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Measurement Theory
For background to this section, reread Griffiths Ch. 4 on spin and Stern-Gerlach
experiment.
We went through the structure of standard ”Copenhagen” interpretation of quantum mechanics last semester. Many elements of the theory are nonintuitive from
point of view of classical physics, but we argued that classical intuition is useless
or even misleading when applied on an atomic scale. The internal consistency of
quantum mechanics required the phenomenon of collapse of wave function, wherein
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measurement of Q in a physical system represented by ψ = n an ψn yielded qn with
probability |an |2 (Q̂ψn ≡ qn ψn ), with the implication that wave function immediately after that particular measurement was ψn . Now we investigate the process of
measurement more deeply, show that apparent inconsistencies arise when we try
to apply ideas to macroscopic scale.
1.1
Linearity
Because quantum mechanics supposed to be based on Schrödinger eqn, which is
linear diffeq., superposition principle supposed to hold: if |ψi and |φi are allowed
states of physical system, so is combination α|ψi + β|φi. The state vectors evolve
according to S.’s eqn,
∂
ih̄ |ψ(t)i = H|ψ(t)i,
(1)
∂t
so since both |ψi and |φi are solns so is α|ψ(t)i + β|φ(t)i.
Example 1: 2-slit expt.
Go back to 2-slit expt. with electron gun. Recall our explanation for interference
fringes which appeared when both slits were opened had to do with the fact that
probabilities don’t add, probability amplitudes do.
Curve plotted on the “screen” at the right in each figure is probability distribution
of particle positions x, e.g.
dPA
= |hx|Ai|2
dx
(2)
for state |Ai, etc. (Recall |xi is state with particle definitely at position x.) “Copenhagen” QM says we don’t add probabilities in |Ai and |Bi to get probability in
|Ci, but rather prob. amplitudes
1
detector
detector
detector
Figure 1: |Ai is state with slit 1 closed, |Bi is state with slit 2 closed, |Ci is state with both slits opened.
1
|Ci = √ (|Ai + |Bi),
2
dPC
= |hx|Ci|2
dx
(3)
So dPC /dx contains not only |hx|Ai|2 and |hx|Bi|2 but interference terms:
dPC
dx
=
{|hx|Ai|2 + |hx|Bi|2
+ hx|AihB|xi + hx|BihA|xi}
|
{z
}
(4)
“interference”
Why do these terms give rise to interference pattern? Because the wave function
ψA (x) ≡ hx|Ai has oscillatory character like a wave amplitude, roughly eikri /ri (ri
measured from slit i, i = 1, 2!). So 1st two terms in (4) are consts., 2nd two vary
as
1 1 ikr1 −ikr2
1 1
∼
(e e
+ eikr2 e−ikr1 ) ∼
cos k(r1 − r2 )
(5)
r1 r2
r1 r2
i.e., classical interference pattern depending only on path difference r1 − r2 .
?Point: linear superposition principle crucial to understanding of this expt.
Example 2: Stern-Gerlach apparatus:
simple device for spatial separation of different-spin particles. For illustration consider neutral spin-1/2 particles, e.g. neutrons, place in inhomogeneous magnetic
field B(r). Recall energy of spin-1/2 with moment ~µ in magnetic field is
U = −~µ · B
(6)
Compare energetics with classical case, where any energy between ±µB is allowed.
2
For quantum spin-1/2 particle, since spin is quantized to point either parallel or
antiparallel to B, only allowed values are ±µB.
Now if B is inhomogeneous there is a classical force on a magnetic moment ~µ equal
to F = −∇U = −µ(±∇B). So assume incoming beam in figure is mixture of spins
k and anti-k to field (B k x̂), experience forces in opposite directions. Can separate
spins, reverse ∇B, recombine as shown.
