The document provides an overview of quantum mechanics, beginning with classical mechanics and the idea of a deterministic "clockwork universe." It then discusses early hints of quantum theory and how the field developed in the 1920s. Key aspects of quantum mechanics are introduced, such as the wave function, superposition, interference, entanglement, and decoherence. Measurement in quantum mechanics is discussed, as are different interpretations like Copenhagen and Many Worlds. The document uses examples like the behavior of a cat to help illustrate various quantum concepts.

1. The Many Worlds of
Quantum Mechanics
Sean Carroll
http://preposterousuniverse.com/

2. Before there was quantum mechanics:
classical (Newtonian) mechanics
Classical mechanics describes the world in terms of:
• Space
• Time
• Stuff
• Motion
• Laws of physics
(e.g. forces, F = ma)

3. Examples
• Billiard balls, pendulums,
inclined planes, spinning tops…
• Moons, planets, rockets.
• Air, water, stone, metal – solids and fluids.
• Electric and magnetic fields.
• Spacetime itself (general relativity).

4. A clockwork universe
If we knew the complete state of the
universe at any one moment;
and knew the laws of physics exactly;
and had infinite computing power;
we would know the state of the
universe at every other moment.
State of the universe
= position and velocity
of every particle/field
“Laplace’s Demon”

5. Quantum Mechanics:
early hints, dawn of the 20th century
Max Planck:
blackbody
radiation
Albert
Einstein:
photons
Henri Becquerel,
Marie & Pierre Curie:
radioactivity

6. Niels Bohr:
electrons in atoms can’t be located just anywhere;
they only have certain allowed orbits

7. 1920’s: quantum mechanics becomes
a full-blown theory
Werner
Heisenberg:
“matrix
mechanics”
Paul Dirac:
Heisenberg’s and
Schrödinger’s
formalisms are
equivalent
Erwin
Schrödinger:
“wave
mechanics”
Newton’s Laws are
replaced by the
Schrödinger Equation
for the quantum state:

8. Quantum mechanics:
the secret
What we observe is much less
than what actually exists.

9. What is the “state” of a system?
Classical mechanics: position and velocity.
Quantum mechanics: the wave function.
Position and velocity are what you can observe.
But until you measure them, they don’t exist.
Only the wave function does.
The wave function tells you the probability of
measuring different values of position or velocity.

10. Electrons in atoms
Cartoon: circling
in orbits
Reality: a static
wave function

11. Simple example: Miss Kitty,
a two-state system (“qubit”)
When we look for Miss Kitty, we only ever find her
under the table, or on the sofa.
Her wave function might give us a 50% chance of finding
her under the table, 50% of finding her on the sofa.

12. Classically: Miss Kitty is either under the table or on
the sofa, we just don’t know which one.
Quantum mechanics: she is actually in a superposition
of both possibilities, until we look.
Reason why: interference.

13. Interference
Imagine that we see Miss Kitty stop by either her food
bowl or her scratching post on the way to the table or sofa.
50%
50%
50%
50%
Either way, we find a 50/50 chance to see her on the
table or the sofa at the end of her journey.

14. But sometimes we don’t watch. She goes by either her
food bowl or scratching post, we don’t know which.
0%
100%
In that case, it turns out we always find her ending
up on the sofa, never under the table!
What’s going on?

15. • For every possible
observable outcome,
the wave function has
a value.
wave function
The wave function tells us the probability,
but it’s not equal to the probability.
position
(or other
observable)
• Wave functions can be positive or negative *. Different
contributions to the wave function can therefore
either reinforce, or cancel each other out.
• Probability of observing an outcome = (wave function) 2.
* More precisely: wave functions are complex numbers, ψ = a + ib, and the probability is given by |ψ|2 = a2 + b2.

16. 1926

17. So if the wave function for Miss Kitty to be under the
table is 0.71, the probability of finding her there
is (0.71)2 = 0.50, or 50%.
0.71
0.71
But, crucially, if the wave function had been -0.71,
we would have the same probability, since
(-0.71)2 = 0.50 also.
That’s what happened in our interference
experiment.

18. (0.71)2
= 0.5
(-0.71)2
= 0.5
(0.71)2
= 0.5
(0.71)2
= 0.5
½(.71-.71)2
=0
½(.71+.71)2
=1
Interference is at the heart of quantum mechanics.

19. The Measurement Problem
What actually happens when we observe properties
of a quantum-mechanical system?
Why should “measurement” or “observation” play a
crucial role in a physical theory at all?
Thus, “interpretations of quantum mechanics.”

20. The Copenhagen (textbook)
interpretation of quantum mechanics
• The “quantum realm” is distinct from the
macroscopic “classical realm” of observers.
• Observations occur when the two realms interact.
• Unobserved wave functions evolve smoothly,
deterministically, via the Schrödinger equation.
• Observed wave functions collapse instantly onto
possible measurement outcomes.
• After collapse, the new quantum state is
concentrated on that outcome.

21. In Copenhagen, our observation of Miss Kitty along her
path collapsed her wave function onto “scratching post”
or “food bowl,” eliminating future interference.

