States, operators and matrices Starting with the most basic form of the Schrödinger equation, and...

States, operators and matricesStarting with the most basic form of the Schrödinger equation, and the wave function ():

2

22

2 xmti

tkxitkxi BeAe The state of a quantum mechanical system is . This state can be expanded to a vector and the system can be more complicated (e.g. with spin, etc.):

jmjmjmjm M Here, a hermitian matrix M was introduced, to extract properties from the wave function (hermitian matrices have real eigenvalues).

Dirac notation

jmjmMjm jmjm ˆ

Note: to extract the expectation value of a property, one needs to sum over the expanded state:

mj

jmmjmj

jmjmmjmj

jmmj cMcc,

2,

,

*2,

,,

mj

jmmjmj

mj cjmMjmcM,

2,

,

2,

ˆˆ

Combining operators

Operator T, transforms the original state.

originaldtransformejmTjm ˆ

Hence, to get the expectation value:

orgorgtrftrfjmTMTjmjmMjm ˆˆˆˆ 1

1ˆˆˆˆ TMTM

Adding spin (1)

• Projection is known (m quantum number)

• Length of the two spins is known (j1 and j2)

• Several possibilities to construct projection by adding the two spins

< j1, j1 - 2 | j2, j2 >

Example: m = jmax - 2

jmax - 2

j1 - 2 j2 < j1, m1 | j2, m2 > =

m

Adding spin (1)




< j1, j1 - 1 | j2, j2 - 1 >


jmax - 2

j1 - 1 j2-1 < j1, m1 | j2, m2 > =

m

Adding spin (1)




<j1, j1 | j2, j2 - 2 >


jmax - 2

j1 j2- 2 < j1, m1 | j2, m2 > =

m

Adding spin (2)


• Length of the maximum total spin known

jmax = j1 + j2• Several possibilities to construct projection

from different sizes of total spin j

jmax - 2

j = j1 + j2 -1

j = j 1 + j 2

j = j1 + j2 -2

< j, m > =< j, m > =< j, m > = < j1 + j2, j1 + j2 - 2 >

< j1 + j2 - 1, j1 + j2 - 2 >

< j1 + j2 - 2, j1 + j2 - 2 >} m ≥ jm

j1 j2

j

| j1 –j2 | ≤ j ≤ j1 + j2

Note:j1, j2 and j can be interchanged. However, changing the composite state with one of the constituent states is not trivial and requires re-weighting of the constituent states

Adding spin (3)

• Note:- j1 m1 j1 , thus m1 has (2 . j1+1) possible values and m2 (2 . j2 +1).

Each combination shows up exactly ones in the second column of the tableso the total number of states is (2 . j1+1) (2 . j2 +1).

• The third column has the same amount of states as the second column.The quantum number j is a vector addition, thus it will never be lower than |j1 – j2|, which is called jmin.

• For m < jmin the amount of states is jmax – jmin + 1 = (j1+j2) – |j1 – j2| + 1 for each m.This situation occurs for - |j1 – j2| m |j1 – j2| , thus (2 |j1 – j2| + 1) times.

Number of states: (2 |j1 – j2| + 1) . [(j1+j2)-|j1 – j2| + 1]

• For m -jmin and m jmin the amount of states is jmax m +1. This results in the following sum:

Number of states:

• Together these contributions add up to (2 . j1 + 1) (2 . j2 + 1)

m=jmax m1=j1 & m2=j2 j=j1+j2 & m=jmax

m=jmax-1 m1=j1 & m2=j2-1

m1=j1-1 & m2=j2

j=j1+j2 & m=jmax-1

j=j1+j2-1 & m=jmax-1

m=jmax-2 m1=j1 & m2=j2-2

m1=j1-1 & m2=j2-1

m1=j1-2 & m2=j2

j=j1+j2 & m=jmax-2



1221

21 121

mjjjjm

jjm

jmax - 2

j = j1 + j2 -1

j = j 1 + j 2

j = j1 + j2 -2

m

Adding spin – Clebsch GordanExample: m = jmax - 2

jmax - 2

j1 j2- 2

m

21

1

21

21

,11,,

21,

21,,,

,:

::,

mmmMJ

mmmmMJ

mMmCJMor

mmMandMJwithmmCJM

213212211 ,21,12,2,1 jjCjjCjjCJJ e.g.

