Evaluate the trace of the square of any component of       for a spin s particle.

8. Calculate the polarization vector explicitly for an arbitrary pure state of a spin 1 particle. Show

that the length of this vector is less than or equal to unity. Find the condition on the state such that

the length is equal to unity.


9. Suppose a spin-1/2 particle is in the state


                                 1 1  i 
                                       .
                                  6  2 

What are the probability of getting  / 2 and  / 2 , if you measure S z and S x ?


10. An electron is in the spin state


                                  3i 
                            A 
                               4 

    a) Determine the normalization constant A.
    b) Find the expectation value of S x , S y , and S z .


    c) Find the “uncertainties”  S x ,       S y , and  S z .(Note: These sigmas are standard
    deviations, not Pauli matrices! )


    d) Confirm that your results are consistent with all three uncertainty principles.


11. Find the eigenvalues and eigenspinors of S y . If you measure S y on a particle in the general

state    , what values might you get, and what is the probability of each? Check that the
                                                                                         2
probabilities add up to 1. (Note: a and b need not be real!). If you measured S y , what values

might you get, and with what probabilities?


12. Construct the spin matrices for a particle of spin 1. (Hint: How many eigenstates of S z are

there?) Determine the action of S z , S  , and S  on each of these states. Follow the procedure
used in the text for spin 1/2.


13. If you measured the component of spin angular momentum along the x-direction, at time t,

what is the probability that you would get  / 2 ? Same question, but for y-component and the

z-component.


14. Apply S  to 10 , and conform that you get              2 1  1 , and apply S  to 00 , and
                                                      and 1  1
                                                                                             2
conform that you get zero. Show that 11                             are eigenstates of S ,with the

appropriate eigenvalue.


15. Quarks carry spin 1/2. Three quarks bind together to make a baryon (such as the proton or

neutron); two quarks (or more precisely a quark and an antiquark) bind together to make a meson

(such as the pion or the kaon). Assume the quarks are in the ground state (so the orbital angular

momentum is zero).


  a) What spins are possible for baryons?


  b) What spins are possible for mesons?


16. A particle of spin 1 and a particle of spin 2 are at rest in a configuration such that the total spin

is 3, and it z-component is      . If you measured the z-component of the angular momentum of the

spin 2 particle, what values might you get, and what is the probability of each one?


17. An electron with spin down is in state  510 of the hydrogen atom. If you could measure the

total angular momentum squared of the electron alone (not including the proton spin), what values

might you get, and what is the probability of each?


18. Suppose a hydrogen atom is in the state


                              1                         
                              2 R21 (r )   Y11 ( ,  ) 
                                                      .
                                3                       
                                 R21 (r ) Y10 ( ,  ) 
                              2                         


  Find the average values of Lz and S z


19. Construct the wave function for hydrogen in the state n  4 , l  3 , m  3 . Express your
answer as a function of the spherical coordinates r ,  , and  . Find the expectation value of r

in this state. If you could somehow measure the observable Lx  Ly on an atom in this state,
                                                                     2   2



what value (or values) could you get, and what is the probability of each?


20. It is obvious that two nonidentical spin 1 with no orbital angular momenta can form j  0 ,

 j  1 , and j  2 . Suppose, however, that the two particles are identical. What restrictions do we
get?


21. N identical spin 1/2 particles are subjected to a one-dimensional simple harmonic oscillator

potential. What is the ground-state energy? What is the Fermi energy?


22. What are the ground state and Fermi energies if we ignore the mutual interactions and assume

N to be very large?


23. Two identical spin 1/2 fermions move in one dimension under the influence of the infinite-wall

potential V   for x  0, x  L , and V  0 for 0  x  L .


  a) Write the ground-state wave function and the ground-state energy when the two particles are

constrained to a triplet spin state (ortho state).


  b) Repeat a) when they are in a singlet spin state (para state).


  c) Let us now suppose that the two particles interact mutually via a very shor-range attractive

potential that can be approximated by V   ( x1  x2 ) (   0 ).