22. A more complicated world
Imagine we have both a cat and a dog:
We can observe Ms. Kitty in one of two possible states:
(on the sofa) or (under the table).
We can also observe Mr. Dog in one of two possible states:
(in the yard) or (in the doghouse).
Classically, we describe the systems separately. In
quantum mechanics, we describe them both at once.

23. Entanglement
There is one wave function for the combined cat+dog
system. It has four possible “basis states”:
sofa
(sofa,
yard) =
sofa
table
(sofa,
doghouse) =
yard doghouse
sofa
(table,
yard) =
yard doghouse
table
yard doghouse
table
sofa
(table,
doghouse) =
table
yard doghouse

24. Consider a state:
½(sofa, yard) + ½(sofa, doghouse)
(cat, dog) =
+ ½(table, yard) + ½(table, doghouse)
Each specific outcome has a probability
(1/2)2 = 1/4 = 25%
So Ms. Kitty has a total probability for (sofa) of 50%,
likewise for (table); similarly for Mr. Dog.
Knowing about Ms. Kitty tells us nothing about Mr. Dog,
and vice-versa.

25. Now instead, consider a state:
(cat, dog) = 0.71(sofa, yard) + 0.71(table, doghouse)
Each specific outcome has a probability
(0.71)2 = 0.50 = 50%
Again, Ms. Kitty has a total probability for (sofa) of
50%, likewise for (table); similarly for Mr. Dog.
But now, if we observe Ms. Kitty under the table, we
know Mr. Dog is in the yard with 100% probability –
without even looking at him!

26. sofa
(sofa,
yard) =
yard doghouse
sofa
(table,
yard) =
sofa
table
(sofa,
doghouse) =
yard doghouse
table
yard doghouse
table
sofa
(table,
doghouse) =
table
yard doghouse
Entanglement establishes correlations between
different possible measurement outcomes.

27. Decoherence
Consider again an entangled state of Ms. Kitty and Mr. Dog:
(cat, dog) = 0.71(sofa, yard) + 0.71(table, doghouse)
But imagine we know nothing about Mr. Dog (and won’t).
How do we describe the state of Ms. Kitty by herself?
You might guess:
(cat) = 0.71(sofa) + 0.71(table)
But that turns out to be wrong.

28. Entanglement can prevent interference
With no entanglement, different contributions to Ms. Kitty’s
path lead to the same final states (sofa or table):
Since final states are the same, they can add or subtract
(and thus interfere):
via post: (cat) = 0.5(sofa) + 0.5(table)
via bowl: (cat) = -0.5(sofa) + 0.5(table)
total: (cat) = 0(sofa) + 1.0(table)

29. Now imagine Ms. Kitty’s path becomes entangled with
the state of Mr. Dog.
(yard)
(doghouse)
Now the final states of the wave function are not the same
(Mr. Dog and Ms. Kitty are entangled), and interference fails:
via post: (cat, dog) = 0.5(sofa, yard) + 0.5(table, yard)
via bowl: (cat, dog) = -0.5(sofa, doghouse) + 0.5(table, doghouse)
total: (cat, dog) = 0.5(sofa, yard) + 0.5(table, yard)
- 0.5(sofa, doghouse) + 0.5(table, doghouse)

30. Upshot: when a quantum system becomes entangled
with the outside world, different possibilities decohere
and can no longer interfere with each other.
If a system becomes entangled with a messy, permanent,
external environment, its different possibilities will
never interfere with each other, nor affect each other
in any way.
It’s as if they have become part of separate worlds.

31. Many-Worlds Interpretation of
Quantum Mechanics
Hugh Everett, 1957
• There is no “classical realm.” Everything is quantum,
including you, the observer.
• Wave functions never “collapse.” Only smooth,
deterministic evolution.
• Apparent collapse due to entanglement/decoherence.
• Unobserved possibilities – other “worlds” – still exist.

32. Schrödinger’s Cat:
Copenhagen version
Ms. Kitty is in a superposition of (awake) and (asleep),
then observed.

33. Schrödinger’s Cat:
Many-Worlds version
Now we have Ms. Kitty, an observer, and an environment.

34. Silly objections to Many-Worlds
1. That’s too many universes!
The number of possible quantum states remains
fixed. The wave function contains the same amount
of information at any time. You’re really saying
“I think there are too many quantum states.”
2. This can’t be tested!
Many-worlds is just QM without a collapse postulate
or hidden variables. It’s tested every time we observe
interference. If you have an alternative with explicit
collapses or hidden variables, we can test that!

35. Reasonable questions for Many-Worlds
1. How do classical worlds emerge?
The “preferred basis problem.” Roughly, the answer
is because interactions are local in space, allowing
some configurations to be robust and not others.
2. Why are probabilities given by the square of the
wave function?
For that matter, why are there probabilities
at all? The theory is completely deterministic.

36. “Despite the unrivaled empirical success of quantum
theory, the very suggestion that it may be literally true
as a description of nature is still greeted with cynicism,
incomprehension, and even anger.”
- David Deutsch