System Measurement and Preparation

System Measurement and Preparation

Quantum Postulates (Claims)

The quantum probability density

probability flow

Uncertainty relation

System-detector interaction

Copenhagen interpretation of QM

Paradoxes, entanglement

Alternative interpretations

Case study: Exploring the electron spin

Stern-Gerlach polarimeters,

Electron spinor states

W. Udo Schröder, 2019

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Quantum Postulates for Microscopic Systems

Copenhagen (orthodox) interpretation of QMSystem: Ensemble of microscopic particles and force (el-weak, nuclear) fields.

Relativistic or classical, specific: QED, QCD (not gravitation yet!)

I. A system ensemble is completely (!?) described by a smooth wave function y ({qi},t) . Independent coordinates qi:

II. Any observable A corresponds to quantal operator which extracts the

associated information from the wave function (representing vector in H).

III. A system can only be observed in an eigen-state of the corresponding indication operator Â, → Âya = a·ya. Orthonormal set = basis

IV. A series of ensemble measurements of observables A, B →

probability distribution P(A), mean and variance are given by

V. System wave functions evolve in time according to the t-dependent Schrödinger Equation,

( ) ( )

22 2

1 2ˆ ... ( , ) ( , )

ˆˆ ˆˆ ˆ ˆ, : , 0

ˆ ˆ, 0.

ˆ

2

ˆ ˆ

,

a

A

i

B

iA dq dq q t q t and

Incompatible observables A B AB BA A B

ComHeisenbe lc patibt

A A

a

A

i en At Brg Un er y Relation

y y

y y

= = −

→

=

→

y y = ˆi t H

(( q )) qi ii iy =

†A A=

ˆ ˆ, 0If A H A const = → =

ay

Quantum Probability Density

Non-relativistic differential equation for proper wave function (generally t-dependent Hamiltonian):


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( ) ( ) ( )

( ) ( ) ( )

( ) ( )

22 2 2

ˆ ˆ( , ) ( ) 0 ;

( , ) ( ,0) ( ,0) .

:

ˆFor stationary solution ( , ) ( , ), ;

( , ) ( ,0

(

con

)

st

, ) ( , 0)

y y y

y y

y y

y y

−

−

−

→ → = → =

→

=

=

= = =

=

=

=

i S

i S t

x t

i E t

x

For V x t V x H t H t S real

probability density t x t e x

l

x t e

x

Examp e

H t x t t

x

x t e x tE x

22with ( , )( , ) ( , ) ( , ) ; 1 y y

+

−

= = = x t ddP x t x t dx x t xdx

Max Born(188-1970)

( )2 2

2ˆ( , ) ( , ) ( , ) ( , )

2y y y

− = = +

i x t H t x t V x t x tt m x

Interpretation of wave function y by Max Born →

Probability to find (measure) system (particles) at time t in volume element dx. → No statement about reality of y !

Inherent statistics of quantum theory: Probability defined by ensemble average over many similarly prepared systems. → No prediction for any single particle!

Structure of set of all such square-integrable wave functions → L2 vector space

Probability Flow

Consider arbitrary finite domains, where wfs can be normalized.


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( ) ( )

( )*

2 2

2

*

2

**

*

*

ˆ ˆ

)2

,

)

2

( , y y y y

y y

y

y y y y

+ +

− −

+

−

+ +

− −

= − =

− = − =

= − + = −

L L

L L

L

L

L L

L L

x

H H

x x

jx x x x

x ti

dx dxt

idx

m

idx x t d

m

22; w h( , ( i ,) , ( ) 1) t yy

+

− = = L

L

x t dx x t d t xx x d

** * *( , ) y y y y y y y y

+ + +

− − −

= + = −

L L L

L L L

x xi

dx d dxt t t t

t i it

( )( , ) , 0

+ =

Continuity Equ n

x t j xx

a i

tt

t o

* *

2y y y y

−

=

Probability curr t

x xm

en

ij

ሻ𝜌(𝑥, 𝑡inj outj

Probability is Conserved

→ →Use Schrödinger Equation to transpose t x

Microscopic System Ensembles


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( ),P x y

xy

( / ):

, ,...

large ensemble of independent objects particleM s quantoni s

All degrees

t

of

c

fr

s

e

i

e

cros op c ys em

dom x y

= =

( ), ,...; nProbability densit d fu oe rx oy bp sa et rvy m lit atio

( ) ( )

( )

( )

2.