Spin eigenstates for B k x̂ are

1
1
χ1 = √ 
2 1



1
1
χ2 = √ 
2 −1
;

,
(7)
where labels 1 and 2 corresponding to paths followed by spins in figure. Note the
spin quantization axis is ẑ as usual although we’ve taken B k x̂. What happens if
particle with “spin up”,

χ=
1
0


is injected into field gradient? Can write as linear combination






1
1
1
1
1

χ= =  + 
0
2 1
2 −1
(8)
(9)
This is strange: there is only 1 particle, but “part” of it must move along path
1 and the other “part” along path 2: there is a nonzero prob. amplitude for it
to take each path. Could it be that the particle is “really” in χ1 or χ2 , and χ
just represents
our ignorance? No, for when we recombine, spin will be up again,
√
χ = (1/ 2)(χ1 + χ2 ). If it were “really” χ1 , say, it would leave as mixture of spin
up and down.
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(Note for E& M purists: I’ve taken ∇B, B to point in same direction in figure, which could be
arranged approximately, e.g. with a coil whose winding density varies along the axis direction. But
of course to keep ∇ · B = 0 the system will generate some small transverse gradients as well. This
doesn’t affect the argument, as you’ll be able to work out in prob. set. See also Griffiths, p. 181 et
seq.)
Example 3: Ammonia molecule N H3
N H3 molecule is simple example of “2-level system”. Consists of triangle of Hatoms and N -atom out of plane in minimum energy configuration. Since Hamiltonian rotationally invariant, given fixed H-triangle there is no reason for N to be
above rather than below, or v.v. In fact two states |topi and |boti must be degenerate. Neither one is ground state, however: rather than break symmetry, nature
chooses symmetric mixture:
1
|0i = √ (|topi + |boti)
(10)
2
and 1st excited state is the asymmetric combination
1
|1i = √ (|topi − |boti)
(11)
2
Here prob. ampl. for finding N atom at given position z relative to 3 H-atom
plane plotted schematically for 4 states.
? N.B. |topi and |boti not energy eigenstates. If molecule is in, e.g. |topi at t = 0
it does not stay there, but “tunnels” into |boti with time.
This phenomenon is observable, & very similar to 2-slit expt. Imagine we do x-ray
scattering experiment off N H3 molecule as shown below.
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If system is in state |topi, only one source for sph.
wave, so no interference in scattering pattern.
If system is in ground state |0i, there are 2 scattered
waves which interfere, causing fringes on photographic
plate.
Example 4: Schrödinger’s cat paradox
We may be tempted to accept notion of molecule in superposition of 2 different
configurations as mysteries of life at atomic scale, but harder to swallow similar implications at macroscopic scales. Famous Gedanken expt. propsed by Schrödinger:
suppose at t = 0 box is filled with a) gun; b) atom in excited state c) cat; and d)
device to detect when atom decays to grnd state and fire gun at cat.
Atom is not in stationary state, therefore system is not, will evolve in time into
admixture of state with excited atom and live cat |1, alivei and state with grnd.
state atom and dead cat |0, deadi (Just as in N H3 case, where |topi evolves after
some time into an admixture of |topi and |boti). Therefore at later time t cat is
neither alive nor dead, but some admixture of two? What happens when box is
opened? Then you “measure” system, determine if cat is alive or dead–collapse
wave function. Observation itself is responsible for killing cat or keeping it alive.
Seems absurd—leave as question for now.
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1.2
Measurement and collapse of wave function
Go back to Stern-Gerlach apparatus, and see what happens if we try to determine
path particle takes. We’ll put special neutron-sensitive TV cameras a and b along
paths 1 and 2 corresponding to spin parallel and antiparallel to x̂.
If particle with spin along x̂ axis enters, it certainly is detected by camera a. If
a particle with spin up (k ẑ) enters, according to rules, probability it’s detected by
camera a is


µ
¶
1
1
Pa = |hχ1 |χi|2 = | √12 √12   |2 =
(12)
0
2
Now when particle leaves the apparatus it is definitely k to x̂, not ẑ, since we know
it went through arm 1, as only particles with spins k x̂ do. Act of measurement
has changed spin state from χ to χ1 .
Slightly more subtle: suppose we had only put camera b in arm 2, and it didn’t
register anything. If the camera is perfect this means with probability 1 the particle
was in arm 1 and wave function is collapsed anyway, even though it was never
“directly” observed.
