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35C3 - Quantum Mechanics

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    Herald Angel: And, so he studied physics
    and I'm thinking we just all need a lot
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    better understanding of quantum mechanics,
    because he sees this theory being misused
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    a lot by some weird esoteric theories,
    kind of abusing it to just justify
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    everything and anything. So he wants to
    change that and he wants to have people
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    with some understanding of this very
    important theory and so he will start
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    today with all of us here and try to
    explain to us the wonders of quantum
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    mechanics. Have a go.
    [Filler, please remove in amara]
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    Sebastian Riese: Well thank you for a warm
    welcome. It will be about quantum
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    mechanics. We will see whether the gentle
    introduction will be a lie depending on
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    how good you can follow me. So at first
    there will be a short introduction, a bit
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    meta discussion about physical theories
    and what is the aim of this talk. And then
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    we will discuss the experiments. Most of
    this is high school physics, you've
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    probably seen it before. And then it will
    get ugly because we'll do the theory and
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    we'll really do the theory, we'll write
    down the equations of quantum mechanics
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    and try to make them plausible and
    hopefully understandable to a lot of
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    people. And finally some applications will
    be discussed. So what is the concept of
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    this talk. The key experiments will be
    reviewed as said, and but we will not do
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    it in historical fashion. We will look at
    the experiments as physical facts and
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    derive the theory from them. And since
    quantum mechanics is rather abstract and
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    not, as I said in German and in science
    theory "anschaulich", we will need
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    mathematics and most of this will be
    linear algebra. So a lot of quantum
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    mechanics is just linear algebra on
    steroids, that means in infinite
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    dimensions. And in doing so we'll try to
    find a certain post classical
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    "Anschaulichkeit" or lividness to
    understand the theory. Since there'll be a
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    lot of math as the allergy advice said,
    there will be crash courses driven in to
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    explain mathematical facts. Sorry for the
    mathematicians that are here they probably
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    suffer because I lie a lot. So at first:
    How do scientific theories work? To really
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    understand quantum mechanics we must
    understand the setting and setting where
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    it was created and how scientific theories
    are created in general. A scientific
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    theory is a net of interdependent
    propositions so we have one proposition
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    for example "F = M times a" in classical
    mechanics and we have another proposition
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    that the gravitational force equals is
    proportional to the product of the masses
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    divided by the distance between the masses
    squared, so something like this. And when
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    we go around, make experiments, look into
    nature, develop theories, calculate, we
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    test those we test hypotheses, different
    hypotheses and try to determine which one
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    describes our experimental results best.
    And if the hypothesis stands the
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    experimental tests they're added to the
    theory. But what happens if there's an
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    experimental result that totally
    contradicts what we've seen before? And
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    that happened in the late 19th and early
    20th century. There are new results that
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    could not be explained. So if such
    inconsistent results are found then our
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    old theory has been falsified. This term
    is due to Popper who said that a theory is
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    scientific as long as it can be falsified,
    that is at least as long as we can prove
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    that it's not true and we can never prove
    a theory true but only prove it wrong. And
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    all that we have not yet proven wrong are
    at least some approximation to truth. And
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    if this happens we have to amend our old
    theory and we have to use care there and
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    find a minimal amendment. This principle
    is Occam's Razor. One could also say the
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    principle of least surprise from software
    engineering. And then we try that our
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    theory is again consistent with the
    experimental results. And of course the
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    new theory must explain why the hell that,
    for example Newtonian mechanics work for
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    two hundred years if it's absolutely
    wrong. And so the old theory must in some
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    limit contain the new one. And now how
    does it begin with quantum mechanics. As
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    already said the time frame is the late
    19th and early 20th century. And there
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    were three or four fundamental theories of
    physics known then: Classical mechanics,
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    which is just governed by the single
    equation the force equals mass times the
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    acceleration with given forces. And two
    known force laws: The immediate distance
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    action Newtonian gravitation and the
    Maxwell electro dynamics, this funny
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    equation here. This funny equation here is
    a way of writing down the Maxwell
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    equations that basically contain all the
    known electromagnetic effects. And finally
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    there were the beginnings of the Maxwell
    Boltzmann statistical physics, but
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    classical statistical physics is a pain,
    doesn't really work. So several
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    experimental results I said could not be
    explained by classical theories. For
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    example the photoelectric effect
    discovered by Hertz and Hallwachs in 1887,
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    or the discrete spectral lines of atoms
    first shown by Fraunhofer in the spectrum
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    of the sun and then studied by Bunsen and
    Kirchhoff with the so-called
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    "Bunsenbrenner", you all know it from the
    chemistry classes. And further,
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    radioactive rays were really a mystery
    nobody understood: How can it happen that
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    something just decays at random intervals?
