1. Week X The Copenhagen Interpretation
Philosophy of Physics
Week X
The Copenhagen Interpretation

2. Making sense of superposition
The state of a system is represented by a vector ψ in a Hilbert space H.
Measurement outcomes are always eigenvalues, a1, a2,… corresponding to the eigenvectors, α1, α2,…
The probability of getting a particular outcome, say a2, depends on the amount of “overlap” of the state vector ψ with the eigenvector α2.
When ψ does not exactly correspond to any eigenvector, it’s in a superposition of eigenstates.
What’s going on?
Mathematically, superpositions are easy to understand – it’s just basic linear algebra.
What does it mean, physically?

3. Consider the “spin” of a photon.
Will it make it through a Polaroid filter oriented in the x direction?
If it does, then it has “spin up in the x-direction”; if not, then it has “spin down in the x direction”.

4. And will measuring tell us which?
Is the system really one way or the other (ie, really spin up or really spin down in the x-direction)?
And will measuring tell us which?
Or is it neither until it is measured? α1 ψ
(α1,ψ) α2
(α2,ψ)

5. Stern-Gerlach
For a nice simulation you can play with, see:

6. Famous Quotes
Those who are not shocked when they first come across quantum theory cannot possibly have understood it. -- Niels Bohr.
If you are not completely confused by quantum mechanics, you do not understand it. -- John Wheeler.
It is safe to say that nobody understands quantum mechanics.
-- Richard Feynman.
If [quantum theory] is correct, it signifies the end of physics as a science. -- Albert Einstein.
I do not like [quantum mechanics], and I am sorry I ever had anything to do with it. -- Erwin Schrödinger.
Quantum mechanics makes absolutely no sense. -- Roger Penrose.

7. Should We Just Give UP ?
The above quotes suggest that maybe the best thing is to give up. We should not try to make sense of QM.
However, nobody likes a quitter, a whiny complainer, an intellectual coward.
Better to go down in flames.
On the other hand, we will see why those famous scientists were so pessimistic.

8. Interpretations Minimal statistical interpretation Realism
Predict observation
Don’t try to do more than this
Deep understanding is impossible; don’t even try
Realism
Science gives us a true description of reality
Measurements discover an independent reality
Anti-realism (one version)
The world is not independent from us
Measurements (in some sense) create, rather than reveal/discover what is already the case.

9. Original Schrödinger interpretation
Problems:
Electrons are localized particles while ψ is spread out
Ψ is complex, many dimensional, thus, cannot be a wave in ordinary space
Ψ is discontinuous on measurement

10. Original Born Interpretation
According to Max Born ψ is a probability wave. It describes the probability density of an ensemble of electrons; it is not a physical wave.
|ψ|2dx is the probability of being in the region dx.
Electrons have positions and momenta at all times.

11. Problems with (original) Born
Single electron (or photon) shows interference effects. (They can be slowed to one a day, not hitting each other.)
How can a non-physical probability wave interfere with itself?

12. The Schrödinger interpretation is ontological in that ψ describes an independent world.
The Born interpretation is epistemic in that ψ describes the state of our knowledge (or ignorance) of the world.
Neither interpretation works. (But the Schrödinger equation and the Born rule are OK.)
The Copenhagen interpretation is a blend of the other two interpretations. Ψ is about the world and about our knowledge of the world.

13. Niels Bohr (1885–1962) Danish, born in Copenhagen
Son of prominent Prof of physiology
Student of Høffding who introduced him to Kant, Kierkegaard, James
1913: Bohr atom
1922: Nobel Prize
Denmark built him the Institute for Theoretical Physics; a Mecca for physics in the 1920s and 30s.
A kind of father figure for Heisenberg, and others who spent much time in Copenhagen.
30 year fight with Einstein over QM
His view is known as the “Copenhagen Interpretation”.

14. Bohr’s Basic Idea
Classical physics has a kind of priority, even though it isn’t true, at least not true in the micro-realm.
But we can only understand things with the help of common sense concepts, even though many common sense beliefs are false.
The ideas of classical physics involve a kind of common sense and we are forced to use those concepts when trying to understand the micro-world.

15. Common Sense
Some common sense beliefs might be right, some might be wrong.
Some of them might be innate, some might be learned.
Things we have learned to believe (some right, some wrong):
If the earth were round, then Australians would fall off.
Sun doesn’t move through the sky; we turn.
Things don’t come naturally to rest; conservation of momentum/energy.
Innate/instinctive beliefs (some right, some wrong):
Objects have colours and smells.
The primary/secondary property distinction (accepted by philosophy, physics, and psychology), says objects have properties such as mass, charge, velocity, etc., but colours, tastes, smells, etc. are produced by the mind.
Innate (but possibly unavoidable):
The world consists of objects that have properties.