In case of N H3 molecule, imagine we can create a beam of x-rays so tight that
we can determine whether the N atom is above or below triangle of H-atoms, as
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shown:
If molecule is in state |boti, there is certainly a scattering. If molecule is in
state |topi certainly no scattering takes place. If the molecule is in ground state
|0i, scattering is observed with 50% probability. If in given expt. no scattering
is observed, molecule is in state |topi at end of observation. Thus starting from
|0i, may happen that act of observation forced molecule into |topi, although no
scattering takes place. Not just semantics: |topi is a higher energy state than
|0i–where did extra energy come from?
Einstein-Podolsky-Rosen “Paradox”
EPR (1935) suggested that Copenhagen qm was an incomplete theory, because
events could only be predicted in probabilistic sense. Proposed “paradox” designed
to prove not that qm was wrong, but that something was missing. Suppose particle
in angular momentum zero state at rest decays into two spin-1/2 particles, which
must be in a spin singlet state, |ψi = (1/2)(| ↑↓i − | ↓↑i) to conserve ang. mom.
Therefore as particles fly apart, no matter how far apart they are, each must be
considered to be in mixed state of | ↑i & | ↓i!
Now suppose one particle detected on Vulcan & found to be | ↑i (outcome had
prob. 1/2). This collapses wave function instantaneously, such that when the
second particle is detected on Klingon home world it is in a state | ↓i with prob. 1.
Measurement on 1st planet has instantaneously influenced measurement on 2nd =⇒
Copenhagen qm fundamentally nonlocal, apparently violates postulate of relativity!
Copenhagen school response: in fact qm not acausal, doesn’t violate relativity, as
no information or energy can be transferred as a result of collapse of wavefctn.
Reason: observer on Vulcan can’t determine result of measurement beforehand.
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1.3
Role of Observer
All examples: ψ changes in 2 ways. 1) Deterministic evolution according to H|ψi =
ih̄∂|ψi/∂t between observations, and 2) abrupt changes in state upon observation.
2nd change is not deterministic, but probabilistic.
Clearly strange, discontinuous things happen in the observation process in Copenhagen description. Why not account explicitly for role played by observer, try to
describe both observer and experiment in deterministic way? von Neumann suggested idealized model system: physicist measuring z component of spin. Before
measurement (frame 1), observer & expt. well-separated, can therefore describe total state by specifying state of physicist & state of spin individually (|ψi = |Ai|ai).
At later time t1 (frame 2) physicist gets up close & personal with spin, must have
|ψ(t1 )i = Û (t1 )|ψi, where Û is time evolution op. (State is no longer direct product!) She now moves away & records observation in notebook at t2 (frame 3).
State has evolved to Û (t2 )|ψi = |A0 i|a0 i, i.e. the act of observation has potentially
altered both spin and physicist (again well-separated).
Suppose the spin initially in state of definite Sz , e.g., |ai = | ↑i. Suppose further:
observer able to make measurement and leaves spin in up state (|a0 i = | ↑i). Then
final state is
Û (t2 )|ψi = |A+ i| ↑i
(13)
where A+ represents the way in which the observer’s knowledge that the spin is up
has altered her. So far, no problem.
Now assume that spin is initially in linear superposition of up & down. Initial state
vector then
α|Ai| ↑i + β|Ai| ↓i
By linearity principle, this must evolve at later time t2 to
|ψ(t2 )i = Û (t2 ) (α|Ai| ↑i + β|Ai| ↓i)
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(14)
= Û (t2 )α|Ai| ↑i + Û (t2 )β|Ai| ↓i
= α|A+ i| ↑i + β|A− i| ↓i
(15)
(16)
So after measurement physicist is in linear combination of states |A+ i and |A− i,
i.e. the notebook doesn’t contain a definite entry on the spin state. This is absurd,
so seems to be impossible to arrange for purely deterministic quantum mechanics
without concept of wave function collapse (see below, however). No need to require
direct observation by a person (a bit anthropocentric!), sufficient to require collapse
whenever microscopic system induces change in macroscopic object which can be
described by quantum mechanics, e.g. detector of some kind. Hmmm... someone
needs to read the detector, though...