    It was unclear. And then the people looked
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    into the atom, Rutherford using alpha
    particles to bombard a gold foil and saw
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    there must be positively charged nucleii
    and they already knew that they were
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    negatively charged, what we now call
    electrons, particles in the atom. So this
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    was really strange that atoms are stable
    at composed like this and I will explain
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    why a bit later. But now to more detail to
    the experiments. The really big
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    breakthrough in this time, experimentally
    speaking, were vacuum tubes, so you took a
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    piece of glass and pumped the air out and
    closed it off and put all sorts of devices
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    in there. And now one thing is this nice
    cathode ray experiment. We have here a so-
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    called electron gun and this is a heated
    electrode, so here flows the current that
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    heats it, so that the electrons get energy
    and seep out into the vacuum. Then we have
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    an electrode that goes around and a plate
    in front that is positively charged. So we
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    accelerate our electrons towards the
    plate. There's a pinhole in the plate and
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    we get a beam of electrons. And now we had
    those evacuated tubes and those electron
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    guns. So we put the electron gun in the
    evacuated tube, perhaps left a bit of gas
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    in because then it glowed when it when the
    atoms in the gas were hit by the electrons
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    so we could see the cathode ray, and then
    we play around. We take magnetic fields
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    and see how does it react to magnetic
    fields. We take electric fields. How does
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    it react to electric fields and so on. And
    what we find out is we somehow must have
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    negatively charged particles that flow
    nicely around in our almost vacuum. And
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    because atoms are neutral which is just
    known macroscopically there must be a
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    positively charged component in the atom
    as well. And this positively charged
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    component was first thought to be kind of
    a plum pudding or so with the electrons
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    sitting in there. But the Rutherford-
    Marsden-Geiger experiment, so it was
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    Rutherford invented the idea and Marsden
    and Geiger actually performed the
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    experimental work, showed that if you had
    a really thin gold foil, really only a few
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    hundred layers of atoms, thats the nice
    thing about gold, you can just hammer it
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    out to really, really thin sheets, if you
    had that and then shot alpha particles
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    that is helium nuclei that are created by
    the radioactive decay of many heavy
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    elements for example, most uranium
    isotopes decay by alpha decay, then they
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    were deflected strongly. If the charge
    would have been spaced throughout the
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    atoms then this could not have happened.
    You can calculate, you can you can
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    estimate the possible deflections with an
    extended charge and with a concentrated
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    charge, and you see the only explanation
    for this is that there is a massive and
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    really, really small positive thing in
    those atoms. So atoms are small,
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    positively charged nucleus as Rutherford
    called it and around it there's a cloud of
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    electrons or, he thought, orbiting
    electrons. But orbiting electrons atoms
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    are stable, this doesn't really make sense
    in classical physics, because in classical
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    physics all accelerator charges must
    radiate energy and be slowed by this
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    process. And this means atoms that are
    stable and composed of some strange
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    electrons and having nuclei they're just
    not possible. It's a no go, so at least at
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    this moment it was completely clear
    classical physics as they knew it up until
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    then is wrong. And the next experiment in
    this direction was the photoelectric
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    effect. What's shown there is a schematic
    of a phototube. And a phototube is again a
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    vacuum tube out of glass and there is a
    for example cesium layer in in the tube at
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    one side and there is a ring electrode
    removed from it. And if we shine light on
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    this there flows a current. But the
    peculiar thing is that if we do the bias
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    voltage across the two terminals of this
    tube to stop the electrons, we see that
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    the bias voltage that completely stops the
    flow is not proportional to the intensity
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    of the light that is incident onto the
    tube, but it's proportional to the
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    frequency of the light that's incident on
    the phototube. And that was again really
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    weird for the people of the time because
    the frequency shouldn't make any
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    difference for the energy. And this was
    when Einstein derived that, or thought of
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    that there must be some kind of energy
    portions in the electric field, from this
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    simple experiment, which is often done in
    physics classes even at the high school
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    level. So it's, from today's view it's not
    a complicated experiment. And to go even
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    further those weird stable atoms had
    discrete, had discrete lines of emission
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    and absorption of light. And here we have
    again a very simplified experimental set
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    up of a so-called discharge tube, where we
    have high voltage between the terminals
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    and a thin gas and then a current will
    flow, will excite the atoms. The atoms
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    will relax and emit light and this light
    will have a specific spectrum with sharp
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    frequencies that are, that have strong
    emission and we can see this with a
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    diffraction grating that sorts light out
    according to its wavelength and then look
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    on the screen or view some more fancy
    optical instrument to do precision
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    measurements as Bunsen and Kirchhoff did.