,y,.

,...; ,.. ;

. ( ): ,y,...

..:

;,

; 1

P

t

robability density dP x x dV

Partial prob d

Normalized

d

x

t t

x t

d dy dP x

P dy P x

t

=

=

2

2

A 0ˆ ( , ) ; ,ˆ ( , ) ( , )

( , ) ( , ) 1: , ) (

Aa

a a a aa a

a aOperator A has Eigen Functions x t with

x t form

A x t a x t

orts x t c xhonormal basis t

a

t wi h c

y =

=

=

=

=

( )

*

0

22

*

0

ˆ

ˆ ˆ ...

)

,

( , ,...; ...) ( , ,...; ...)

( , ,...; ...) ( , ,...ˆ ...

ˆ

; .

ˆ

..)

(A

Any observable operator projects a info out of

Mean x y x y dx

x y x y dx

Variance

a

i

A

A

define a Gaussian probab lity P a

a A dy

A dy

A A

y y

y y

y

= = =

=

= −

Any measurement of observable A returns one possible value a, with probability P(a)=|ca|2

“Collapse” of wave function (stays in a) → Measurement in QM (?, still debated).

“Fourier” expansion

have quantum-statistical uncertainty

and

Math Example: Prob. Moments from Generating Functions


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( )

( ) ( )

( ) ( )

−

=

=

(

:

)

: s x

ik x

s dx e P x Laplace transformation of

Character

P

k dx e

istic functio

P x Fou

ns of P

rier t

x same info o

ransformat

n system

ion of P

x=x 0

( ) ( ) ( ) ( )2*

1 ,

1 . . .

: 1 : particExample

spatia

simple dimensional system

dP x dxl d

lem

o f x

a

x x

s m

x

s−

= = →

( ) ( ) ( ) ( )= =

− = = → = − 0 0

: 1n

nn

ns s

d ds dx x P x x similar x s

ds ds

( ) ( ) ( ) ( )

( ) ( )

*

But must be Her

: ( )

1( )

: ( , ) mitian

ik xIf P x x x dk e P k known Fourier transform

P x k P k k mean wave number momentumi x

Quantal operators work on amplitudes x ty

= =

= → =

→

(Set x=0)

( ) ( ) ( ) ( )

( ) ( )

− − = = −

= →

0 0

:( :

: [ ]

)n n

ns x n s x

n n

nn n

n

d ds dx e P x dx x e P x

ds ds

ds Laplace transform of

Derivatives of s

x P x xds

(Set s=0)

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Precision Limits: Uncertainty Relation (Incomp. Observables)

( )2

2 3 22 2

p xxn = −

Heisenberg Uncertainty

Relation

Heisenberg, 1924

a=10 fm a=30 fm

Probability distributions for position x and momentum px are anti-correlated, minimum widths. → {x},{px} = conjugate spaces, like n and t in classical Fourier analysis

Task: measure position of (catch) particle in an ideal 1D box and measure its speed (momentum px=conjugate to x).

position x

momentum px

pro

bability d

ensity

( ) cos sinn an bn

Particle in a Box wave functions

x xx c n c n

a ay

−

= +

:

ˆ ˆ: , 0

( '

, 0

)

ˆx

General feature for observable A and conju

No Cloning

u

a

Theor

Incomp tib xle observables

Role of comm tato

e

r

i

ga

Can t mak system cop es

te

e

p

B

m

i

B

A

→

=

= −

→

→

−

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Classical Wave UR: Fourier Transforms

T0 2=

Fourier Transforms

Uncertainty of time vs. frequency : t ~ 1/f → t · f > 0

time → time →

frequency → frequency →

inte

nsity →

inte

nsity →

inte

nsity →

inte

nsity →




probability flow










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Expectation: Measurement/Preparation (CI)


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Postulate: qm system can only be observed in an eigen-function (state) of the corresponding indication operator Â → Âya = a·ya. Other states are not realized ! Orthonormal set {ya} forms “basis.”

