Download Quantum Mechanics

The Relativistic Quantum World
A lecture series on Relativity Theory and Quantum Mechanics
Marcel Merk
University of Maastricht, Sept 24 – Oct 15, 2014
The Relativistic Quantum World
Standard
Model
Quantum
Mechanics
Relativity
Sept 24:
Lecture 1: The Principle of Relativity and the Speed of Light
Lecture 2: Time Dilation and Lorentz Contraction
Oct 1:
Lecture 3: The Lorentz Transformation
Lecture 4: The Early Quantum Theory
Oct 8:
Lecture 5: The Double Slit Experiment
Lecture 6: Quantum Reality
Oct 15:
Lecture 7: The Standard Model
Lecture 8: The Large Hadron Collider
Lecture notes, written for this course, are available: www.nikhef.nl/~i93/Teaching/
Literature used: see lecture notes.
Prerequisite for the course: High school level mathematics.
Lecture 6
Quantum Reality
“Your theory is crazy, but not crazy enough to be true.”
-Niels Bohr
“I don’t like it, and I’m sorry I ever had anything to do with it.”
-Erwin Schrödinger
The Double Slit Experiment
Case 5:
The Delayed Choice Experiment
Case 4: Watch the Electrons
Consider again the double slit experiment in which we watch the electrons.
D1
D2
Can we try to fool the electron?
Wheeler’s Suggestion (1978)
John Wheeler (1911 – 2008):
Famous for work on gravitation
(Black holes – quantum gravity)
Replace detectors D1 and D2 with telescopes T1 and T2
which are focused on slits 1 and 2
What happens if we afterwards would reconstruct
whether the electron went through slit 1 or slit 2?
TT 1
1#
TT 2
2#
Wheeler’s Delayed Choice Experiment
We can suddenly decide to make the back-screen transparent to electrons.
We decide whether or not to make it transparent after the electrons passed the slits!
T
T1# 1
T
T2# 2
What will we see?
An wave interference pattern or a bullet-like non-interference pattern?
Answer: “Bullets”. We still have killed the interference.
Wheeler’s idea
What if we make the distance from slits to screen very long?
The double slit experiment can also be done replacing electrons by photons.
In this case Wheeler uses “gravitational lensing” as a “double slit”.
Star
One screen or
two screens
T1
Galaxy
T2
Then, either: Project image of T1 and T2 on separate screens,
Or:
Combine the image of T1 and T2 on one screen
 QM: no interference!
 QM: interference!
Crucial point: it must be impossible to know which path the photon took to see interference!
Delayed Choice…?
The Experiment of Aspect (2007)
Alain Aspect and his team have done the experiment
in yet another way: using photons in the lab.
They used beam-splitters to create two alternative routes
For a photon to the same place. Path 1 = Path 2 = 48 m
Beam-splitter:
Photon has 50% chance to pass through
and 50% chance to reflect.
Like 2-slits: the quantum can do both!
.
Three Equivalent Experiments
T1
T2
The Experiment of Aspect (2007)
Situation 1: “Are you a particle?” (open BSoutput)
Answer: “Yes!”
(Photon never on 2 paths)
Make the choice to open or close BSoutput
well after the photon has passed BSinput!
Situation 2: “Are you a wave?” (closed BSoutput)
Answer: “Yes!”
(Photon on 2 paths)
The Experiment of Aspect (2007)
Situation 1: “Are you a particle?” (open BSoutput)
Answer: “Yes!”
(Photon never on 2 paths)
Make the choice to open or close BSoutput
well after the photon has passed BSinput!
Situation 2: “Are you a wave?” (closed BSoutput)
Answer: “Yes!”
(Photon on 2 paths)
The Experiment of Aspect (2007)
Situation 1: “Are you a particle?” (open BSoutput)
Answer: “Yes!”
(Photon never on 2 paths)
“Thus one decides the photon shall have come by one route or by
both routes after it has already done its travel”
- John A. Wheeler
Situation 2: “Are you a wave?” (closed BSoutput)
Answer: “Yes!”
(Photon on 2 paths)
Apparently the quantum wave function includes both possibilities.
The observation makes one of them a reality via the collapse of the
wave function.