    So what we knew up until now was that
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    something was really weird and our
    physical theories didn't make sense. And
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    then it got worse. Someone took an
    electron gun and pointed it at a
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    monocrystalline surface. And such a
    monocrystalline surface is just like a
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    diffraction grating: A periodically
    arranged thing. And off periodically
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    arranged things there does happen regular
    interference pattern creation. So they saw
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    interference pattern with electrons. But
    electrons aren't that particles? How can
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    particles, so what was thought of then,
    since the times of Newton as a little hard
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    ball, how can a little hard ball flowing
    around create interference patterns? It
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    was really weird. And there's even more
    and as already mentioned radioactivity
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    with the random decay of a nucleus. This
    doesn't make sense in classical physics,
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    so it was really, really bad. And here
    I've added some modern facts that we'll
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    need later on. Namely that if we measure,
    if we try to measure the position of a
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    particle and use different position
    sensors to do so, only one of them, so at
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    only at one position will the single
    particle register, but it will
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    nevertheless show an interference pattern
    if I do this experiment with many many
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    electrons. So there must somehow be a
    strange divide between the free space
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    propagation of particles and measuring the
    particles. And you can do really weird
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    stuff and record the information through
    which slit the particle went. And if you
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    do this, the interference pattern
    vanishes. And then you can even destroy
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    this information in a coherent manner and
    the interference pattern appears again. So
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    what we know up until now is that quantum
    mechanics is really, really weird and
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    really different from classical mechanics.
    And now that we've talked about those
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    experiments, we'll begin with the theory,
    and the theory will begin with a lot of
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    mathematics. The first one is simple.
    Complex numbers. Who doesn't know complex
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    numbers? Okay. Sorry I'll have to ignore
    you for the sake of getting to the next
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    points. *laughter* So I'll just say
    complex numbers are two components of, two
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    componented objects with real numbers. And
    one of them is multiplied by a imaginary
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    number i. And if we square the number i it
    gets -1. And this makes many things really
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    beautiful. For example all algebraic
    equations have exactly the number of
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    degrees solutions in complex numbers, and
    if you count them correctly. And if you
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    work with complex functions it's really
    beautiful. A function that once
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    differential is infinitely many times
    differential and it's, it's nice. So now
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    we had complex numbers. You've all said
    you know them. *laughter* So we go onto
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    vector spaces, which probably also a lot
    of you know. Just to revisit it, a vector
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    space is a space of objects called
    vectors, above some scalars that must be a
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    field. And here we only use complex
    numbers as the underlying fields. There is
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    a null vector, we can add vectors, we can
    invert vectors and we can multiply vectors
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    by real numbers. So we can say three that
    five times this vector and just scale the
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    arrow and these operations interact nicely
    so that we have those distributive laws.
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    And now it gets interesting. Even more
    maths: L2 spaces. L2 spaces are in a way
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    an infinite dimensional or one form of an
    infinite dimensional extension of vector
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    spaces. Instead of having just three
    directions x, y, z, we have directions at
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    each point of a function. So we have an
    analogy here. We have vectors which have
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    three discrete components given by x index
    i on the right side and we have this
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    function and each component is the value
    of the function at one point along the
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    axis x. And then we can just as for
    vectors define a norm on those L2
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    functions which is just the integral over
    the absolute value squared of this
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    function f and the nice thing about this
    choice of norm. There are other choices of
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    the norm. This norm is induced by a scalar
    product and this little asterisk that is
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    there at the f denotes the complex
    conjugate, so flipping i to minus i and
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    all complex values. And if you just plug
    in f and f into the scalar product you
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    will see that it's the integral over the
    squared absolute value. And this space,
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    this L2 space is a Hilbert space and the
    Hilbert Space is a complete vector space
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    with a scalar product where complete means
    that uh... It's mathematical nonsense.
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    Forget it. So but the nice surprise is
    that most things carry over from finite
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    dimensional space. What we know from
    finite dimensional space is we can always
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    diagonalize matrices with certain
    properties and this more or less works.
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    And the mathematicians really, really,
    really do a lot of work for this but for
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    physicists we just know when to be careful
    and how and don't care about it otherwise.
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    So just works for us and that's nice. And
    now that we have those complex numbers we
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    can begin to discuss how particles are
    modeled in quantum mechanics. And as we
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    know from the Davisson-Germer experiments
    there's diffraction of electrons but
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    there's nothing in electrons that
    corresponds to an electric field in some
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    direction or so. Some other periodicity
    has, so periodicity of electrons during
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    propagation has never been directly
    observed. And De Broglie said particles
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    have a wavelength that's related to their
    momentum. And he was motivated primarily
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    by the Bohr theory of the atom to do so.