QM makes no specific predictions for any single measurement of observable A, except: In any measurement, system will be found in one of the possible eigen states of Â with probability amplitude existing at time of measurement.

→Many independent measurements are distributed “statistically.”

BUT: Immediate repeat measurement of A on the same system give identical results, not a distribution!

→Wave function “collapses” (is suddenly reduced) to one component and frozen in that state.

→By measurement of A and selection of eigen value a (→state ya), a system can be “prepared” in that state ya. Repeat measurements yield same a.

Discontinuous change in t-dependent behavior of wf is not described by Schrödinger Equation.

Ideal Expectation: System-Apparatus Int.

Debate from the start of QM: Measure dynamical variable W (observable) of system S with an experimental apparatus A.


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S

A

Info in state vector/wave function S

Projected with some W sensitivity projector P

Argument assumes discrete eigen spectra

) )

ˆ ˆSystem eigen states of : ;n ;n

ˆ ˆApparatus : eigen states of A : A ;m ;m

indicator variable(e.g. position of needle)

m,n auxiliary qu.#s

W W =

=

=

)

)

; ;

( ) ; ;

S A states LinComb of n m

Assume pure initialsimple n m

=

( ) ( ) ( )

( ) ) )

,..., ,...

, ,

ˆ ˆ ˆ:

" "

,,ˆ ; ; ; ;

,,n m

Time t after measurement unitary operator t p t t

entanglement because interaction S with A mixes states

nnt n m p n m

mm

P = P P

P = →

correlation → , to make A proper indicator

Measurement : System-Apparatus

Transition amplitude p. Coherent superposition of A states, emphasis on eigen value (by construction of apparatus).

Macroscopic state of system at time t, after measurement:


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( ) ( ) ( )( ),

ˆ ; , , ;t

n

S t n p n n n

= P = →

S

A

) ( ) )

), ,

ˆ ;

,,;

,,

t

n m

t m

nnp m

m

A

m

= P

= →

Macroscopic state of measurement apparatus at time t, after measurement:

Macroscopic state of system: different from initial state, no longer Weigenstate, but a coherent superposition → subsequent measurements of the same observable on the same event would produce different outcomes →

Not consistent with postulated “collapse of the wave function” attributed to measurement process. → future t solutions??

Density matrix approach more appropriate?QM prep and measurement




probability flow










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Quantum Measurement

W. U

do

Sch

röder,

20

1914

QM States: Philosophical Access

Positivism: Nature revealed through perception, useless to search for

reality that cannot be sensed empirically (e.g., absolute space, time,..), Scientific theory: understand relationships via logical deduction verifiable through experiment, (Ernst Mach, Schlick, and “Vienna Circle”)

Realism: There is an observer-independent reality (unprovable !),

(Schrödinger), local reality.

Objectivity: Observations are independent of observer, verifiable

by others in the same fashion; e.g., Einstein’s special relativity shows that there are no absolute reference frames.

Locality: Physical processes occurring at one place should have no

immediate (faster than light) effect on the elements of reality at another (distant) location (no “spooky action” at a distance).

CI → Paradox Schrödinger’s Cat

Cat in a box connected to a poisoned gas volume. Gas is released upon decay of radioactive nucleus and kills cat.

While the box is closed: It is not known if decay has occurred → nucleus is in superposition of original and decayed state →Cat is both dead and alive at the same time → contradicts classical science/experience

Electronic release triggered by rad. (statistical) decay

Cat in a superposition of basis states

1 2 1 2: , , ; ( ,, ) nucl cat u dG aener ia al ve del w df y y y y = + = =

11 2

2

, , ,atom cat u d

entangled state

either

or

y y y

y

= + →

Measurement (observation):

Reduction (“Collapse”) of wave function. → How and when?

After: Different system, different Schrödinger Equ.

