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What Does It Mean?

From the “Afterword” of D J Griffith’s Introduction to Quantum Mechanics

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Da Big Question

Did the physical system “actually have” the eigenvalue in question prior to the measurement? (REALIST)

OR Did the act of measurement “create” the

eigenvalue (constrained by the wavefunction)? (ORTHODOX)

OR Can we completely duck the question?

(AGNOSTIC)

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Realist View

If the realist view is true, QM is an incomplete theory because:Even if you know everything that QM has to

tell you about the system, you STILL cannot determine all of its features!

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Orthodox Position

Measurement forces the system to “make a stand” helping create an attribute that was not there previously

Since repeated measurements yield the same result, the act of measurement collapses the wavefunctions.

This is strange but not mystical

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Agnostic Response

I refuse to answerI ignore these problems

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EPR Paradox

1935- Einstein, Podolsky, and Rosen came up with the EPR Paradox

Designed to prove that the realist postion is the only possible one

A simplified version goes like this: Assume that a pion at rest decays into an electron

and positron 0->e- + e+

The positron and electron fly off in opposite directions and the pion has spin=0

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EPR Cont’d

Since pion is spin 0, then electron and positron are in singlet configuration |00>

So if the positron is spin up then electron must be down or vice versa.

QM can’t tell you which you will get, but that they will be correlated

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1

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Movin’ On Up to the Big Time, the Deluxe Apartment in the Sky!

Now move the electron far apart (pick one)

10 meters10 light years

Now say you measure electron spin down, then IMMEDIATELY you know that the positron is spin up

BTW, this is an INSTANTANEOUS knowledge!

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The 3 views again

Realist: This is not surprising, the positron was always up since the decay

Orthodox: Hmmm… electron was neither up or down until measurement. The measurement caused the wave function to collapse and that “knowledge” was transmitted instantaneously across the gulf to the corresponding particle.

Agnostic: I don’t have an opinion

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EPR Says

Orthodox view is “spooky action at a distance”

Ergo, says EPR, the realist have it correct and the spins were predetermined at the decay

Of course, this is all predicated on the argument that nothing, not even information, can go faster than the speed of light– called the principle of locality

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Bell’s Thm

EPR did not doubt that QM is correct, just incomplete

Some other “hidden variable” is needed to completely quantify the system. The hidden variable could be a single number or a

whole collection of numbers; it doesn’t matter J. S. Bell proved that ANY hidden variable

theory is INCOMPATIBLE with QM

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The Gedunken

Instead of having the electron and positron detectors along the same direction, allow them to be rotated independently

The first detector measures the component of the electron spin in the direction of unit vector a and the second along the direction of b

For simplicity, we will record the spins as +1 (up) and -1(down)

a b

e- e+

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The Results of the Experiment

Bell proposed to calculate the average value of the product of the spins called P(a,b)

If detectors are parallel, then we have original EPR configuration so always +1 and -1 and therefore P(a,a)= -1 (and so is the average)

electron positron product

+1 -1 -1

+1 +1 +1

-1 +1 -1

+1 -1 -1

-1 -1 +1

… … …

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However

If they are anti-parallel, then P(a,-a)=+1For any arbitrary orientation then

P(a,b)=-a ·bThis result is IMPOSSIBLE for any

hidden variable theory

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Why?

Assume, that the hidden variable is called k k varies in some way that we neither

understand nor control from one decay to the next

Suppose that the outcome of the electron measurement is independent of the orientation (b) of the positron detector. b is chosen after the decay but before measurement of electron and thus is hindered by speed of light (locality condition).

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Let’s do some math

So there is a function A(a,k) which gives the result of the electron measurement and B(b,k) which gives the result of the positron measurement

A(a,k)=+/- 1 and B(b,k)=+/-1 When detectors are aligned, the results

are perfectly anti-correlated A(a,k)=-B(b,k) for all k

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So

dkkbBkaAkbaP ),(),()(),(

Where rho is the probability density of k

Now let’s eliminate B(b,k) by using our correlation function

dkkaAkaAkbaP ),(),()(),(

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If c is an other unit vector then

),(1),(),(

)],(),(1)[(),(),(

0)],(),(1)[(1)],(),([1

),(),()],(),(1)[(),(),(

:1)],([

)],(),(),(),()[(),(),(2

cbPcaPbaP

dkkcAkbAkcaPbaP

kcAkbAkandkbAkaASince

dkkbAkaAkcAkbAkcaPbaP

kbASince

dkkcAkaAkbAkaAkcaPbaP

Bell’s Inequality:

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|P(a,b)-P(a,c)|<= 1+P(b,c)

It is easy to show that P(a,b)=-a·b (the QM prediction) is incompatible with Bell’s Inequality

Suppose all three vectors, a, b, and c lie in a plane with c at 450 to a and b (a perpendicular to b)

Then P(a,b)=0 and P(a,c)=P(b,c)=-.707Obviously .707 is not greater than 1-

0.707 (.293)

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But What does it Mean?

If EPR is correct, then QM is completely WRONG! On the other hand, NO hidden variable is going to

rescue us from the nonlocality that Einstein considered preposterous

Many experiments were performed to test Bell’s inequality: the results were compatible with QM and incompatible with Bell’s Inequality

In other words, the realists are wrong and there is spooky action at a distance

Or in the lingo: there is the possibility of superluminal influences

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Supernaturally Superluminal

A causal influence that propagates faster than light is bad news Because relativity says that anything going faster

than light is going backward in time! Faster than light things: any geometric point

We don’t get upset that geometric points move faster than light

So is this influence casual or is just information like a geometric point?

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Two types of influence

Causal: subluminal or luminal“ethereal”: neither energy or information

and for which the only evidence is a correlation in the data of two different subsystems

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Here, kitty, kitty ….

What is a measurement and why is it so different from other physical processes? And how can we tell when a measurement has occurred?

Schroedinger attempted to answer this in his famous thought experiment formally titled “The cat paradox”

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The Cat Paradox

A cat is placed in steel chamber, together with the following hellish contraption: In a Geiger counter there is a tiny amount of radioactive material so tiny that maybe within 1 hour one of the atoms decays but equally probable none of them decays. If one decays then the counter triggers and via a relay activates a little hammer which breaks a container of cyanide. If one has let this entire system for one hour, then one would say the cat is living if no atom has decayed. The first decay would have poisoned it. The wave function of the entire system would express this by containing equal parts of the living and dead cat. At then end of an hour, the wave function of the cat has the form of

)(2

1DeadAlive

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It’s ALIVE! It’s Dead! Etc. etc.

So according to the paradox, the cat is trapped between life and death: a linear combination until

You look! And the cat is:ALIVE: You saved it by measuring itDEAD: You killed it by measuring it

Schroedinger thought this whole blame game was nonsense

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Out of the Paradox

Most widely accepted answer: the Geiger counter is the measurer, not you and it will make the statistical determination, not you

Wigner and others say that it is the intervention of human consciousness that constitutes measurement in QM

Wigner was Dirac’s brother in law and was associated with the Maharishi Institute and the Natural Law party.

Unfortunately, the term measurement seems to imply human intervention but it does is not necessarily so

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My Answer to All These Difficult Dilemmas

I refuse to answer.