16. Common Sense (Continued)
Here are some assumptions we all make, at least in ordinary circumstances.
The world consists of definite objects.
A bird is distinct from the air it flies in
A tree is distinct from the ground
A photon is distinct from a polaroid filter
These objects (tree, ground, photon, filter) are distinct and our concepts of these things are also distinct.

17. Contrasts Y N Y ? Y ? Common sense Classical physics Bohr
The world consists of physically distinct elements Y N
The world consists of conceptually distinct elements
Observable macro-world and hidden micro-world are ontologically the same
Y ?
The world (both micro- and macro-) exists independently of observers and measuring instruments. Measurements discover what objectively exists
Determinism. The state at time t determines the state at all later times
Y ?

18. Some Main Bohr principles
These points are all disputable. Bohr was not a clear writer; most find him very obscure.
1. All experience is to be accounted for by means of classical concepts.
By classical, Bohr meant the concepts of classical physics, which he took to be refinements of common sense concepts.
We cannot avoid these concepts and must use them when we describe our experience of the micro-realm.

19. For example: location in space, location in time, momentum, mass, etc.
We need these concepts in order to have meaningful experience.
We also need them for communication.
Without concepts we could experience nothing. (Everything would be a “buzzing, blooming confusion.” – William James)
Parallels with Kant?
We contribute causation when we experience A causing B
We contribute space when we see objects in space.

20. 2. The application of any concept depends on the presence of suitable physical conditions. Micro-objects have properties only in relation to the macro-world.
For instance, a photon has a spin component in the x-direction (ie, spin up or spin down), iff there is a macro-device (eg, a polaroid filter) to which it is appropriately related.
Without such a setup, there would be no spin at all.
In such a setup (ie, a measurement in the x-direction), there are no spin components in the y-direction (neither up nor down).

21. In the beam of light, a photon has a position in the x-direction (ie, at the hole in the screen), but it has no momentum in the x direction. (NB. Not zero momentum, but no momentum.)

22. 3. There is a finite quantum of action associated with the measuring process in the micro-world. This quantum of action links the micro-system and the macro-measuring device into an indivisible and uncontrollable unity.
It is here, says Bohr, that causation breaks down and determinism fails.

23. 4. The concepts of classical physics (eg, position, momentum) are not all simultaneously applicable. Which of a pair of complementary concepts is applicable depends on the whole physical arrangement.
If you are measuring spin in the x-direction, then you cannot measure it in the y-direction. If you are measuring position in the x-direction, then you cannot measure momentum in the x-direction.

24. Werner Heisenberg (1901-1976) Matrix mechanics 1925
Uncertainty principle 1927
Nobel prize 1932
Many contributions after that
Lead German program to build atomic bomb. This remains quite controversial today.
Frayn, Copenhagen is an excellent play (and film) about the Bohr-Heisenberg relation during the war.

25. Heisenberg on the uncertainty principle

27. How should we understand this?
Realist: the uncertainty is mere ignorance; position and momentum both exist, but we can only know one. After all, how can we “disturb” the momentum unless there is a momentum to disturb. Heisenberg’s thought experiment presupposes realism.
Bohr type anti-realist: the uncertainty should be understood ontologically; if there is a position, then no momentum exists.
The terms “uncertainty principle” and “indeterminacy principle” are, respectively, often used to mark this distinction. (But these terms are not universal.)

28. Heisenberg’s account
“If one wants to clarify what is meant by ‘position of an object,’ for example, of an electron, he has to describe an experiment by which the ‘position of an electron’ can be measured; otherwise this term has no meaning at all.” --Heisenberg
He seems to claim that ignorance in principle of X implies the non-existence of X. This is a form of verificationism or operationalism.

29. Heisenberg introduced the term “observables” for the properties of quantum entities, eg, position, momentum, spin, etc. are observables.
Why (according to someone like Heisenberg) is “path” or “trajectory” not an observable (and hence, not a legitimate concept)?
First, ask yourself, What is a path? In other words, what do you need to know in order to describe the path of, say, an electron?

30. Heisenberg on determinism
According to most people, QM is not deterministic. Why?
Heisenberg characterizes determinism or the principle of causality as follows: “If we have exact knowledge of the present, then the future can be calculated.”
H says this is false, since we cannot have exact knowledge of the present.
Is this a good argument? No.
But H might perhaps say “knowledge of the present” is meaningless, so the causal principle (ie, determinism) is not applicable.

31. Later Heisenberg
When a quantum system is in a state of superposition of eigenstates of, say, momentum, then it has no momentum, but it has a potential for various different momenta. α1 ψ
(α1,ψ) α2
(α2,ψ)

32. Reality consists of potentials that become actualized in measurements.
potential actual
The idea is from Aristotle (acorns are potential oak trees).

33. How is potentiality turned into actuality?
This the same as asking How does the wave function collapse?
And this is equivalent to asking How does reality become determinate?
Heisenberg hasn’t solved any problem at all, just put it in a new form.

