C4 Lecture 3 - Jim Libby 1
Lecture 3 summary• Frames of reference• Invariance under transformations
• Rotation of a H wave function: d-functions
• Example: e+ e-→μ+ μ-
• Angular momentum as a rotation generator• Euler angles
• Generic translations, conservation laws and Noether’s theorem
C4 Lecture 3 - Jim Libby 2
Frames of reference• Consider a frame of reference O in which a generic state is described
i.e. the H wave function ψ(r) of an e- in the 2p state.• If O’ is a different frame of reference connected to O by
where G is a group of transformations i.e. translations, rotations or Lorentz transformations of the coordinate system
• The wave function in O’ will in general be ψ´(r´) with
where ψγ is an orthonormal basis
GggD where)( rr
)()(
))(()()(
O and Oin same theiesprobabilit as )()(1
22
rr
rrr
rr
a
gD
C4 Lecture 3 - Jim Libby 3
Example: infinite 1D well
• For the observer O the ground state is given by
• Now translate x´=D(a)x=x+a. In O´ the ground state is not ψ(x´) but
• In analogous fashion in O´´ where the translation is D(a/2)
-a a
0 2a
-a/2 3a/2
ψ(x)
ψ(x´)
ψ(x´´)
x
x´
x´´
axBx 2cos
ax
aax
axaD
ax
BB
BBxx
22)(
2)'(
2
sincos
coscos)()(1
a
xaxB
ax
aax
ax
BB
Bxx
222
422)2/(
2
sincos
sincos
cos)()(
The physics is invariant
Different eigenfunction
Summation over eigenfunctions
C4 Lecture 3 - Jim Libby 4
Some atomic physics
• We will consider the H atom in one of its excited states• The eigenfunctions are:
• We define a new reference frame O´(x´,y´,z´) which is rotated about the y axis of the original frame by an angle β
– r´=Ry(β)r
• The Hamiltonian and L2 are unchanged as are their respective eigenvalues n and l. In other words they commute with Ry.
• However, the z direction has changed so m is not the same, it is now projected on a new z´-axis
)( ofcomponent theof eigenvalue theis
)1( momentumangular of eigenvalue theis )1(
eigenvalueenergy theis
where)(
22
21
mLzm
llLll
En
u
n
En
nlm
r
2p wavefunction
C4 Lecture 3 - Jim Libby 5
Rotation H wavefunction• The new wave function u´nlm(r´) can be expressed by a superposition
of wavefunctions with the same n and l but with different m´
• Now we will use the 2p state (n=2,l=1,m=0) as an example:
)()()(
tocompared becan which )( )()(
rrr
rrr
a
uRduu mnlm
yl
mmnlmnlm
,
:satisfy tscoefficien The
sin),( and sin),( ,cos
are thesecase 2 For the harmonics. spherical theare where),()(
and on dependenceonly consider thereforecontributenot doescomponent radial The
1
1
1
1010
10
83
1183
1143
10
21
1
1
10210210
mm
m
m
ii
lmlmnlm
mm
ym
YdY
d
eYeYY
pYYrRu
uRduu
r
rrr
C4 Lecture 3 - Jim Libby 6
H continued
),(),(sinsincossin and cos
harmonics spherical theof in terms cossin and cosfor sexpression find then We
cossinsincoscoscos
:givesrotation under
, modulus, theof invariance given the which s,coordinate spherical tomovingby
cossinsincoscossincoscos
relation thegets one
sincos
implies Noting
sincos
sincos
:rotation theofn descriptio a is require n weinformatio of piece final The
111132
21
1034
YYeeY
rr
rxzr
xzz
R
xzz
yy
zxx
R
ii
y
y
rr
rr
C4 Lecture 3 - Jim Libby 7
H continued• We use the previous results to express Y10 in terms the β and
spherical harmonics in the rotated frame
• Comparing to the general expression
we get the d coefficients
• In a similar fashion all dlm′m coefficients can be calculated
– Somewhat labourious – neater method later
• Work out the probability that rotated state is in an eigenstate
),(),(sin cos
cossinsincoscos
cos
111121
10
43
43
10
YYY
Y
,, 1111011
11010
10010 YdYdYdY
sin and sin ,cos2
11102
1110
100 ddd
210'
2
111
0'
2
101 , , , , mmmmm dYYdYYP
C4 Lecture 3 - Jim Libby 9
Example: e+e-→μ+μ-
e+ e-
Spins in the relativistic limitμ+
μ-
Only photon exchange in relativistic limit (Mμ <<CoM energy<<MZ0)
z
z′
θ
Left-handed electron annihilates with right-handed electron from helicity conservation.Therefore, final state particles have opposite helicity as well
Two amplitudes must be of equal intensity, ∫|A|2dcosθ, because of parity conservation
e+ e−
μ+
μ−
z
z′
θ
Initial state Jz=+1Final state Jz′=+1
Initial state Jz=+1Final state Jz′=−1
RH
RH
LH
LH
RH LH
LH
RH
221
22
211
1,1
211
1,1
cos1
cos
cos1
cos1
LRRLRLRL
LRRL
RLRL
eeee
ee
ee
AAd
d
dA
dA
C4 Lecture 3 - Jim Libby 10
Example: e+e-→μ+μ-
1+cos2θ
Weak, Parity violatingeffects distort the distribution
C4 Lecture 3 - Jim Libby 11
Angular momentum operators as generators of rotations
zx
xziJJ
xa
za
xa
za
aaaaa
aaaaaR
R
xzz
yy
zxx
R
xzz
yy
zxx
R
Ry
yyi
zxzx
xzyzx
xzyzxy
y
yy
y
because 1
expanding)(Taylor
0)(for ),,(
)sincos,,sincos(
point specific a take As
coordinate same at the and between iprelationsh a find want toWe
sincos
sincos
inverse and
sincos
sincos
'
:axis about therotation aconsider willAgain we
1
1
1
aa
aaaa
aa
a
aa
arrrr
aaa
rrrr
rr
Jy is the generator of rotations about the y axis.
Similar results for Jx and Jz
C4 Lecture 3 - Jim Libby 12
Angular momentum operators as generators of rotations
• If ψ is a solution of the Schroedinger equation is ?
)(
1
RU
yi J
0, ifsolution a is Therefore,
)()(
)()(
)()(
left with from Operate
)()(
isequation er Schroeding The
HUHHUU
tHUUtdt
di
tUHUUtUdt
di
tHUtdt
dUi
U
tHtdt
di
RRR
RR
RRRR
RR
R
Requires [Jy,H]=0
Unitary operator
C4 Lecture 3 - Jim Libby 13
Angular momentum operators as generators of rotations
• We will now consider time variation of matrix elements of Jy
• Then matrix element is invariant with time and the eigenvalues of Jy are constant. This implies:
– the y projection of angular momentum is conserved– wavefunction is invariant under rotations
tHJHJti
tt
Jt
tt
Jttt
JttJt
ttJt
t
yyy
yy
yy
0],[ and timeoft independen is If
and , used have weWhere
y
*
yJHJ
tttHtdt
ditHt
dt
di
C4 Lecture 3 - Jim Libby 14
Finite rotations
mjJi
mjd
mjdmjJi
Ji
U
Ji
Ji
Ji
U
yj
mm
m
jmmy
yR
y
n
y
n
y
n
nR
,exp,
,,exp
Recalling
exp
Therefore,
expexp1lim
rotations malinfinitisi of nsapplicatio successive
by generated becan angle ofrotation finiteA
0
J
iU R
ˆ.exp
General case where α is a vector in the direction of the rotation axis
with a magnitude equal to the angle of rotation
C4 Lecture 3 - Jim Libby 15
d matrices from rotation operators
1,11,11,1 and 0,11,1
1,10,10,120,12
1,11,11,11,11,1
again and
1,11,11,11,10,10,11,1
again with Operate
0,11,1
2)1()1( where0,11,1
fact that theuse and examplean as 1 Using
tscoefficien findcan we
,, and ,,for relation findcan weIf
1exp
)exp(
212
2
12
24
2221
213
211010
221
221
2
2
2
11112
2
212
33!3
22!2
1
ny
iny
yii
iiyy
yi
y
y
iy
jmiy
iy
jmm
mm
ny
mm
ny
yi
yyy
yj
mm
JJ
J
JJJJJJ
CCJJJJ
J
J
CmmjjCCJ
JJJj
d
mjcmjJmjcmjJ
JJJiJi
jmJimjd
C4 Lecture 3 - Jim Libby 16
d matrices from rotation operators
cos
1,1)exp(0,120,1)exp(0,1
sin
01,11,0 and 1,11,0 as 0,10,1
1,110,11,1)exp(0,1
cos111
1,11,-1 and 01,11,-1 as 1,11,1
1,111,11,1)exp(1,1
cos111
1,11,1 and 01,11,1 as 1,111,1
1,111,11,1)exp(1,1
.calculated becan tscoefficien d the1,11,11,1 and 0,11,1 relations With the
100
213
!31
21
2
2
123!322
33!3
22!2
1101
214
!412
!21
21
212124
!41
212
!21
21
33!3
22!2
1111
214
!412
!21
21
212124
!41
212
!21
21
33!3
22!2
1111
212
2
12
yyy
ny
iny
iii
yi
yyy
ny
ny
yi
yyy
ny
ny
yi
yyy
ny
iny
JJiiJid
JJi
JJJiJid
JJ
JJJiJid
JJ
JJJiJid
JJ
C4 Lecture 3 - Jim Libby 17
Euler angles
• Generic rotations described by Euler angles
• Define three successive rotations:– angle α about the Z axis
– angle β about the y0 axis
– angle γ about the new z axis
• Can be recast as first γ about original z, β about the original y and α about the original z
• The rotations of wavefunctions can be represented by D matrices
• Using angular momentum operators as generators of the rotations and
(z0)
x0
y0
jmm
zyzj
mm
dmmi
mjJi
Ji
Ji
mjRD
)(exp
,expexpexp,
C4 Lecture 3 - Jim Libby 19
Euler angles
1) -γ about current z2) -β about current y3) -α about current z
C4 Lecture 3 - Jim Libby 20
Translations
• Similar analysis can be applied to translations:
• Assume a is infinitesimal:
)()()()( arrar STU
11)(
:asresult general thecan write We
:isoperator momentum that theRecalling
1)(
),,(),,(),,(),,(),,()(
paaa
p
aa
ar
iU
i
U
zyxz
azyxy
azyxx
azyxazayax
ST
ST
zyxzyx
C4 Lecture 3 - Jim Libby 21
Translations
• So for finite translations:
• Invariance of the wave equation under translations ↔conservation of momentum
• Similarly, for time if Hamiltonian, H, is time independent:
• Invariance of the wavefunction with respect to time translations ↔ conservation of energy
H
iUTT
exp)(
pAA
i
U ST exp)(
C4 Lecture 3 - Jim Libby 22
Symmetry Principles
• An invariance or symmetry principle exists for a physical system, S, and transformation, g G, if the physical laws, expressed for S by the observer O in his coordinate system, also hold good for the same system S in the coordinate system of the observer O’
• Noether’s theorem:– Symmetry principle ↔ Invariance of theory ↔ Conservation law
'
O')(
O)(
:same theare nsHamiltonia then thesame thearemotion of equations theIf
on.ansformatiunitary tr induced theis )( where
'
OO
gUgD
HH
gU
OO
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