Download Tutorial 1 - NUS Physics

PC3130/3201 - Quantum Mechanics
Tutorial 1
1.
Find the eigenfunction and the energy spectrum of a particle in the potential well
given by V ( x)  0 if x  a and V (x)    if x  a .
2.
A one-dimensional harmonic oscillator is in a state such that at t = 0,
 ( x )  A exp (x
2
/2a 2 ) exp ( ipx /  )
If the wave function  ( x ) is normalised, find : (a)  ( x)
2
(b) AA *
(c) the expectation value  x  of the position operator x,
(d) (x) 2   ( x  x ) 2    x 2    x  2 , where x is the position operator.
(e) the expectation value of momentum,  p 
(f) (p ) 2   p 2    p  2 , where p is the momentum operator.
(g) the expectation value of the potential energy.
Is x p  ( / 2) [Heisenberg’s uncertainty principle] satisfied?
3.
At time t = 0, the normalised wave function for the electron in a hydrogen atom is
 ( x1 , x2 , x3 )  A exp [( x12  x2 2  x3 2 )1 / 2 / a]
1
2
2
2
Find : A,
  where r  ( x1  x2  x3 )1 / 2 ,
 x1  ,
r
and
4.
(x1 ) 2   x1  -  x1  2 .
2
Show that the differential operator
P
 d
,
i dx
is Hermitian in the space of all differentiable wavefunctions  x    ( x) , say,
which vanish at both ends of an interval [a, b].
1
5.
Let a1 and a 2 form a set of orthonormalised base vectors. A vector  is given
by
  3 a1  4 a2
Is  normalised? If not, find the normalised  . Consider a different set of
orthonormalised base vectors
e
1

, e2
e
1

, e2
which are related to the base vectors
as follows:-
ei   cij a j
j
where
2

c ij  1
for i = 1,2.
j1
Express  in terms of the base ei . If a system is in the state  , what is the
probability amplitude of observing the system in the state ei and what is the
corresponding probability?
6(a).
Consider an operator  3 in a Hilbert space and two complete base vectors  and
 . Let
3  =
3  = - 
 ,
Using the  and  as a base, find the matrix representing the operator  3 .
Here  3 is the Pauli matrix.
(b).
Suppose the matrix elements of an operator A with respect to the base
Aij . Consider another complete set of base vectors
original base
 
 
i
are
   which are related to the
i
i
 i   t il  l
l
where t ij are known coefficients. Find the matrix elements of A with respect to the
base
   in terms of A
i
ij
and t ij .
2
7.
In the abstract Hilbert space H, the energy eigenvectors of a particle in an infinite
square well of width L are designated by E1 , E2 , E3 ,, for the ground state, the
first excited state, the second excited state, and so on. Suppose that at a given time a
particle is in the state
 
1
3
E1 
E3
2
2
a)
b)
c)
d)
e)
Express this state in the coordinate x representation.
Express this state in the momentum p representation.
Express this state in the energy representation.
Write down the energy operator in each of these three representations.
Calculate the expectation value of the energy. Do this calculation three times, once in
each of the representations.
f) Using whichever representation you like best, find the rms deviation of the energy
from the mean.
3