34. Schrödinger’s cat This problem arises from two key assumptions.
QM applies to both micro- and macro-objects
The de Broglie relation λ = h/p applies to electrons and to trucks.
Exercise: check value for h, and value for p for an electron at 10 m/s and a truck at the same speed, then calculate λ for each.
Measurement is a physical process; hence it should be described by QM

35. Measurement
Let’s suppose a photon has the property spin in the x direction and there are two possible outcomes: up and down. It’s called a two-state system.
A device to measure this might be a polaroid filter with a photosensitive detector that flashes green or red depending on whether the photon passed the filter or not.
Thus, the measuring device is also a two-state system.

36. We can represent the photon as in a state ψp= u + d.
The measuring device also has two eigenstates, g and r for green flash and red flash.
Features of QM imply that the correct representation of the photon and the measuring device is the so-called tensor product: ψp+m = ψp ⊗ ψm. This, too, is a vector in a Hilbert space, constructed out of the photon’s and the measuring device’s separate Hilbert spaces.
In general, ψp+m = (u ⊗ g) + (d ⊗ r).
This means, the measuring device also goes into a state of superposition.

37. The consequence is bizarre.
A cat is in a box with a radio-active atom; if it decays, the cat will be poisoned. All are hidden from view.
Is the cat alive or dead before looking?
Or is the cat neither (ie, it’s in a state of superposition?
It seems to be the latter.

38. In other words, the cat seems to be in a state of superposition of being alive and dead:
Ψatom+cat = (energeticatom ⊗ alivecat) + (decayedatom ⊗ deadcat)
energetic ⊗ alive
Ψatom+cat
decayed ⊗ dead

39. Responses
Schrödinger: this is absurd; QM is wrong (or at least the Copenhagen interpretation is wrong).
Bohr: there is a difference between the micro- and the macro-worlds; QM applies only to the micro-.
Decoherence: The cat does go into a state of superposition, but only for a very, very short time, so don’t worry.
Wigner: Conscious minds are crucial.

40. Eugene Wigner (1903-1995) Promoter of “symmetry” in physics
Nobel prize 1963
Developed the view of von Neumann and others that consciousness is what collapses the wave function.

41. Two questions
Do macroscopic objects such as cats really go into superpositions?
Yes, QM applies to big and small (there is only one physical world)
When we look we only see eigenvalues (of eigenstates), never superpositions. How is this possible? What collapses the wave function?

43. Wigner: We need something to which QM does NOT apply when measuring.
None of these measuring devices can collapse the wave function, ie, put the system into one of the eigenstates.
But it happens. How?
Wigner: We need something to which QM does NOT apply when measuring.
It would have to be non-physical, since QM applies to everything physical
Are there any obvious candidates? Yes, the mind.
Consciousness makes measurements, collapses the wave function, and makes reality determinate.

44. Ridiculous? Entails mind-body dualism.
Is dualism plausible?
Be tolerant: QM is very weird, so give serious consideration to views that seem very farfetched.
Note that the rivals are just as farfetched. (Wait until you hear about nonlocality, many worlds, etc.)

45. Delayed choice
John Wheeler (an outstanding physicist and friend of Wigner) proposed an even weirder possibility: changing the past.

46. Wheeler: We have a choice between “wave” detecting and
“particle” detecting devices
If we choose the interference detector (wave), the state will be
ψtop + ψbottom, which is a superposition.
If we choose the particle detector, the state will be one of either
ψtop or ψbottom, neither of which is a superposition.
The experimental arrangement determines the state.
BUT, the photon passed the galaxy long ago; the choice of which
experimental setup to have is delayed until after the photon is on
its way.

47. “It is wrong to think of the past as ‘already existing’ in all detail
“It is wrong to think of the past as ‘already existing’ in all detail. The ‘past’ is theory. The past has no existence except as it is recorded in the present. By deciding what questions our quantum registering equipment shall put in the present we have an undeniable choice in what we have a right to say about the past.”
– Wheeler
Thus, our present choices (in some important sense) create past reality. Our choice today determines whether the photon went one way or went both ways billions of years ago.

48. Further Reading
A very nice book is: Murdoch, Niels Bohr’s Philosophy of Physics
A good online survey:
A brief biography of Bohr can be found at:
A longer biography is: Pais, Niels Bohr’s Times in Physics, Philosophy, and Polity
There is a short film clip of some of the main people at the famous Solvay Conference in 1927: (Some of the names are mispronounced.)
Michael Frayn, Copenhagen is a wonderful play (and film) about a meeting between Heisenberg and Bohr in Nazi-occupied Denmark.

49. Further Reading (more)
Heisenberg, Physics and Philosophy
Wigner, Symmetries and Reflections (chapters on quantum measurement)
Online survey:
The measurement problem:
Schrödinger’s Cat:
Consciousness:
Wikipedia articles on Heisenberg:
and on Wigner:
and
and on Wheeler: