8/9/2019 CHM102
1/3
CHEM106: Assessment 4Wave Functions and Probability
Answer ey
1! "e#ine t$e com%le& con'u(ate o# t$e #ollowin(
#unctions!
)$e com%le& con'u(ate #or a com%le& number * + a , ib is
de#ined as *- + a − ib! .n
(eneral/ to #ind t$e com%le& con'u(ate o# any #unction/ all
we need to do is toreverse t$e si(n o# t$e ima(inary com%onent!
A! ibZ += 3*
!ix
e xf =)(*
C! )2sin()(* x xf = / noticin(
t$at #or a real #unction/ )()(
* xf xf =
! W$at are t$e re2uirements #or an acce%table wave#unction3
ince ΨΨ=Ψ *2
re%resents t$e %robability density #or #indin( t$e
%article/ t$e
wave#unctionΨ itsel# must meet t$e #ollowin( re2uirements:
51 #inite 5or2uadratically inte(rable7 5 sin(le valued7 and 58
continuous!
8! W$ic$ o# t$e #ollowin( is an acce%table wave #unction over
t$e indicated interval3
A!∞≤≤ xwith
x0
1
x
1
is not an acce%table wave#unction since it a%%roac$es
in#inity as & + 0!
! 2cos( x ) with − ∞
≤ x ≤ 0 2cos( x ) is an acce%table
wave#unction since it meets all t$ree re2uirements listedin
2uestion in t$e s%eci#ied domain!
C! ∞≤≤∞−− xwithe x
xe−
is not an acce%table wave#unction since it a%%roac$es
in#inity as∞−approaches x !
4! W$ic$ o# t$e #ollowin( #unctions are normali9able over t$e
indicated intervals3ormali9e t$ose #unctions t$at can be
normali9ed! 5Hint: .n normali9in(
wave#unctions/ t$e inte(ration is over all s%ace in w$ic$
t$e wave #unction isde#ined!
A! π φ φ φ
20)( ≤≤= −ief
)$e #unction is #inite everyw$ere/ and t$us normali9able!
8/9/2019 CHM102
2/3
;iven
π φ φ φ φ φ φ
π
π π
φ φ
π
2)()( 2
0
2
0
2
0
2
0
* ===== ∫ ∫ ∫ −+
ddeedf f M ii
/
t$e normali9ation constant is M
1
!
)$us/ t$e normali9ed #unction is
φ
π φ i N ef
−=2
1)(
!
! ∞≤≤∞−= xwithe xf x2)(
)$e #unction xe xf 2)( =
is in#inite at ∞= x / and t$us not normali9able!
C! L xwith x
L xf ≤≤= 0)
2sin()( π
)$e #unction is #inite everyw$ere/ and t$us normali9able!
;iven
xd x L
xd x L
x L
xd xf xf M
L L L
∫ ∫ ∫ ===0
2
00
* )2(sin)
2sin()
2sin()()(
π π π
/
loo< u% t$e inte(ration table/ and #ind t$at
)2sin(4
1
2
1)(sin
2 x x xd x α α
α −=∫ !
)$us/ 2)]0sin()
4[sin(
82)
4sin(
82
10
L L
L
L L x
L
L x M
L=−−=−=
π
π
π
π !
)$e normali9ation constant is
M
1
!
)$us/ t$e normali9ed #unction is)
2sin(
2)( x
L L xf N
π =
!
=! Consider t$e #ollowin( normali9ed wave#unction)
2
3cos(
1)( x
L L x
π ψ =
/ #or a
one>dimensional system con#ined in t$e re(ion
L x L ≤≤− !
A. Calculate t$e %robability t$at t$e %article will be #ound in
t$e re(ion between & + 0
and & + ?@!
xd x L L
xd x x P L L
∫ ∫ = ==2/
0
22/
0
*)
2
3(cos
1)()(
π ψ ψ
?oo< u% t$e inte(ration table/ and #ind t$at
)2sin(4
1
2
1)(cos
2 x x xd x α α
α +=∫ !
