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Homework assignment (113A, week 4)
Due: Thursday, 2/11/16
1. The operators which satisfies
*(x) A(x)
dx= (x) A(x) *
dx
for all
well-behaved functions (a single valued, the second derivative
exists) called
Hermitian Operators. All uantu! !echanical operators should be
"er!itian
operators. #h$%
&. 'ind the result operating with operator !
d&
dx& x&
on the function
eax&
.
'or what values of a* wille
ax&
be an eigenfunction of %
+. The displace!ent operator
Ois defined b$ the euation
Of(x)=f(x+ a)
how that the eigenfunctions of
O
are of the for!
(x)= exg(x)
where, g(xa)g(x), and is an$ co!plex nu!ber.
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#hat is the eigenvalue corresponding to
%
. /onsider the entangled wave function for two photons,
1&= 1
&1(H)&(V)+1(V)& (H)( )
where " represents a hori0ontal polari0ation and represents a
vertical
polari0ation
Assu!e that the polari0ation operator
i
P
has the properties
Pi
i(H)=
i(H)
and
( ) ( )i i i
P V V = +
where
1 or &.i i= =
a how that1&
is not an eigenfunction of1P
or&P
.
b how that each of two ter!s in1&
is an eigenfunction of the polari0ation
operators1P
and&P
.
c #hat is the average value if the polari0ation1P
that $ou will !easure on
identicall$ prepared s$ste!s% 2t is not necessar$ to do a
calculation toanswer this uestion.
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3. 2f the wave function describing a s$ste! is not an
eigenfunction of the
operator
B
, !easure!ents on identicall$ prepared s$ste!s will give
different
results. The variance of this set of results is defined in error
anal$sis as
( )&&
B B B =
,where 4 is the value of the observable in a single
!easure!ent and
B
is the average of all !easure!ents. 5sing the definition
of the average value fro! the uantu! !echanical postulates,
* ( ) ( )A x A x dx = , show that
&& &
B B B =
.
6. /onsider the one-di!ensional proble! of a particle of !ass !
in a potential 7
{ V(x )=,for xa
a. how that the bound state energies ( Eiven =, !, and En(0
)
, find the eigenvalues of ".
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?. /onsider the one-di!ensional wave function
(x)=A
(x
x0
)
n
e
xx
0
, where A, n, and
x0
are constants.
5sing chr@dinger*s euation, find the potential (x) and the
energ$ ,
for which this wave function is an eigenfunction. (Assu!e that
as
x ,V(x )0 )
B. A particle of !ass ! !oving in one di!ension is confined to
the region CDxD: b$ an
infinite suare well potential. 2n addition, the particle
experiences a delta function
potential of strength = located at the center of the well. (ee
'ig.). The chrodinger
euation which describes this s$ste! is, within the well,
2
2m2
(x)x
2 +(x
L2 )(x )=E(x ) ,0
