The Many Worlds ofQuantum Mechanics

Sean Carrollhttp://preposterousuniverse.com/

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)

Examples

• Billiard balls, pendulums, inclined planes, spinning tops…

• Air, water, stone, metal – solids and fluids.

• Moons, planets, rockets.

• Electric and magnetic fields.

• Spacetime itself (general relativity).

A clockwork universe

If we knew the complete state of theuniverse at any one moment;and knew the laws of physics exactly;and had infinite computing power;we would know the state of theuniverse at every other moment.

State of the universe= position and velocityof every particle/field

“Laplace’s Demon”

Quantum Mechanics:early hints, dawn of the 20th century

Max Planck:blackbodyradiation

Albert Einstein:photons

Henri Becquerel,Marie & Pierre Curie:radioactivity

Niels Bohr:electrons in atoms can’t be located just anywhere;

they only have certain allowed orbits

1920’s: quantum mechanics becomesa full-blown theory

WernerHeisenberg:“matrixmechanics”

ErwinSchrödinger:“wavemechanics”

Paul Dirac:Heisenberg’s and Schrödinger’s formalisms are equivalent

Newton’s Laws arereplaced by theSchrödinger Equationfor the quantum state:

Quantum mechanics:the secret

What we observe is much lessthan what actually exists.

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.

Electrons in atoms

Cartoon: circlingin orbits

Reality: a staticwave function

Simple example: Miss Kitty,a two-state system (“qubit”)

When we look for Miss Kitty, we only ever find herunder the table, or on the sofa.

Her wave function might give us a 50% chance of findingher under the table, 50% of finding her on the sofa.

Classically: Miss Kitty is either under the table or onthe sofa, we just don’t know which one.

Quantum mechanics: she is actually in a superpositionof both possibilities, until we look.

Reason why: interference.

Interference

Imagine that we see Miss Kitty stop by either her foodbowl or her scratching post on the way to the table or sofa.

Either way, we find a 50/50 chance to see her on the table or the sofa at the end of her journey.

50%

50%

50%

50%

But sometimes we don’t watch. She goes by either herfood bowl or scratching post, we don’t know which.

In that case, it turns out we always find her endingup on the sofa, never under the table!

What’s going on?

0%

100%

position(or other

observable)

• For every possibleobservable outcome,the wave function hasa value.

* More precisely: wave functions are complex numbers, = a + ib, and the probability is given by ||2 = a2 + b2.

The wave function tells us the probability, but it’s not equal to the probability.

wav

e fu

nctio

n

• Probability of observing an outcome = (wave function)2.

• Wave functions can be positive or negative*. Differentcontributions to the wave function can therefore either reinforce, or cancel each other out.

1926

So if the wave function for Miss Kitty to be under thetable is 0.71, the probability of finding her thereis (0.71)2 = 0.50, or 50%.

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.

0.71

0.71

(0.71)2

= 0.5(0.71)2

= 0.5

½(.71+.71)2

= 1

½(.71-.71)2

= 0

(-0.71)2

= 0.5(0.71)2

= 0.5

Interference is at the heart of quantum mechanics.

The Measurement Problem

What actually happens when we observe propertiesof a quantum-mechanical system?

Why should “measurement” or “observation” play acrucial role in a physical theory at all?

Thus, “interpretations of quantum mechanics.”

The Copenhagen (textbook)interpretation of quantum mechanics

• The “quantum realm” is distinct from themacroscopic “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 ontopossible measurement outcomes.

• After collapse, the new quantum state isconcentrated on that outcome.

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

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. Inquantum mechanics, we describe them both at once.

Entanglement

There is one wave function for the combined cat+dogsystem. It has four possible “basis states”:

yard doghouse

sofa table

yard doghouse

sofa table

yard doghouse

sofa table

yard doghouse

sofa table

(sofa, yard) =

(sofa, doghouse) =

(table, doghouse) =

(table, yard) =

Consider a state:

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.

½(sofa, yard) + ½(sofa, doghouse)+ ½(table, yard) + ½(table, doghouse)

(cat, dog) =

(cat, dog) = 0.71(sofa, yard) + 0.71(table, doghouse)

Now instead, consider a state:

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, weknow Mr. Dog is in the yard with 100% probability – without even looking at him!

yard doghouse

sofa table

yard doghouse

sofa table

yard doghouse

sofa table

yard doghouse

sofa table

(sofa, yard) =

(sofa, doghouse) =

(table, doghouse) =

(table, yard) =

Entanglement establishes correlations betweendifferent possible measurement outcomes.

(cat, dog) = 0.71(sofa, yard) + 0.71(table, doghouse)

Consider again an entangled state of Ms. Kitty and Mr. Dog:

Decoherence

But imagine we know nothing about Mr. Dog (and won’t).How do we describe the state of Ms. Kitty by herself?

(cat) = 0.71(sofa) + 0.71(table)

You might guess:

But that turns out to be wrong.

With no entanglement, different contributions to Ms. Kitty’spath lead to the same final states (sofa or table):

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)

Since final states are the same, they can add or subtract(and thus interfere):

Entanglement can prevent interference

Now imagine Ms. Kitty’s path becomes entangled with the state of Mr. Dog.

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)

(yard)

(doghouse)

Upshot: when a quantum system becomes entangledwith the outside world, different possibilities decohereand can no longer interfere with each other.

It’s as if they have become part of separate worlds.

If a system becomes entangled with a messy, permanent,external environment, its different possibilities willnever interfere with each other, nor affect each otherin any way.

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.

Schrödinger’s Cat:Copenhagen version

Ms. Kitty is in a superposition of (awake) and (asleep),then observed.

Schrödinger’s Cat:Many-Worlds version

Now we have Ms. Kitty, an observer, and an environment.

Silly objections to Many-Worlds

1. That’s too many universes!

The number of possible quantum states remainsfixed. The wave function contains the same amountof 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 postulateor hidden variables. It’s tested every time we observeinterference. If you have an alternative with explicitcollapses or hidden variables, we can test that!

Reasonable questions for Many-Worlds

1. How do classical worlds emerge?

The “preferred basis problem.” Roughly, the answeris because interactions are local in space, allowingsome configurations to be robust and not others.

2. Why are probabilities given by the square of thewave function?

For that matter, why are there probabilities at all? The theory is completely deterministic.

“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