 |
An electron is moving freely in the x-direction.
At t=0
the electron is described by the wave function (neglect spin)
y (x)=Aexp{-x2/2b2}exp{ip0x/(h/2p )}.
 | (a) Compute the constant A such that  . |
| (b) Compute Dx at t=0. |
| (c) Compute Dp at t=0
and show that  for the electron. |
| (d) Assume that the electron has a position uncertainty of Dx=10-10m.
Compute its velocity uncertainty
compared to the speed of light. (me=9.1´10-31kg,
 =1.05´10-34J-s,
c=3´108m/s).
Hint:  .
To obtain, for example,  , differentiate
with respect to b and then set b=g=0.
|
|
|
| A particle of mass m is confined to an infinite one
dimensional potential well of width L, i.e. V(x)=0, 0<x<L, V(x)=¥ everywhere else.
At t=0 the particle is equally likely
to be found in the ground state or the firsts excited state.
 | (a) What is the expectation value of the energy of the system? |
| (b) Write down a properly normalized wave
function to describe the
system at subsequent times. |
| (c) Find <px> for times t>0.
|
|
|
|
| Consider a quantum system for which the exact Hamiltonian is H.
Assume the quantum system is of bounded spatial extend, so that it is known rigorously
that the eigenstates of H, {|yn>},
are complete.
| (a) Show that if |yn>
and |ym> are two eigenstates of H
with eigenvalues En and Em with En¹Em , then <yn|ym>=0. |
| (b) Suppose En=Em, with n¹m.
Can we still have <yn|ym>=0 ? |
| (c) The problem H|y>=E|y> is very complicated but it is suggested that we use a trial
function |ytrial> for |y> and approximate E by E=<ytrial|H|ytrial>/<ytrial|ytrial>.
Show that E>E0, where E0 is the lowest eigenvalue
of H.
|
|
|
|
| Assume that the Hamiltonian H for a quantum system is
Hermitian.
| (a) Show that its eigenvalues E are real. |
| (b) Show that the eigenvectors |E> and |E’>
corresponding to different eigenvalues E¹E’ are
orthogonal. |
| (c) If the square of the angular momentum operator L2
and its z component Lz have eigenvectors |Elm> that are
simultaneous eigenvectors of H, i.e.
and these eigenvectors form a complete set of states, show that [H,L2]=[H,Lz]=0. |
|
|