From symmetry relations and ortho-normality, the C coefficients can be calculated. The first few:

½ x ½ 1, 1 1, 0 0, 0 1, -1

+½, +½ 1 0 0 0

+½, -½ 0 ½ ½ 0

-½, +½ 0 ½ ½ 0

-½, -½ 0 0 0 1

JM

m1,m2

Combining spin and boostLorentz transformations:

For:

Jackson (section 11.7) calculates the corresponding operator:

Extracting rotations:

Allowing definition of the canonical state:

w

E

w

p

E

p

pLz

)cosh()sinh()tanh(

)cosh(00)sinh(

0100

0010

)sinh(00)cosh(

zRpppEp ˆ0,,,

piepL ˆ

22 pEw

0,,ˆˆ0,,ˆˆ 1 RpLRpL z

jmpjmpL ,ˆ

Intermezzo: rotation properties– The total spin commutes with rotation

– However, the projection is affected with a phase.Consider the rotation around the quantization axis

– Euler rotations, convention: z y’ z’’Advantage: quantization axis used twice for rotation

– Rotations are unitary operators. The rotation around y’ includes a transformation of the previous rotation.

– This results in:

jmJni

jmJni

jmJni ejjJeeJ

122

mimm

JiJni

mjm

Jnijm

ejmejmjmejmge

jmejmR

zz

'

''

''..

'

zyz JiJiJi eeeR '''ˆ

'1 yzyzy JiJiJiJiJi eeeeeUU zyz JiJiJi eeeR ˆ i.e. Rotations can all be carried out in same

coordinate system when order is inverted.

Intermezzo: the rotation matrixSummary of the result from the previous page

– Euler z y z convention makes left and right term easy

– The expression for djm´m is complicated, but is used mainly for the

deduction of symmetry relations:

' ''''

''

'

'

m m

jmmjm

jmm

mjm

JiJiJijm

jmDD

jmeeejmR zyz

mijmm

mimiJimi

JiJiJi

edeejmejme

jmeeejmy

zyz

''' '

'

jmm

mmjmm dd '

''

jmm

jmm dd ','

Note 1: Inverse rotation is accomplished by performing the rotations through negative angles in opposite order:

( Djm’m(, , ) )-1 = Dj

m’m(-,-,-)

Note 2: Since rotations are hermitic, the conjugate matrix is:

Djm’m(-,-,-)=Dj*

mm’()

Note 3: Combination of all this gives: Dj*

m’m()=(-)m’-mDj-m’,-m()

Rotations, boosts and spin

jmRpLRRjmpLRjmpR z100111ˆˆˆˆˆˆ,ˆ

',',

'ˆ'ˆ

ˆˆˆˆˆˆˆˆˆˆ

11,''

11,'

'11,'

'1,'1

110,10,11

11

1001

jmpRDjmpRD

jmpRLDjmDpRL

jmRRpLRjmRRRpLRR

jmm

m

jmm

m

jmm

m

jmm

zz

Describing the rotation of a canonical state:

i.e. rotation of a canonical state rotates the boost and affects the spin state in the same way as it would in the `particle at rest’ state.

'

1,'11 'ˆm

jmmjm jmDjmRR

Remember:

Helicity jpjRpLjp ,ˆˆ, 0

The helicity state is defined with:

',ˆˆˆˆˆˆ

',ˆˆ

0,'01000

0,'0

jmpDjmRRLRjmLR

jmpDjmRpL

m

jmmzz

m

jmm

Compared to the same operations on the spin state:

Hence: jmpDjpm

jm ,, 0,

In other words: helicity () is the spin component (m) along the direction of the momentum.