1.4
“Resolution” of measurement paradoxes
Success of qm forces us, reluctantly, to believe that an N H3 molecule can be put in
a superposition of different states, but it’s harder to swallow that a cat can be in
such a lin. comb. Saw there is a logical inconsistency as well if we allow measuring
device (“physicist” of sec. 1.3) can be put in superposition–how can measurement
be completed?
Some ways out: (no generally accepted answers!)
1. Copenhagen approach (most common). Relax requirement that every element
of physical theory correspond to element of “reality”. Only goal of physical
theory should be to systematize our knowledge, increase it, and make predictions which agree with experiments. Wave function ψ serves as device used in
computation of probabilities of events to be recorded in macroscopic notebooks
or macroscopic brains, and this is all we can hope to know.
Collapse of wave function no problem: once measurement on microscopic system is made, we have new knowledge & this alters all probabilities for all
subsequent measurements. Prescription which describes all microscopic qm:
after measurement, start computing events with new wave function. Question
of whether N atom in N H3 problem is “really” in two places at once in ground
state is ill-posed–“at once” is an experience-laden term which is irrelevant to
question of what happens in a measurement, which qm tells us, albeit probabilistically.
Measurement process at macroscopic level: Schrödinger’s cat, Wigner’s friend.
Macroscopic system itself not really allowed to be in lin. comb. of distinct
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states. We were sloppy when we described the measurement process, which
actually occurs 1st time microscopic system interacts with macroscopic object.
In case of S’s cat this was when atomic decay triggered gun.
EPR “paradox” no problem: nonlocal influences do exist in nature, but are of
a sort where no information is transferred, consistent with relativity.
2. Wigner approach
• Linearity an approximation only valid at microscopic level–new rules must
be found to describe macroscopic physics.
• Application of human consciousness which constitutes measurement. Consciousness must be considered external to qm, accounted for in description
of measurement process.
3. Many-worlds approach (Everett) Here idea is bizarre and sci-fi like. When
physicist measures spin in mixed state, instead of being placed in mixed state
herself, the universe forks into two copies of itself (you may call them “parallel”
if you wish, à la Star Trek). In 1st universe she measures | ↑i , in 2nd, | ↓i
with probability 1. Two questions I don’t understand: 1) what happens to the
amplitude factors α and β weighting the two pure states in the microscopic
wave function? If |α| ¿ |β| is one universe less likely? 2) How does this work
at the microscopic level? In the N H3 case we don’t want the universe to split
into one copy with |topi and one with |boti. The real ground state (confirmed
by x-ray expts.) is the mixture |0i. How does nature decide when to split and
when not?
4. Hidden variables approach (Einstein, Bohm) Some other variables ζ are assumed to characterize system completely, in addition to wave fctn. ψ. No
idea how to measure ζ =⇒ “hidden” variable. For example, EPR “paradox”
now resolved by saying, 1st particle on Vulcan had spin | ↑i all along since
its creation, and 2nd one had | ↓i all along. During one such decay, ζ might
have one value (as determined by the hidden variables of the initial state &
presumably some conservation laws), determining the spin of the particle on
Vulcan to be | ↑i , etc. During another decay, it might have a different value
leading to | ↓i on Vulcan. Local means ζ was set at the site of the decay, and
the information is carried with travelling particles. Information then obviously
travels at sublight speeds, no problem with causality.
Bell’s Theorem
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? Bell: If local hidden variables theory exists, must satisfy Bell’s inequality (see
discussion below, based on Griffiths p. 377-8). But QM predictions violate
inequality =⇒ QM is not just incomplete, but wrong. Reverse implication:
if QM is right (i.e., confirmed by all expts), no local hidden variable theory
is allowed. Surprising further implication: qm inherently nonlocal (but not
acausal, because no information can be transferred due to collapse of wave
fctn.!)
pf.: EPR expt., let alignment of 2 detectors be general (measure spin component along â, b̂
on two planets. Each detector can only measure ±1 in units of h̄/2. Product of measurement
A(â) on Vulcan and B(b̂) on Klingon home world is ±1 since A, B = ±1.