24. Consider three weakly interacting, identical spin 1 particles.


  a) Suppose the space part of the state vector is known to be symmetric under interchange of any

pair. Using notation  0            for particle 1 in ms  1 , particle 2 in ms  0 , particle 3 in

ms  1 , and so on, construct the normalized spin states in the following three cases:

  (1) All three of them in  ; (2) Two of them in  , one in 0 ; (3) All three in different

spin states. What is the total spin in each case?


 b) Attempt to do the same problem when the space part is antisymmetric under interchange of

any pair.
       Part 6     Perturbation theory (time-independent and time-dependent)

1. The unperturbed Hamiltonian of a two-state system is represented by


                                        E10      0 
                                  H0             0
                                                       .
                                        0        E2 

There is, in addition, a time-dependent perturbation


                                            a b
                                   V (t )               (both a and b are real)
                                            b a

Find the energy using perturbation theory up to second order.


2. Suppose we put a delta-function bump in the center of the infinite square well:


                                H '   ( x  a / 2) , where  is a constant.

  a) Find the first-order correction to the allowed energies. Explain why the energies are not

perturbed for even n.


  b) Find the first three nonzero terms in the expansion of correction to the ground state,    11 .

3. Two identical bosons are placed in an infinite square well. They interact weakly with one

another, via the potential


                             V ( x1 , x2 )  aV0 ( x1  x2 )

(where V0 is a constant with the dimensions of energy, and a is the width of the well).


  a) First, ignoring the interaction between the particles, find the ground state and the first

       excited state (both the wave function and the associated energies).


  b) Use first-order perturbation theory to estimate the effect of the particle-particle interaction

       on the energy of the ground state and the first excited state.


4. Consider a particle of mass m that is free to move in a one-dimensional region of length L that
closes on itself. Show that the stationary states can be written in the form


                                         1 2 inx / L
                                 n        e         , ( L / 2  x  L / 2 )
                                          L


                                                                           2  n 
Where n  0, 1, 2,            , and the allowed energies are En                . Notice that, with the
                                                                           m L 

exception of the ground state ( n  0 ), these are all doubly degenerate.


5. Suppose we perturb the infinite cubical well by putting a delta function “bump” at the point

( a / 4, a / 2,3a / 4 ):


                           H '  a3V0 ( x  a / 4) ( y  a / 2) ( z  3a / 4) .

Find the first-order corrections to the energy of the ground state and the (triply degenetate)first

excited state.


6. Consider a quantum system with just three linearly independent states. Suppose the Hamiltonian,

in matrix form, is


                              1        0 0
                                            
                           H  0         1         where V0 is a constant, and        is some small
                               0          2
                                            

number (   1 ).


  a) Write down the eigenvectors and eigenvalues of the unperturbed Hamiltonian (   0 ).


  b) Solve for the exact eigenvalues of H. Expand each of them as a power series in                , up to
second order.


  c) Use first- and second-order nondegenerate perturbation theory to find the approximate

eigenvalue for the state that grows out of the nondegenerate eigenvector of H0. Compare the exact

result, from (a).


  d) Use degenerate perturbation theory to find the first-order correction to the two initially

degenerate eigenvalues. Compare the exact results.
7. Consider the isotropic three-dimensional harmonic oscillator. Discuss the effect (in first order)

of the perturbation    H '   x 2 yz (for some constant  ) on

  a) the ground state;


  b) the (triply degenerate) first excited state.


8. Consider the (eight) n  2 states, 2ljm j . Find the energy of each state, under weak-fiel

Zeeman splitting and construct a diagram to show how the energies evolve as Bext increases. Label

each line clearly, and indicate its slope.


9. If l  0 , then j  s , m j  ms , and the “good” states are the same for weak and strong fields.

Determine EZ and the fine structure energies, and write down the general result for the l  0
           1


Zeeman effect (regardless of the strength of the field). Show that the strong field formula

reproduces this result, provided that we interpret the indeterminate term in square brackets as 1.


10. Analyze the Zeeman effect for the n  3 states of hydrogen, in the weak, strong, and

intermediate field regimes. Construct a table of energies, plot them as function of the external field,

and check that the intermediate-field results reduce properly in the two limiting cases.