Does the state exist as objective reality?cat

Poison gas

Cat hidden in box

Quantum Measurement

W. U

do

Sch

röder,

20

1916

CI → EPR Paradox: Decay Experiment Violates HUR

Radioactive Decay

Nucleus → collinear correlated products A & B.

Q=xA-xB and P=pA+pB→ HUR [Q,P] ≠ 0 !

1) Measure pA -pB of A with DpA=0 (2 slits)2) Measure xB -xA of B with DxB=0 (1 slit)→ Q(t=0), P(t=0) with DQ=DP=0 Contradictionunless 1) causes immediately DxB→ ∞

→Requires instantaneous (spooky) action at a distance (xA →∞) between 2 local realities, forbidden by Special Relativity.….OR

→ Quantum Mechanics is not a complete theory → Hidden variables to make a system

an objective reality (deBroglie/Bohm). ….OR

→ Microscopic systems are non-local. ….OR…?

→No evidence (yet) for hidden local variables.

QM: Entangled state, incommensurable relative position and relativemomentum → − − 2p p x xA B A B

Einstein, Podolski, Rosen: PR 47, 777(1935)

A B

pBpA

xxBxA

P=(pA+pB)=0

Spin-Spin Correlations Test Local Realism

Experimental setup: Two polarimeters (I,II) in orientations a and b, perform dichotomic measurements of linear polarization of photons n1 and n2. Polarimeters can rotate independently about the axis of the incident photon beam. Electronics counts singles and the coincidences. (after A. Aspect et al., PRL 49, 91 (1982))


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a b

J=0

J=0

J=1

40Ca atomic E1-cascade

n1

n2

Glass cube polarimeters split beams into 2 each, linearly polarized components →

detected simultaneously with Det 1-Det 4

( )I a ( )II bDet1 Det2

Det3

Det4

40Ca atomic E1-cascade emits two (spin-1) photons with anticorrelated polarizations, coupling to J=0.

Spin-Spin Correlations


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( ) ( ),

.

Photons emitted in opposite directions measure spin S correlations with

Oriented polarimeters I a and II b

y y y y y y= =

=

= = + = −

=

12 12 12 12 12 12

2 1,.

" " 1) :

..,

ˆ ˆ1 1abn n

ba

correlated spin orientatio

Expected for local reality for parallel

ns in each of n N events

S S S Sa

orientation a b

b

a

a

b

b

ab

ab

Co-planar Detector setup

( )( )

= →

−1

, : n nn

Spin spin correlation C a b a bN

( ) ( )

−

→

=− →

2) , :

, co, s

1

,, . .

n n

Arbitrary orientation a b

Speci

anti correlation in each event

QM predictions

depend on a b g

fic

C a beometry

a

e ag

b

b

“Entangled” wave functions

( ) ( )y y+ − − + + − − += + = +12 12

1 1/

2 2a a a a and or b b b bQM:

bosons

|S=0,Sz=0>= (qu#1 qu#2) (qu#1 qu#2)

I II

Spin-Spin Correlations Reject Local Realism


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( ) ( )

= = + + − →

→ = =

→ = → →

1: ;

1

2 2

n n n n n n n n n nn

n n n n n n n n n n

n

for spin orientations

C g g a b a b a b a bN

all a b and a b a b a b a b

Assuming local reali

g C

ty

( ) ( ) ( ) ( ) + + − = , , , , 2C a b C a b C a b C a b C

( )!!

cos c: os: abSpecific setup QM C =→ += − 1 2 2

QM Prediction

Expt. vs. Prediction

A. Aspect et al., PRL 49, 91 (1982)

( ) ( ) ( ) ( )

3) , , , :

, , , , , , , ?" "

Consider a a b b

Do genera realisti

pairs of polarimeter or

l constraints exist on C a b C a b C a b

ientatio

a

n

C

s

bc

QM spin formalism agrees with many experiments but violates Bell-type inequalities (constraints).→ Initial spin correlations are not defined in reality

(→ not conserved).

→ Spin correlations arise at time of distant (non-local) measurements, according to QM rules (collapse of entangled wf).