Schrödinger’s Cat
The Copenhagen Interpretation
Niels Bohr and Albert Einstein debates at Solvay conf.
Niels Bohr:
• Uncertainty relation
• Complementary, collapse of
the wave function.
1927
Albert Einstein:
• “God does not play dice”
• Objective Reality
Photo: Paul Ehrenfest
(December 1925)
Particle-Wave duality: one of the great mysteries of quantum mechanics.
Complementarity: A quantum object is both a particle and a wave.
A measurement can illustrate either particle or wave nature but not both at the same
time, because the object is affected by the act of measurement.
Schrodinger’s Cat
Paradox (thought experiment) invented by Erwin Schrödinger in 1935
to demonstrate that the Copenhagen interpretation makes no sense.
Compare quantum choice with double slit situation.
In a radioactive source, a single random quantum event has 50% probability
to trigger a lever arm and break a flask containing deadly poison.
Is the cat both dead and alive before we open the box to observe?
Who is observer? When does the wave function collapse? Does it require consciousness?
Is it the cat? The Experimenter? The press reporter? Or you when you hear the news?
Schrodinger’s Cat
Paradox (though experiment) invented by Erwin Schrödinger in 1935
to demonstrate that the Copenhagen interpretation makes no sense.
In a radioactive source, a single random quantum event has 50% probability
to trigger a lever arm and break a flask containing deadly poison.
Is the cat both dead and alive before we open the box to observe?
Who is observer? When does the wave function collapse? Does it require consciousness?
Is it the cat? The Experimenter? The press reporter? Or you when you hear the news?
The EPR Paradox
The EPR Paradox (1935)
EPR = Albert Einstein,
Boris Podolsky,
Nathan Rosen
Bohr et al.: Quantum Mechanics:
The wave function can be precisely calculated, but a measurement of mutually
exclusive quantities is driven by pure chance.
Einstein et al.: Local Reality:
There must exist hidden variables (hidden to us) in which the outcome of the
measurement is encoded such that effectively it only looks as if it is driven by
chance.
Local Realism vs Quantum Entanglement:
EPR: What if the wave function is very large and a measurement at one end can
influence the other end via some “unreasonable spooky interaction”.
Propose a measurement to test quantum entanglement of particles
The EPR Paradox
Two particles produced with known total momentum Ptotal, and fly far away.
Alice can not measure at the same time position (x1) and momentum (p1) of particle 1.
Bob can not measure at the same time position (x2) and momentum (p2) of particle 2.
∆ x1 ∆ p1 ≥
~
2
∆ x2 ∆ p2 ≥
~
2
pt ot al = p1 + p2
But:
If Alice measures p1, then automatically p2 is known, since p1+p2= ptotal
If Alice measures x1, then p1 is unknown and therefore also p2 is unknown.
How can a decision of Alice to measure x1 or p1 affect the quantum state of Bob’s particle
(x2 or p2 ) at the same time over a long distance?
Communication with speed faster than the speed of light? Contradiction with causality?
Is there “local realism” or “spooky action at a distance”?
An EPR Experiment
Produce two particle with an opposite spin quantum state.
Heisenberg uncertainty: an electron cannot have well defined
spin along two different directions, eg. z and x
1: z-Spin=
+
–
2: z-Spin=
–
+
Quantum wave function: total spin = 0.
If Alice measures spin of her particle along the z-direction,
Then also Bob’s particle’s spin points (oppositely) along the z-direction!
An EPR Experiment
Produce two particle with an opposite spin quantum state.
Heisenberg uncertainty: an electron cannot have well defined
spin along two different directions, eg. z and x
1: x-Spin=
+
–
2: x-Spin=
–
+
Quantum wave function: total spin = 0.
If Alice measures spin of her particle along the x-direction,
Then also Bob’s particle’s spin points (oppositely) along the x-direction!
An EPR Experiment
Produce two particle with an opposite spin quantum state.
Heisenberg uncertainty: an electron cannot have well defined
spin along two different directions, eg. z and x
Alice
Bob
1: z-Spin=
+
–
2: z-Spin=
–
+
Alice
Bob
1: x-Spin=
+
–
2: x-Spin=
–
+
Quantum wave function: total spin = 0.