11. A weak electric field is applied to an electron in an infinitely deep one-dimensional square well

extending from z  0 to z  L , producing a perturbation V  e ( z  L / 2) . Find the

first-order corrections to the ground state energy of the electron.


12. Calculate the Stark effect on the n  2 level in hydrogen (ignoring spin-orbit coupling) due

to an electric field in the z-direction (which produces a potential) V'  e r cos  .


13. Find the first-order Stark effect for a hydrogen atom in the state n  3 . Sketch the

arrangement of the levels and state the quantum numbers associated with each.


14. A very slowly increasing magnetic field in the z-direction is applied to an atom in an energy

eigenstate that is also an eigenstate of orbital angular momentum and of spin, with quantum

numbers L  1 , S  0 .Sketch the behavior of the energy eigenvalues with time.


15. If the first n  1 eigenfunctions of a particular Hamiltonian are known, write a formal

expression for a variation-method trial function that could be used to get an upper limit on the
nth energy level.

16. A system that has three unperturbed states can be represented by the perturbed hamiltorian

matrix


                                E1    0       a
                                                 
                               0     E1       b
                                a*   b*       E2 
                                                 

Where E2  E1 . The quantities a and b are to be regarded as perturbation that are of the same

order and are small compared with E2  E1 . Use the second-order nondegenerate perturbation

theory to calculate the perturbed eigenvalues (is this procedure correct?). Then diagonalize the

matrix to find the exact eigenvalues. Finally, use the second-order degenerate perturbation theory.

Compare the three results obtained.

                                                                                                     3
17. A one-dimensional harmonic oscillator is perturbed by an extra potential energy bx .
Calculate the change in each energy level to second order in the perturbation.


18. A hydrogen atom is placed in a uniform electric field of strength E. Calculate the total electric

field at the nucleus. Give a qualitative discussion of your answer in physical terms.


19. Show that the WKB approximation gives the correct energy eigenvalues for all states of the

harmic oscillator.


20. We have encountered stimulated emission, (stimulated) absorption, and spontaneous emission.

How come there is no such thing as spontaneous absorption?


21. A hydrogen atom in its ground state is placed between the places of a capacitor field that has
                                            t /
the time dependence E=0 for t  0 , E  E0e       for t  0 . Find the first-order probability that

the atom is in the 2S state after a long time. What is the corresponding probability that it is in each

of the 2P states?


22. Suppose the perturbation takes the form of a delta function (in time): H '  U  (t ) , assume

that U aa  U bb  0 , and let U ab  U ba   . If ca ()  1 and cb ()  0 , find ca (t ) and
                                           *



cb (t ) , and check that | ca (t ) |2  | cb (t ) |2  1 . What is the net probability that a transition
occurs?
23. A hydrogen atom in its first excited (2P) state is placed in a cavity. At what temperature of the

cavity are the transition probabilities for spontaneous and induced emission equal?


24. What is the spontaneous transition probability per unit time, expressed in sec-1, of a hydrogen

atom in its first excited state?


25. What is the selection rule for allowed transitions of a linear harmonic oscillator? What is the

spontaneous transition probability per unit time, expressed in sec-1, of an oscillator in its first

excited state, when e, m, and       are the same as in Prob. 8?




                                      Part 7    Scattering theory

1. An incident particle of charge q1 and kinetic energy E scatters off a heavy stationary particle if

charge q 2 .


  a) Derive the formula relating the impact parameter to the scattering angle.


  b) Determine the differential scattering cross-section.


  c) Show that the total cross-section for Rutherford scattering is infinite.


2. The Lippmann-Schwinger formalism can also be applied to a one-dimensional

transmission-reflection problem with a finite-rang potential, V ( x)  0 for 0 | x | a only.

Suppose we have an incident wave coming from the left:             x |   eikx / 2 . Consider the
special case of an attractive  -function potential


                                2 
                          V        ( x)        (  0 ) .
                                2m 

Solve the integral equation to obtain the transmission and reflection amplitudes. The

one-dimensional  -function potential with   0 admits one bound state for any value of  .

Show that the transmission and reflection amplitudes you computed have bound-state poles at the

expected positions when k is regarded as a complex variable.