Recent EPR Experiments


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( ) =

→

>

22 2

2

(0.12 0.01)

( .

2

2:

)

r pB rB

HUR for uncorr particles should be

cf paper

Lixiang Chen et al., Ph. Rev. Letters 123, 060403 (2019)

Produce correlated photons by simultaneous parametric down conversion of light (example: light splitting @ Calcite).

→ Entangled (correlated) photons

Measure correlations & uncertainties in radial position and radial momentum.

EPR Setup


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Quantum Measurement

W. U

do

Sch

röder,

20

192

2

Possibility (?): Many Worlds Scenario

Conjecture: All possible qm outcomes for measurement of single system are realized (somewhere).

Multiverse postulate: System evolves always smoothly, no collapse. At t of measurement, system is in one of the many possible eigen states of Â in our Universe.

Other eigen states will appear in other universes →Multi-Universes (Multiverses). → Many paradoxes (e.g.,

time travel, causality)! Scientific viability?

Reduction of to one of its components of Universeat t0 = time of measurement Different

Universes

( ) ( )

( )

( )

( ) ( )

( )

1

2

00 0

3

, ..........

, .........., ( ) ,

, , ....

, ..........

n

n

n nn n

n

x t

x tx t t c t x t

x t t x t

x t

→

→ = →

= →

→...

Our Universe




probability flow










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Peculiarities of QM Measurement

Heisenberg-type uncertainty has classical analog for waves and certain pairs of observables (frequency and time, wavelength and trajectory,…). But not for massive particles!


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Demonstrate quantum properties with Stern-Gerlach experiments on spin-½ electron. To what extent can a quantum state be controlled (“prepared”) for further use, e.g., fixate spin-up or spin-down (qubits in quantum computer) ?

S

BNon-classical (?) particle phenomenon: Spin- ½ electron (etc.) magnetic moment, spatial orientation (polarization)? (→ Compare polarized elm. waves, cf. later).

Spin: Associated with mechanical angular momentum (Einstein-de Haas experiment on magnetized Fe bar.)

Follow e.g. Townsend’s Ch. 4, etc.

Spin and Magnetic Moment of Ag Atom

Stern and Gerlach (1922): Splitting of Ag beam in inhomogeneous B. B field exerts force on e- magnetic dipole

Ag atom: outer unpaired s1/2 electron.

Experiment: Only 2 orientations of e- spin/dipole → 2S+1=2

→ S = 1/2.

= − = = +

( )z zz s s B s

B BEF g m

z z zOtto Stern 1888-1969

Walther Gerlach1889-1979

Prof. Gerlach’s postcard with the announcement to Bohr in Copenhagen.

Bx

y

z = 410zB z T m

Stern-Gerlach Spin Polarimeter

Magnetic field B-spin s=1/2 interaction → Hamiltonian


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int

0 0

ˆ 0

0z

outside magnet t

H c z inside t T

outside magnet T t

= −

=

= − = − = − int

ˆ /2

z z s z z

z z zS Pauli spin operator matr

H B S B

ix

B g

0

:

(

/

()

0

: )z x

Inhomogeneous magnetic field B Maxwell Gauss

Example and B x x BB z B z B

=

== + −

+ −

−

+ −

= + =

= + → =

=

= =0 0

1( , , , ) ; ,

2

1( , , , ) 2 0

2

0 0y

y y

ip y

p

i i i ic z T p y c z T p

p z

x

y

zx y z r p z z with r p e with

x y z e e z e e z component

t

T s pt

p p

Block path of one trajectory by absorbing “slit” → S-G Spin PolarimeterRotate ( /2) to deflect in x direction.→ Preparation of spin state:

+

pr zp

Stern-Gerlach Thought Experiments

Classical: z longitudinal (spin) polarization → no component perpendicular to z(e.g., Townsend, Modern Approach to QM, Ch. 1)

Beam of N0 Ag atoms: S-G setup filters (projects) Ag with in positive magnetic-field direction → Spin Polarimeter


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( ) ( )y y

+ −

+ −= + → =

2

0

: 2

1 2 1

Initial Ag beam is unpolarized states z and z

z z initial intensity N

:

1

0

Orthonormalized basis

z z z z and

z z z z

+ + − −

+ − − +

= =

= =

1st S-G filter polarizes in +z direction (inhibiting z-). 2nd S-G tests polarization in x direction.