Note that the helicity state does not change with rotations:

jpRjRpRLjRRLR

jLRRjRpLRjpR

z

z

,ˆˆˆˆˆˆ

ˆˆˆˆˆˆ,ˆ

1011011010

01011

Note that the helicity state does not change with boosts (as long as the direction is not reversed):

jpRjRpLjRRpLR

jRRpLRRpLRjppL

z

zz

,ˆˆˆˆˆˆ

ˆˆˆˆˆˆˆ,ˆ

10010010100

01000

101001

jLRjRRLRjRpL zzˆˆˆˆˆˆˆˆ

001000

Note:

Discrete symmetriesParity

JpxJ

pdt

dxp

xx

Commutation relations:

ˆˆˆˆ

ˆˆˆˆ

pLpL

RR

Left-handed Left-handed

Mirror analogy

Charge conjugation

lesantiparticparticlesCˆ

c

0 MiDirac:

Conjugating+transposing gives:

0 TTMi

C is the matrix doing the transformation: 1CC

T

Resulting in:

0

0

011

11

T

T

TT

CMi

CCMCiC

CCMiCC

antiparticle

particle

bbC e.g.

Time reversalb c

cs

W+

Vcb*

Vcs

b

c

c

s

W+

Vcb*

Vcs

Note that time reversal changes t in –t and input states

in output states (in other words: < bra | to | ket > ).

ˆˆˆ ˆ

ˆ

ti

tiH

tiH

'i.e. the transformed state does not obey the description of motion of the

Hamiltonian, it needs an extra ‘–’ sign.

Another way to show this:

The solution is to make time reversal anti-unitary:

tt

i

ti

tiH

*

*

ˆˆˆ ˆ

Note: this can alsobe shown with the

commutation relation: '

e.g.

ixpixp ,,

Time reversal continued

*ˆ..ˆ AeiAA

Next, the the time reversal operator is split in a unitary part

and a complex conjugation.The m states consist of real numbers, i.e. projections.

mm

m

mmmKm

mmKK

*

ˆ~

',

1''~~

mm

mmmm

Hence a time reversed expectation value can be described with:

Calculating the expectation value of operator Â:

*1

1**

ˆˆ~ˆ~

ˆ~ˆ~

~~ˆ

ˆˆ

ˆˆˆˆ

AAA

AA

A

And combining with the time reversal operator:

AAA ˆ~ˆ~ 1ˆˆSince a second time reversal should

restore the original equation:

AAA ˆ~~ˆ~~ 22 ˆˆ

Hermitic AAAA ˆˆˆˆ 122 ˆˆˆˆ

~~~~

Time reversal and spin

kijkjikijkji JiJJJiJJ

,,ˆ

From the commutation relation:

Consistent with:

JpxJ

pdt

dxp

xx

And: 0ˆ,ˆ R

"''ˆ

"'ˆ'ˆˆˆ

"'

*

'

**

'

*

''

"''''

*

ˆˆˆ

ˆ

jmTDjmDjmDjmDjmR

jmDTjmRTjmTRjmRjmR

jmm

jmmtt

jmm

jmm

jmm

jmm

jmm

jmm

jmmtt

j

mmmjj

mmmmj

mmmjj

mmmmj

mmjmm

jmm

mjjmmmm

mjjmm

jmm

DDDTD

DDDT

",",'2

',"'

','

"'

*

'

","','"''

jmm

jmm

jmm

jmm TDDT "'

*

'"'' The equation: holds if:

jmm

mmmjj

mm

d

T

'

,''

Time reversal and spin continued

mjpjmpormjp

jmpjmpTjmpjmpmjmj

mmmjj

mmtt

,,,,,:,,

',',,,

ˆ

ˆ ,''

*

Time reversal of a canonical state:

Time reversal of a helicity state:

jpDjmpjmpDjp jm

m

jm ,,,,

*

j

mjimj

mmj

m

jm

mjmj

jm

jm

DeDDwith

jpDmjp

andjpDjpDjmp

,

*

,

*

,

*

,,:

,,,,,,,

,,, ˆˆˆ

,,,, ,ˆ jpDejpD j

mjimmjj

m

Should give: jpeDjpD ijm

jm ,,,ˆ

Hence:

Parity and spin

jmpjmpjmp ,,,,ˆ The parity operation on a canonical state:

The parity operation on a helicity state:

jpDjmpjmpDjp jm

m

jm ,,,,

*

j

mjimj

mmj

m

jm

jm

mjm

jm

DeDDwith

jpDjmp

andjpDjpDjpDjmp

,,

*

*

,

**

,,:

,,,,,

,,,, ˆˆˆˆ

jpDejpD j

mjimj

mm ,,, ,, ˆ

Should give: ,,,,ˆ jpejp ji Hence:

Note: jpjp ,,, jmpjmp ,,, but:

Since helicity states include rotational properties: jLRjRpLjp zˆˆˆˆ, 00

Composite states

b-quark s-quark

Bs-meson

21,2

121

21,2

1210,0

Bs ground state:

This ground state can decay to two vector mesons:

b-quark s-quark

Bs-meson

s-quarks-quark

c-quark c-quarkW+

J/-meson

-meson

(easy to detect)

Isospin = 0Spin = 1Parity = -(C-parity = -)

Isospin = 0Spin = 1Parity = -(C-parity = -)

Isospin = 0Spin = 0Parity = -

21

21

21 ,21,,

,2121

mmmmS

mm

mmCmmSMmmSM

Note: M = m1 + m2

Two body decay properties

1,1310,03

11,1310,0 Spin states of Bs, J/ and :

2121 ||, mpmpamm

Not the complete story… Consider momentum in two particle decay (Bs rest frame):

Normalization

21,, mm=

21

21,

21,, ,,mm

mmS mmCSM Hence:

'

' 'ˆm

jmm jmDjmR

Remember:

'

' ',,ˆm

jmm SMDSMR

Still no complete story… Consider angular momentum:

SMYdSMlm lm ,,

Angular momentum

Total spin

With: 0,4

12 *

0

lm

lm D

lY

21

21

21 ,21,,

,2121

mmmmS

mm

mmCmmSMmmSM

Missing: Formalism that describes angular momentum and Yml states.

Rotation with angular momentum ',',ˆ,ˆ

' SMDYdSMRYdSMlmR SMM

lm

lm

','0,4

12','

*

0'' SMDDdl

SMYDd lm

SMM

lm

SMM

Split up the Yml state, to match with new rotation:

'12

4

0,'0,'?0,

'

*

0'1*

'

*

0'

*

'

*

0

lm

lmm

lm

lmm

lm

lmm

lm

Yl

D

DRDDDD

','',',ˆ''''' SMlmDDSMYDDdSMlmR lmm

SMM

lm

lmm

SMM

i.e. the transformation is a product of the rotation of two rest states:

',',ˆ,ˆ'' SMlmDDSMlmRSMlmR lmm

SMM

21

21,

21,, ,mm

mmS mmC

Angular momentum and spinWe can also express them as states with a sum of angular momentum and spin:

S

S

S

S

S

S

S

S

S

Mm mm