Note:
1) A does not depend on b̂, etc. because we will hypothesize locality, i.e. just before Vulcan
measurement is made, experimenter on Klingon home world may pick favorite orientation b̂
for his detector, such that signal with this information will never make it to Vulcan in time to
influence outcome.
2) Only if â = b̂ do we have A = −B and p ≡ A · B = −1 with 100% certainty.
Now suppose that given decay is characterized by value of hidden variable ζ. The value
of the measurements on the two planets will now be assumed to depend not only on the
detector orientation, but also on ζ, A = A(â, ζ), B = B(b̂, ζ). (Somehow ζ must arrange
for antisymmetry of total wave fctn.!) Define average value of product of spins over many
measurements to be
Z
P (â, b̂) =
dζρ(ζ)A(â, ζ)B(b̂, ζ)
(17)
where ρ is arbitrary distribution fctn. for hidden variable. Now if detectors aligned, A and B
must be perfectly anticorrelated, A(â, ζ) = −B(â, ζ). So can write
Z
P (â, b̂) = −
dζρ(ζ)A(â, ζ)A(b̂, ζ)
(18)
so for any other direction ĉ,
Z
P (â, b̂) − P (â, ĉ) = −
h
Z
= −
i
dζρ(ζ) A(â, ζ)A(b̂, ζ) − A(â, ζ)A(ĉ, ζ)
h
i
dζρ(ζ) 1 − A(b̂, ζ)A(ĉ, ζ) A(â, ζ)A(b̂, ζ)
(19)
(20)
since A(b̂, ζ)2 = 1. Note that since A = ±1 we have |A(â, ζ)A(b̂, ζ)| ≤ 1 and ρ(ζ)[1 −
A(â, ζ)A(ĉ, ζ)] ≥ 0, so
Z
|P (â, b̂) − P (â, ĉ)| ≤
h
i
dζρ(ζ) 1 − A(b̂, ζ)A(ĉ, ζ)
= 1 + P (b̂, ĉ)
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(21)
(22)
This is Bell’s inequality applicable to local hidden variable theories.
Now show quantum mechanics gives examples incompatible with (21-22). First note qm =⇒
P (â, b̂) = −â · b̂. (Prove this on prob. set 1!) Example. If detector b is oriented perpendicular
to detector a, although each measurement yields ±1, the average or expectation value of the
product is zero. Write down a few trial sets√of values for â k z and b̂ k x to convince yourself.
And if the detectors are 45◦ apart, P = −1/ 2. Apart from sign, similar
to classical
√
√ polarizers!
◦
But if â k z and b̂ k x, with ĉ at 45 between them, (21-22) says 1/ 2 ≤ 1 − 1/ 2, which isn’t
true.
Endnotes
“Professor Wigner, are there any laws of nature which we cannot know?”
—Anthony Zee, currently Institute of Theoretical Physics, Santa Barbara
“I do not know of any.”
—Eugene Wigner, formerly Princeton University, dec. 1985(?)
“... it is entirely possible that future generations will look back from the vantage point of a more sophisticated
theory, and wonder how we could have been so gullible.”
— Griffiths
References on measurement theory:
1. Bohm, David, Quantum Theory, Dover, NY, 1989. General discussion of measurement theory by adherent of
hidden variables viewpoint.
2. Wigner, Eugene, Symmetries and Reflections, Indiana U. Press, Bloomington 1967. Essays on quantum physics
including discussions of role of observer by one of founders of qm.
3. N.D. Mermin, Physics Today p. 38 (April 1985). Mermin writes often for Physics Today on the foundations of
quantum theory, and he’s always worth reading. His summary of developments regarding hidden variables in
Rev. Mod. Physics 65 (1993), p. 803 is probably the most up-to-date high-level review.
4. M. Jammer, The Philosophy of Quantum Mechanics, Wiley, NY 1974.
5. J.S. Bell, Rev. Mod. Phys. 38, 447 (1966).
6. J. Gribbin, In Search of Schrödinger’s Cat and Schrödinger’s Kittens and the Search for Reality, Little, Brown,
1984 and 1995, respectively. Lay account of measurement paradoxes leaning towards hidden variables interpretations.
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