Quantum Mechanics: Interpretation and Philosophy

Significant content from: “Quantum Mechanics and Experience” by David Z. Albert, Harvard University Press (1992).

Main Concepts:

-- complementarity-- the uncertainty principle-- superposition-- collapse of the wavefunction-- the measurement problem

Color

Hardness s

hw

b

Assume electrons have a “color” and a “hardness” property (which we can accurately measure).

Assume the “color” of an electron can be either black (b) or white (w).

Assume the “hardness” of an electron can be either hard (h) or soft (s).

(discrete values -- quantization)

Color:

Hardness:

€

ˆ A

€

ˆ B

€

ˆ A ̂ B - ˆ B ̂ A ≠ 0

measurement

measurement

Lets do something easy:

say we measure that an electron is whitee- e-

then if we measure its color again, we will find it is white(same for black, same for “hardness” properties)

(our measuring device is reliable)

Are the color and hardness properties of an electron correlated?

Hardness s

h

No: 50% of white electrons are hard

w

50%

50%

knowing the color of an electron tells us nothing about its hardness (and no additional information helps)

Now assume a white electron emerges from the soft aperture of a hardness box (50% will do so).

e-

Let us measure the color

e-

50%

50%

somehow the hardness box has tampered with the color value !

Colorand

Hardnessb+h

w+h

b+s

w+s

Can we build a “color and hardness” box?

This box would need to consist of a hardness box and a color box. But the hardness measurement scrambles the color, and vice versa.

To say “the color of this electron is now such-and-such and the hardness of this electron is now such-and-such” seems to be fundamentally beyond our means (uncertainly principle: color and hardness are incompatible physical properties)

the “black box” is just a fancy mirror that makes the two paths coincide

If we feed an s (or an h) electron in, it emerges along the h and s path, unchanged.

Use a w electron and measure hardness.

w

Find 50% h and 50% s

Use a h electron and measure color.

h

Find 50% w and 50% b

Use a w electron and measure color.

w

This device is just a fancy hardness box (it is a hardness box with a few harmless mirrors), and we know a hardness measurement scrambles the color.

Use a w electron and measure color.

w

Find 100% w !!!

Let us add a movable wall that absorbs electrons

Slide the wall into place:

1.) 50% reduction in the number of electrons emerging along the h and s path

2.) Of the electrons that emerge, their color is now scrambled: 50% w and 50% b.

w

What can possibly be going on ?

Consider an electron which passes through the apparatus when the sliding wall is out.

Does it take route h ? No, because h electrons have 50/50 b/w color statistics.

w

w

Does it take route s ? No, same reason.

Can it somehow have taken both routes ? No: if we look to see where the electron is inside the apparatus, we find that 50% of the time it is on route h, and 50% of the time it is on route s. We never find two electrons inside, or two halves of a single, split electron, or anything like that. There isn’t any sense in which the electron seems to be taking both routes.

Can it have taken neither route? No: if we put sliding walls in place to block both routes, nothing gets through at all.

But these are all the logical possibilities that we have any notion whatever of how to entertain !

What can these electrons be doing? Electrons would seem to have modes of being available to them which are quite unlike what we know how to think about.

The name of this new mode (which is just a name for something we don’t understand) is superposition.

What we say about an initially white electron which is now passing through our apparatus (with the wall out) is that it’s not on h and not on s and not on both and not on neither, but, rather, that it’s in a superposition of being on h and being on s. And what this means (other than “none of the above”) we don’t know.

We know, by experiment, that electrons emerge from the hard aperture of a hardness box if and only if they’re hard electrons when they enter that box.

When a white electron is fed into a hardness box, it emerges neither through the hard aperture nor through the soft one nor through both nor through neither. So, it follows that a white electron can’t be a hard one, or a soft one, or (somehow) both, or neither. To say that an electron is white must be just the same as to say that it’s in a superposition of being hard and soft.