lmmmSMmJ

Mm mmSMmJ

MmSMmJ

MmSMmJ

MmSJSJ

mmYdCC

mmMmmlmCSMlmC

SMlmCSMlmJMSMlmlSJM

, ,21,,,,

, ,2121,,

,,,

,,,

,

21

21

21

,

,

,,,,Note: MJ = m + MS

Note: MS = m1 + m2

',',ˆ'' Slmm

SMMS SMlmDDSMlmRSS

Since:

And: JJ

JJSSMM

JMMSJSJ

lmm

SMM DmMMmMMDD

,',''' '''

lSJMDlSJMR JS

MMJ JJ,',ˆ

' The state can be rotated with:

Note that (as expected) the sum of angular momentum and spin (J) is not affected by the rotation, neither are the angular momentum (l) and the total

spin (S). This result is the equivalent of the non-relativistic L-S coupling.

(see 1 page back)

Two body decay and helicity (1)

21

1111

112121

|00ˆ

ˆˆˆˆˆˆˆ

ˆˆˆˆ||

R

pLpLRpLRpLR

RpLRpLppa

zzzz

As with spin, we need to consider momentum in the helicity state:

Angles are zero. Particles are boosted back to back along the positive and negative Z-axis

,,21

*

,,,,

,,21

*

2121

21

**

2121

,

|00ˆ

,,

||,

1212

2

22

1

11

S

SS

S

S

MS

SMSmMS

MS

SMs

sm

sm

DCC

RDSmmSMa

ppDDa

mpmpamm

The link with previous page is provided via the relation between canonical and helicity states:

MS = m1 + m2 = 1 - 2 Why?

21,1 a

p

wa 4

Single state normalization.Rest mass: wMomentum: p


S

J

S

SS

S

J

S

SS

JSSJ

S

S

S

SS

S S

SSS

S

S

Mm mm

JM

MSSmMSmmSMmJ

Mm mm

JM

MSSmMSmmSMmJ

JM

lm

SM

lm

lm

lm

SM

JM

Mm mm

SM

lm

MSSmMSmmSMmJ

Mm mm MS

SMSmMS

lmmmSMmJ

Mm mm

lmmmSMmJJ

DJ

dmlfJ

lCCCC

Dmlfdl

CCCC

DDDJ

lCD

J

lCDD

DDl

dCCCC

DCCYdCC

mmYdCClSJM

, ,21

*

,,,,,,,,,,

, ,21

*

,,,,,,,,,,

***

0

*

0

*

, ,21

**

0,,

,,,,,,,,

, , ,,21

*

,,,,,,,,

, ,21,,,,

21

121221

21

121221

21

121221

21

121221

21

21

,4

12),('

12

12

,),(4

12

12

12

12

12

,4

12

,

,,

21, JJM

Intermezzo, check…

Expressing states with the sum of angular momentum and spin in helicity states:

NJNormalization (later, easier the other way around)


21'

21

*

''

',21

*

'

*

'

21

*

21

*

21

,'

|00'ˆ'

|00'ˆ'

|00ˆ

|00ˆˆ,ˆ

JSMM

SMJ

SMM

M

SM

SMMJ

SMJ

SMJJ

JMD

RDdND

RDDdN

RDdN

RRDdNJMR

JJ

JJJ

J

JJJ

J

J

Check the transformation properties for rotations:

Transforms as it should…Remember: jpRjpR ,,ˆ

1

i.e. Mj transforms, but 1 and 2 do not.

Canonical versus helicity states

21

*

21

*

21 ,4

12,, J

MJMJJ JJ

DJ

dDNdJM

212121

**

,,21

*

,,,,21

,,,

,,

2

22

1

11

2

22

1

11

1212

mmDDDD

DCCmm

sm

sm

sm

sm

MS

SMSmMS

S

SS

(2 pages back)

(3 pages back)

21,

212121

*

0

*

0

*

,,2121

21

*

21

*

21

,012

4

4

12

12

12,

12

4

012

12

,4

12,,

2

22

1

11

2

22

1

11

mmYdmmMllmMMl

J

J

lJM

Yl

D

DllmMMJ

lDD

DmmMDD

mmDDDJ

dDNdJM

lm

MSS

lmJSJ

lm

lm

lm

lmJS

JM

SM

MS

SMS

sm

sm

sm

sm

JM

JMJJ

S

JS

S

S

JJ

lSJM J ,

Combine

(4 pages back)

Normalization

Sl mM

lmSJSJ

S

mmYdmmMlmMMlJ

lJM

,

21212121 ,012

12,

Decay amplitudes

JJJ

JJ

J

JMJMJMp

w

JMp

wJMa

JMppA

m

mm