AgN0

z+ N0/2 x+ N0/4

x- N0/4ˆ

zP z z

+ + += ˆ

xP x x

+ + +=

: ˆz

Projector on z P z z+ + +

=

1 2

Classically unexpected results of measurement: All N0/2 events entering S-G 2 produce a result at the exit of S-G 2. 50% of them show spin in +x direction, 50% in –x.On average no net spin polarization in x direction (undetermined):

02 2

xMean values x x S x x

+ − + += → = − =

Result of many Sz-measurements

Stern-Gerlach Thought Experiments

Classical: z longitudinal (spin) polarization → no perpendicular component (e.g., Townsend, Modern Approach to QM, Ch. 1)



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( ) ( )y y

+ −

+ −= + → =

2

0

: 2

1 2 1

Initial Ag beam is unpolarized states z and z

z z initial intensity N

:

1

0

Orthonormalized basis

z z z z and

z z z z

+ + − −

+ − − +

= =

= =

AgN0

z+ N0/2 x+ N0/4

x- N0/4ˆ

zP z z

+ + += ˆ

xP x x

+ + +=

( )

20 0

2

ˆ ˆ ˆ ˆ2 4

11 ( )

2

x z x z

i

N NP P x x z with P P

z x with Phase factor e

y y

+ + + + + + +

+ +

= =

→ = + =

+ + +=: ˆ

xPProjector o xx xn

1 2

Repeat experiment: blocking the spin-up component→ same (random) result →once z (Sz) is measured, x (Sx) is randomized. Incompatible measurements

( ) ( ) + + − −

=→ = = ==+ +2 2

,1

11

2 2phase factx z oz sz x rx

1st S-G filter polarizes in +z direction (inhibiting z-). 2nd S-G tests polarization in x direction.→ No net, but equal

Stern-Gerlach (Thought) Experiments




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Results of measurement after 2 filters: again equal (random) #s with up or down. Appropriate reduction in total intensity of transmitted component.

( )

y y

+ + ++ + ++ + + + + + + +

+ −+ +

= = =

=

→ → = + = =

220 0 0

22 2

ˆ ˆ ˆ ˆ2 2

ˆ

1... 11 2 ( )

2

ˆz zx x zz

N N NP z x z z with P P P

etc x z z with

P z z

Ph

P x

x z ase

x

: ˆz


=

AgN0

z+ N0/2 x+ N0/4 z+ N0/8

z- N0/8

ˆz

P z z+ + +

=ˆz

P z z+ + +

= ˆx

P x x+ + +

=

1 2 3 ≠0!! Random results , also with opposite z,xfilters

≠ 0: Strange basis? Doesn’t behave like vector in 3D space.

Stern-Gerlach (Thought) Experiments




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: ˆz


=

Arbitrary magnet settings 1, 2,…N; splitting and recombining trajectories for different spin orientations. only last (S-G) performs measurement w/r z direction.No events get lost, mean polarization in any direction is zero.

Equal #s spin-up and spin-down, total # N0 conserved.

z+ N0/2

z- N0/2

Magnet 1 Magnet 2S-G

z axisSource Ag

N0

( ) ( ) ( )y

+ − + − + −

+ +

−

= + + +

= = →

→

2 2

, :

" " 1 2;

(

1

)

; 2 ,

?

z x y

i

Found in S G experiments that

z z x x y y

Up to overall phase factors relative phase factors

Strange overlaps x

What can be learned ab

z y x

out properties of s

no

pin states ket

x yt p rp

s

ze

Conclusions From S-G Experiments

Found in S-G experiments: Any (2D-binary ) polarization amplitude

can be expressed in terms of LC of perpendicular components.