But how does Bob’s particle know that Alice measures x-spin or z-spin?
Either the particles are linked because of some hidden variable
(objective reality) or they are QM “entangled” and a
measurement “collapses” the wave function.
An EPR Experiment
Produce two particle with an opposite spin quantum state.
Either the particles are linked because of some hidden variable
(objective reality) or they are QM “entangled” and a
measurement “collapses” the wave function.
Alice
Bob
1: z-Spin=
+
–
2: z-Spin=
–
+
Alice
Bob
1: x-Spin=
+
–
2: x-Spin=
–
+
John Bell: ”inequality equation” (1964):
Measure many times (“n”) spin along 3 different axes “A-B-C”.
Then local reality (hidden variable) requires:
n(A+B+) ≤ n(B+C+) + n(A+C+)
Quantum Mechanics should violate this equation.
Alain Aspect 1982
(A slightly different version (CHSH) of Bells inequality with coincidences is tested)
(CHSH = John Clauser, Michael Horne, Abner Shimony, Richard Holt)
Observations agree with quantum mechanics and not with local reality!
Philosophical?
Hugh Everett (PhD Student of John Wheeler) formulated the
Many Worlds Interpretation of quantum mechanics in 1957
The wave function does not collapse, but at each quantum
decision both states continue to exist is a decoupled world.
Triggered science fiction stories
with “parallel universes”
A tree of quantum
worlds for each
quantum decision.
Still a topic of debate (and theoretical calculations). Several alternatives (“Many Minds”).
Personal opinion: Quantum coherence is related to complexity of the wave-function.
Application 1: Quantum Cryptography
Alice sends a secret message to Bob and prevents Eve to eavesdrop.
First idea by Stephen Wiesner (1970s)
Worked out by Bennet (IBM) and Brassard (1980s)
Quantum Key Distribution (QKD):
1. Public Channel (Internet, email):
send an encrypted message.
2. Quantum Channel (Laser + fiber optics)
send key to decode the public message
3. Eve cannot secretly eavesdrop. She destroys
quantum information and is detected.
Physicsworld.com Sept 2, 2013
“Quantum cryptography coming to mobile phones”
Application2: Quantum Computer
Idea: Yuri Manin and Richard Feynman:
Use superposition and entanglement of quantum states to make a super-fast computer.
Normal computer: bits are either 0 or 1
Quantum computer: qbits are super-positions of two states: 0 and 1
(Eg. spin up and spin down)
Compute with quantum logic.
With 2 bits it can do 4 calculations simultaneously.
With 3 bits 8 calculations, with n bits 2n !
Difficulty: prevent “decoherence”.
Recently, “D-wave systems”:
Claim the first commercial quantum
computer based on 128 qbits.
Not generally accepted by the community
that it really uses quantum mechanics.
Food for thought
Relativity theory:
The finite speed of light means that there is no sharp separation
between space and time. (Think of different observers)
Universal constant: c = 300 000 km/s
Quantum Mechanics:
The finite value of the quantum of action means that there is no
sharp separation between a system and an observer
Universal constant: ħ = 6.6262 × 10-34 Js
John Wheeler:
“Bohr’s principle of complementarity is the most revolutionary
scientific concept of the century.”
Next Week
Next week:
• Quantum Field Theory and Antimatter
• The Standard Model
• The Large Hadron Collider
• The Origin of Mass: Higgs
Paul Dirac
Francois Englert
Richard Feynman
Peter Higgs
The origin of mass and “Higgs”
The particle associated to the origin with mass is predicted by Peter Higgs
The corresponding quantum field is the Brout-Englert-Higgs field
Robert Brout (1928 – 2011)
Francois Englert (1932 )
Peter Higgs (1929)
The mechanism that describes the origin of mass is the
Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism.
Nobel Prize in Physics 2013
“for the theoretical discovery of a mechanism that contributes to our
understanding of the origin of mass of subatomic particles, and which was
recently confirmed through the discovery of the predicted fundamental
particle, by the Atlas and CMS experiments at CERN’s Large Hadron Collider.”
Perhaps time for…