Why can’t we say “the color of this electron is now such-and-such and the hardness of this electron is now such-and-such”? It isn’t at all a matter of our being unable to simultaneously know what the color and the hardness of a certain electron is (that is: it isn’t a matter of ignorance). It’s deeper than that.

Why is quantum mechanics a probabilistic theory?

If it could ever be said of a white electron that a measurement of its hardness will with certainty produce the outcome (say) “soft”, that would be inconsistent with what we know (that it’s in a superposition of hard and soft)

On the other hand, our experience is that every hardness measurement whatsoever either comes out “hard” or it comes out “soft”.

So apparently the outcome of a hardness measurement on a white electron has got to be a matter of probability !

Elitzer-Vaidman bomb testing problem

w w

We can tell whether or not the wall is in path s

Elitzer-Vaidman bomb testing problem

w w

Assume the wall is actually a very sensitive bomb, so sensitive that if an electron hits it, it will explode.

How can we detect the presence of the bomb without setting it off ??

Elitzer-Vaidman bomb testing problem

w

50% of the time (compared to the bomb not being there) no particle emerges along h and s. The bomb explodes.

25% of the time, we get a w electron out at h and s. We learn nothing (same result as bomb being absent)

25% of the time, we get a b electron out at h and s. We have detected the presence of the bomb without touching it !!!! (interaction-free measurement)

Use a w electron and measure color.

w

Find 100% w

Actually, this device does nothing, so of course the color doesn’t change. This device doesn’t measure hardness, because when the electron exits the device, we don’t know which path it followed! We erased the hardness information by recombining the paths.

In this context the bomb acts as a measures device – it tells us which path (hence which hardness value) the electron took.

e- e-

incoming electron

outgoing electron

hardness measurement

ready hard soft

Niels Bohr (Nobel Prize 1922):

Quantum mechanics does not describe how nature is but rather articulates what we can say about nature.

“We must only discuss the outcomes of measurements.”

Expressed in modern language, this means that quantum mechanics is a science of knowledge, of information.

Suppose we accept that collapse happens (since this is consistent with our experience that measurements yield definitive outcomes).

We could then ask, when does collapse occur?

Eugene Wigner (Nobel Prize 1963) proposed an answer in 1961:

Collapse occurs precisely at the level of consciousness, and perhaps, moreover, consciousness is always the agent that brings it about.

(this doesn’t really help)

When does collapse occur?

Suppose that Alice has a theory about collapse:

collapse happens immediately after the electron exits the measurement box.

And suppose that Bob has another theory about collapse:

collapse happens later, for example when a human retina or optic nerve or brain gets involved.

Can we decide who is right empirically, that is, by performing some experiment?

Here’s how to start: Feed a black electron into a hardness device and give it enough time to pass through. If Alice is right, the state of the system is now

either

Can we decide who is right empirically, that is, by performing some experiment?

or

(with 50% prob.)

(with 50% prob.)

€

hard mφhe

€

softmφse

whereas if Bob is right, the state of the system is currently

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12 hard mφh

e + 12 softmφs

e

so all we need to do is to figure out a way to distinguish, by means of a measurement, these two cases: In one case the pointer points in a particular (but as yet unknown) direction, and in the other case the pointer isn’t pointing in any particular direction at all.

What if we measure the position of the tip of the pointer? That is, lets measure where the pointer is pointing. This won’t work.

If Alice is right, of course we will find a 50/50 chance of finding the pointer “pointing-at-hard” vs. “pointing-at-soft”. This is because, according to Alice, the pointer is already in one of those two states.

But if Bob is right, then a measurement of the position of the tip of the pointer will have a 50% change of collapsing the wavefunction of the pointer onto the “pointing-at-hard” state, and 50% change of collapsing it to “pointing-at-soft”.

Therefore the probability of any given outcome of a measurement of the position of the pointer will be the same for both these theories; and so this isn’t the sort of measurement we are looking for.