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( ) ( )( )

( ) ( )( ) ( )( )

+

− ++ −

+ +

− + − ++ −

−

+ + − + −

−− − −

+ + − + − + + −

= + = +

= + = + = +

1

2 2

1:

2 2 2

iii i

i ii ii i

ex e z e z z e z

e ey e z e z z e z Hence y z e z

( ) ( )

( ) ( )( )

( )( )

+ ++ +

+ + − + − +

−−D − D− D D

+ + + − + −

= → D = − D = −

= + + = +

2

1 2 ( : )

122 2

ii iii i

y x abbreviate and

e e ey x z e z z e z e

( ) ( )2

1

2

1dx zaz i znz y

+ −− += =

→ Must have space of vectors with complex components !

( )( ) ( )( ) ( )( )

( )

D − D − D − D

+ +

D = →

= + + = + D − D

→ D − D = → D − D =

=

=

→

D

2 1 1 11 1 2 2cos

2 4 4

cos 0 2. ( . .

1 2

) 0 2

i i

Choose

y ex e

e g

Constructing the Spin Space

Electron spin is an angular-momentum like observable with exactly 2 components w/r to any direction → one Hermitian operator with a

component for each spatial dimension i = (x, y, z). Each has 2 eigen states/kets and eigen values /2.


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2

" ,

ˆ: ,

"

1 1ˆ

1

22 2

:

0

0 1

z z z

Consider z as quantization axis spin operator and eigen kets

S z z and S z z

Choose these kets as members of ortho normalized basis

Represent by column ve t

z S S

c ors z and z

+ + − −

+ −

→ = = −

−

= =

( ) ( )1 2 1 21 1

2 21 2 2

Represent S G spinors in basis kets

x z z and y z i zi

+ − + −

−

= =

2 2 2

ˆ ˆ ˆ: , ,

ˆ ˆ ˆ ˆ ˆ ˆ, 0; , 0; , 0

ˆ ˆ ˆ ˆ ˆ ˆ, , 0; , 0; , 0

x y z

x y x z y z

x y z

S G Expts Incompatible successive measurements in S S S

S S S S S S

However S S S S S S

−

→

= = =

Sˆ

iS ˆ

iS

2 2 2 2ˆ ˆ ˆ ˆ ˆ:x y z

Length or square of S S S S S= + +

2 2ˆ

1ˆ

3

2

4

zS z

z

z

S z

=

=

Polarization Operator & Pauli Spin Matrices


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= → =

−

→ =

− → =

1 01 1 1ˆ ˆ:2 2 20 1

1 2 0 11 1ˆ:2 21 01 2

1 2 01 1ˆ:2 202

ˆ ˆ,

z z z

x

y

y

x

y

x

S z z corresponding operator S

From x corresponding operator S

iFrom y corresponding operator S

ii

S S =

−( 0,ˆ )z

as requiri S and cy ed bycl Si Gc

( )

= = + =

−

− =

c

: ,

os sinˆˆ ˆ ˆsi

,

n2 sin cos

,0

n x

y

z

xS G example unpol Ag beam in y direction n

S S n S S c

n

s

n

o

= ˆˆ

2n n

S S n

z x y

nx y z

n n in

n in n

− = − −

Spin op

Matrix represent.

=

.x

y

z

n

Spin quantization in arbitrary dir n n

n = = + + :n x x y y z z

n n n n

+ + + −= + → = =

:

ˆ ( ) ( )2

.;n n n

eigen value eq

To predict measurement

Solve S n n n up n downu

Wolfgang Pauli1900 - 1958

Experimental Spin Preparation and Measurement


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Preparation of State |z+>

Measurement of State |n+>

n

z

n z

y

Task: Measure the spin polarization in n-direction for a beam that is 100% polarized in z-direction. Remember: An observable like Sn is accessible only via the eigen states of the associated operator and their eigen values.

Spin Operator For Arbitrary Direction

Electronic spin can be polarized in a direction by specific SG setup,

Subsequent measurement confirm polarization → prepared.

Subsequent measurements of components in directions destroy prior measurement (polarization).