3. A particle of mass m and energy E is incident from the left on the potential
                                  0 x   a
                                  
                         V ( x)  V0  a  x  0
                                   x  0
                                  


                                           (where k 
                                     ikx
   a) If the incoming wave is Ae                            2mE / ), find the reflected wave.

   b) Confirm that the reflected wave has the same amplitude as the incident wave.


4. Calculate the total cross-section for scattering from Yukawa potential, in the Born

approximation. Express your answer as a function of E.


5. For the potential in Problem 3,


   a) Calculate f ( ) , D( ) , and        , in the low-energy Born approximation.

   b) Calculate f ( ) for arbitrary energies, in the Born approximation.


   c) Show that your results are consistent with the answer to Problem 3, in the appropriate

regime.


6. Prove the optical theorem, which relates the total cross-section to the imaginary part of the

forward scattering amplitude:


                                         4
                                          Im( f (0))
                                          k

7. Use the one-dimensional Born approximation to compute the transmission coefficient for

scattering from a delta function and from a finite square wall. Compare your results with the exact

answers.


8. Find the scattering amplitude for low-energy soft-sphere scattering in the second Born

approximation.


9. Find the scattering amplitude, in the Born approximation, for soft-sphere scattering at arbitrary

energy. Show that your formula reduces to


                                           m          4      
                        f ( ,  )               V0   a 3  in the low-energy limit.
                                         2    2
                                                      3      
10. Find the differential scattering cross section for a real potential V (r )  V0e r / a , using the

Born approximation. What is the validity criterion in this case, and under what circumstances is it

satisfied?


11. Use the Born approximation to discuss qualitatively the scattering by a crystal lattice of

identical atoms.


12. Use the optical theorem and the Born approximation amplitude to calculate the total cross

section for a real potential. Discuss your result. Repeat for a complex potential.


13. Show that the total scattering cross section by a real potential that falls off at great distances

like r  n is finite if and only if n  2 , first by means of the Born approximation amplitude.


14. Use the perturbation theory to calculate the differential collision cross section for the

1S  2 P excitation of a hydrogen atom by an electron. Show that the total cross section
becomes the expression given below


                                     8 2 a0 e2
                                           2
                            fB          2
                                                  (q 2 a0  9 / 4) 3
                                                        2




At high bombarding energy.


15. A proton is scattered from an atom that can be represented by a screened coulomb potential

( Ze2 / r )e r / a , together with a real attractive short-range square well potential of depth V0 and
radius R that arises from the nucleus. Use the distorted wave Born approximation to calculate

the scattering amplitude. Assume that only the l  0 partial wave is affected by the nuclear

potential; threat this exactly, and treat the coulomb field to first order.


16. Assuming f  f ( ) show that (d / d ) / 2  0 for fermions in the triplet state.


17. A particle is scattered by a potential V (r )  a / r , Find the differential cross section of
                                                               2


s-wave.


18. Show that the differential cross section for the elastic scattering of a fast electron by the

ground state of the hydrogen atom is given by
                                        V r  a
                               V ( x)                      ( E  U 0 ,U 0  0 )
                                        r  a

19. A spinless particle is scattered by a time-dependent potential


                               V (r , t )  V (r ) cos t .

Show that if the potential is treated to first order in the transition amplitude, the energy of the

scattered particle is increased or decreased by       . Obtain d / d  . Discuss qualitatively what
happens if the higher-order terms are taken into account.


20. Show that the differential cross section for the elastic scattering of a fast electron by the

ground state of the hydrogen atom is given by

                                                                         2
                             d  4m 2 e 4   16      
                                  2 4  1      2 2 
                             d   q  [4  (qa0 ) ] 

21. Consider scattering by a repulsive   shell potential:


                             2m 
                             2  V (r )   (r  R) , (   0 ).
                                

Set up an equation that determines the s-wave phase shift  0                        as a function of

k (E     2
              k 2 / 2m ).

22. Consider the scattering of a particle by an impenetrable sphere


                                     0r  a
                            V ( x)  
                                     r  a

    a) Derive an expression for the s-wave ( l  0 ) phase shift.


    b) What is the total cross section in the extreme low-energy limit k  0 ?

				
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