m

212121

2121

21

4

4

,|,

2121 ,|,, ppa

*

2121

*

21

*

212121

,,

,,,,

JMJ

JMJ

JMJJ

JJ

J

DNDNd

DNdJM

Remember:

p

wa 4

2121 ,,|, app

The transition amplitude to a helicitystate is calculated with the matrix element:

With:

Check…

Hence: JJMJJJ

JMJ FDNJMJM

p

wDNA

JJm

21

*

21

*4

Helicity amplitude

Complete setAnd:

Momentum of decay products: pResonance rest mass: w

Helicity amplitude

SlJJ

SlJJ

JJ

SlJJ

JJJJ

JJJ

JMlSJMp

wll

JMlSJMlJ

lJ

p

wFN

JMlSJMlJ

l

p

w

JMlSJMlSJMJMp

w

JMJMp

wF

m

m

m

m

m

,4012

,012

12

4

124

,012

124

,,,4

,4

21

21

21

21

21

21

21

Sl

JJ lSJMlJ

lJM

,012

12, 2121Remember:

Switch to canonical states:

Complete set

412

J

N JAnd:

Partial wave amplitude: als

Canonical states

Bs0 → J/ , tree level

b c

s

c

ss

Bs0

J/

W+

Vcb*

Vcs

s

c

s

c

J/

W-

Vcb*

Vcs

s

b

Bs0 Bs

0

b

s

W W

Oscillation Decay

0

0

0

0

M

M

Schrödinger2

a ad ii M

b bdt

0

0a

b

M

M

0

0a

b

20

20

iM t

iM t

a ea

bb e

a b

a b

M M

Particle and anti-particle:

No mixing, just 2 states

The basics0 0a X b X

en

Eigen states of the Hamiltonian:1 0

0 1

Time evolution of eigen states

2

a ad ii M

b bdt

12

21

12

21

M M

M M

Hermitic:*

12 21*

12 21

M M

The basics0 0s sa B b B

12*12

12*12

M M

M M

Mixing:

Note: to obtain properties with real values, the matrix needs to be Hermitic.

The basics0 0s sa B b B

12 12* *12 12

* *12 12 12 12

* *12 12 12 12

2

02 2 2

02 2 2

M M ai

M M b

i i iM M M

a

bi i iM M M

The matrix equation is 0 if:* *12 12

12 12

2

2

iMq

a p b qip M

Eigen-states: p p

q q

The basics0 0s sp B q B

0 01

0 02

s s

s s

B p B q B

B p B q B

2 2

1p q

Eigen-states: p p

q q

1,2 1,221,2 1,2( )

ii M t

B t B e

Note 1: No particle and anti-particle. Two different masses and decay times

Note: The Hamiltonian describes the quantum mechanical system. Only for the eigen-states of the Hamiltonian Mass and Decay time have meaning

Note 2: What is the meaning of time (t) at this point. This is a composite system, one particle contains two states. The time is calculated in the rest frame of the particle… On the next slide it becomes obvious.

The basics

1 1 2 2

1 1 2 2

0 0 0 02 2

1 20 0 0

0 0 0 02 2

1 20 0 0

2 2

2 2

i ii M t i M t

s s s s

s s st

i ii M t i M t

s s s s

s s st

p B q B e p B q B eB B qB f t B f t B

p p p

p B q B e p B q B eB B pB f t B f t B

q q q

1 1 2 22 2

2

i ii M t i M t

e ef

2 20 0 0 0

22

0 0 0 0

s s s st

s s s st

P B B B B t f t

qP B B B B t f t

p

1 2

1 22

22 1cos

tt tf t e e e M M t

Note: So time (t) is just the decay time of the measured particle.

States, operators and matrices Starting with the most basic form of the Schrödinger equation, and...

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Transcript of States, operators and matrices Starting with the most basic form of the Schrödinger equation, and...