What if we measure the hardness of the electron? This won’t work.

What if we measure the color of the electron? This won’t work.

These arguments establish that different conjectures about precisely where and precisely when collapse occurs cannot be empirically distinguished from one another.

And so the best we can do at present is to try to think of precisely where and precisely when collapses might possibly occur (that is, without contradicting what we do know to be true by experiment). But it turns out to be hard to do even that.

Schroedinger’s CatWigner’s Friend

Quantum Cryptography: how to communicate without eavesdropping

(from S.J. Lomonaco, Jr., http://www/csee.ubmc.edu/~lomonaco)

Key idea: How to determine if Eve is listening to Alice and Bob’s conversation?

Quantum Cryptography: how to communicate without eavesdropping

(from S.J. Lomonaco, Jr., http://www/csee.ubmc.edu/~lomonaco)

Key idea: How to determine if Eve is listening to Alice and Bob’s conversation?

Alice

Bob

Eve

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ih∂ψ

∂t= ˆ H ψ

This is our hardness value (hard/soft)

This is our color value (black/white)

Quantum Teleportation

Classical information can be copied any number of times, but perfect copying of quantum information is impossible, a result known as the no-cloning theorem, which was proved in 1982 by Wootters and Zurek of Los Alamos National Laboratory.

(If we could clone a quantum state, we could use the clones to violate Heisenberg’s Uncertainty Principle.)

Quantum Teleportation

Quantum mechanics seems to make such a teleportation scheme impossible in principle. Heisenberg’s uncertainty principle rules that one cannot know both the precise position of an object and its momentum at the same time.

Thus, one cannot perform a perfect scan of the object to be teleported; the location or velocity of every atom and electron would be subject to errors.

(In Star Trek the “Heisenberg Compensator” somehow miraculously overcomes this difficulty.)

However, in 1993 physicists discovered a theoretical way to use quantum mechanics itself for teleportation. They used entanglement to circumvent the limitations imposed by Heisenberg’s uncertainty principle without violating it.

Entangled Pair

laser beam (photon)

crystal

Occasionally a single UV photon is converted into two photons of lower energy, one vertically polarized and one horizontally polarized.

If these two photons emerge along the cone intersection (green areas),neither of them has a definite polarization. Why? Indistinguishability (7th postulate, photons are bosons).

Feynman(1965 Nobel Prize)’s Rules for interference

If two or more indistinguishable processes can lead to thesame final event (particle could take either path), then add their complexamplitudes and square, to find the probability:

P = |A1+A2 |2 ≈ |eikL1 + eikL2 |2 ≈ 1 + cos k(L1-L2)

If multiple distinguishable processes occur, find the realprobability of each, and then add:

P = |A1|2 + |A2|2 ≈ |eikL1 |2 + |eikL2 |2 ≈ 1

If there is any way – even in principle – to tell which process occurred, then there can be no interference (if you knew which path the particle came from, you’d see a randomized color) !

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ψ(1,2) = 12 c 1⇔2 + 1

2 ⇔1c 2

Entangled Pair

Photon 1 does not have a definite polarization, but its polarization is complementary to that of photon 2.

Measurement of the polarization of photon 1 will collapse photon 2 into the opposite polarization (action at a distance).

Quantum TeleportationNo cloning allowed, so how can we teleport a photon?

Answer: we will use an entangled pair of photons (A and B) to teleport a photon X from Alice to Bob.

Quick summary: Alice begins with photon X but does not know what its properties are (because if she measured it, its properties might change!). She teleports it to Bob, which destroys her “copy”. Bob ends up with photon X.

Quantum Teleportation

Quantum Teleportation

Quantum Teleportation

Hardness s

h

h blackbox

h and s

slidingwall

track

s

general outline:

-David Z. Albert book chapter 1-collapse postulate-Schroedinger’s cat-bomb detection