But: Subsequent polarization measurements in direction produces


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z

z+

or yx

n

, 2,n z

Check cases 0

=

ˆ ˆ2 2 ,2

n nspin orientations relative n S eigen states n n with S n n

Express in above coordinate system

+ − → − =

( )

ˆ ˆ ˆ: cos sin cos sin

cos 0 0 sin cos sinˆ :2 2 20 cos sin 0 sin cos

n z y z y

n

Spin measurements perpendicular to beam axis

n x z plane

S S S

S

− →

= + +

= + =

− −

( ) ( )ˆ. : 2 2 cosn n n z

Exp value n z S n z

= → = =

Determining Polarization Eigen States


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ˆ, ..

)cos sin

sin cos

. : ; :

ˆ ( 1

n

n

In basis z z componentsof S eigen states n n

Solve EV equation S n expect

−

+ −

+ − + −

+

→ − = =

=

→

−

= =

( )

( )

cos sin

sin cos

cos sin 1 cos sin

sin cos 1 cos sin

+ = →

−

+ = → − =

− = → + =

2 2 2 2 2 2

cos sin

sin cos

cos sin cos sin 1 . . .

0

det 0

0

Non trivial solutions for secular determinant

q e d

→

− =

→ =

→ =

−

− −

+ − → =− + =

( )( ) ( )

( )( )( )

( ) ( )

2

2 2

2sin 2 cos 2 cos 2sin

1 cos sin 22sin 2

cos 2 sin 2 1

applying normalization

+

+

= = = →−

+ =

Rotations in 3D and Spinor Spaces


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:

cos si

, ... ; :

( 1)n

sin cos

In basis z z components n n

Solve EV equation expected

+ −

+ − +

−

+ −

→ =

=

= →

−

=

( )

( )

( )

( )

+

−

+ −

−= =

⎯⎯⎯⎯⎯→

cos 2 sin 2: :

sin 2 cos 2

Rotation n nz

zand

Rotation by an angle of of spin quantization axis in 3D vector space corresponds to rotation of spin-1/2 spinor kets in spinor space by /2.

Rotations of objects in normal 3D space have SU(3) symmetry, rotations in spinor space are SU(2) symmetric.

( ) ( ) ( ) ( )sincosP n

If system

n z

prepared

P

in z

and n n z + + + − − +

+

== =

→

=22 2 22 2

Conclusions From S-G Experiments

Electronic spin can be polarized in a direction by specific SG setup,

Subsequent measurement confirm polarization → prepared.

Subsequent measurements of components in directions destroy prior measurement (polarization).

But: Subsequent polarization measurements in direction produces

→ Measurements of any 2 components of electronic spin are incompatible: Fix one → both others are completely uncertain.

Results of 2 subsequent measurements depend on order, corresponding operators do not commute. S-operators change states!


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z

z+

or yx

ˆ ˆ ˆ ˆ ˆ ˆ ˆ, 0,..... .z y z y y z x

S S S S S S i S etc = − = −

( #) 3 .

,

Spin space is different from regular real D vector space

Complex overlap amplitudes important relative phases→

→ Heisenberg Uncertainty Relations for incompatible observables.

n

( ) ( )ˆ. : 2 2 cosn n n z

Exp value n z S n z +

= → = =

Task of Quantum Models

For theoretical modeling of bound microscopic system, assume (conservative) forces (bond potentials) and inertias (m). Use TISE to predict discrete energy spectrum of internal states.

→ Check by experimental absorption/emission spectroscopy

For theoretical modeling of unbound microscopic system of several complex particles, assume interaction forces (interaction potentials) and inertias (m1, m2,..). Use TDSE to predict scattering probabilities as functions of relative velocities.

→ Check by measuring experimental scattering and internal particle excitation

probabilities as functions of angle, relative velocities.


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Claim:All accessible pertinent information about an ensemble of microscopic systems is contained in the qm (well-defined, smooth, square-integrable) wave function y. → Measurement process

Task: Theory must provide immanent answers to questions:→ How to make and interpret measurements? → For which conditions do qm and classical pictures coincide (or mimic e.o.)?


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End of Section

System Measurement and Preparation

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