    And he was shown right by the Davisson-
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    Germer experiments so his relation for the
    wavelength of a particle is older than the
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    experiments showing this, which is
    impressive I think. And now the idea is
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    they have a complex wave function and let
    the squared absolute value of the wave
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    function describe the probability density
    of a particle. So we make particles
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    extended but probability measured objects
    so there isn't no longer the position of
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    the particle as long as we don't measure.
    But we have just some description of a
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    probability where the particle is. And by
    making it complex we have a phase and this
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    phase can allow, still allow, interference
    effects which we need for explaining the
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    interference peaks in the Davisson-Germer
    experiment. And now a lot of textbooks say
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    here there's a wave particle dualism, blah
    blah blah. Uh, distinct nonsense, blah.
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    The point is it doesn't get you far to
    think about quantum objects as either wave
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    or particle, they're just quantum. Neither
    wave nor particle. Doesn't help you either
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    but it doesn't confuse you as much as when
    you tried to think about particles as
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    waves or particles, or about quantum
    particles as waves or particles. And now
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    that we say we have a complex wave
    function what about simply using a plain
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    wave with constant probability as the
    states of definite momentum because we
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    somehow have to describe a particle to say
    that has a certain momentum and we do
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    this. Those have the little problem that
    they are not in the Hilbert space because
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    they're not normalizable. The absolute
    value of Psi is 1 over 2 pi everywhere, so
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    that's bad. But we can write the
    superposition of any state by Fourier
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    transformation those e to the i k dot r
    states are just the basis states of a
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    Fourier transformation. We can write any
    function in terms of this basis. And we
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    can conclude that by Fourier
    transformation of the state Psi of r to
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    some state till the Psi of k we describe
    the same information because we know we
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    can invert the Fourier transformation and
    also this implies the uncertainty
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    relation. And because this is simply
    property of Fourier transformations that
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    either the function can be very
    concentrated in position space or in
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    momentum space. And now that we see that
    we have states of definite momentum. And
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    the other big ingredient in quantum
    mechanics are operators next to the state
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    description. And operators are, just like
    matrices, linear operators on the state
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    space. Just as we can apply a linear
    operator in form of a matrix to a vector,
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    we can apply linear operators to L2
    functions. And when we measure an
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    observable it will be that it's one of the
    eigenvalues of this operator that's the
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    measurement value. you know [?wrap?]. So
    eigenvalues are those values: If a matrix
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    that just scales a vector by a certain
    amount that is an eigenvalue of the matrix
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    and in the same sense we can define
    eigenvalues and eigenvectors for, to
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    functions. And there are some facts such
    as that non-commuting operators have
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    eigenstates that are not common. So we
    can't have a description of the basis of
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    the state space in terms of function that
    are both eigenfunctions of both operators
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    and some examples of operators are the
    momentum operator which is just minus i
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    h-bar Nabla which is the derivation
    operator in three dimensions. So in the x
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    component we have derivation in the
    direction of x and in the y component in
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    direction of y and so on. And the position
    operator which is just the operator that
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    multiplies by the position x in the
    position space representation of the way
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    function. And as for the non-
    communtitivity of operators we can already
  • 24:48 - 24:55
    show that those p and x operators that do
    not commute but fulfill a certain
  • 24:55 - 25:01
    commutation relation. And a commutation
    relation is just a measure for how much
  • 25:01 - 25:07
    two operators do not commute. And the
    commutator is AB minus BA for the objects
  • 25:07 - 25:15
    AB, so if they commute, if AB equals BA
    the commutator simply vanishes. And
  • 25:15 - 25:21
    there's more on operators just to make it
    clear: Linear just means that we can split
  • 25:21 - 25:27
    the argument if it is just some linear
    combinations of vectors and apply the
  • 25:27 - 25:32
    operator to the individual vectors
    occuring, we can define multiplication of
  • 25:32 - 25:38
    operators and this just exactly follows
    the template that is laid down by finite
  • 25:38 - 25:44
    dimensional linear algebra. There's
    nothing new here. And there are inverse
  • 25:44 - 25:49
    operators for some operators, not for all
    of them, that give the identity operator
  • 25:49 - 25:54
    if it's multiplied with the original
    operator. And further there's the so-
  • 25:54 - 26:00
    called adjoint. Our scalar product had
    this little asterisk and this means that
  • 26:00 - 26:05
    it's not linear in the first component. If
    I scale the first component by some
  • 26:05 - 26:11
    complex number alpha the total scalar
    product is not scaled by alpha, but by the
  • 26:11 - 26:18
    complex conjugate of alpha. This kind of
    not quite bi-linearity is sometimes called
  • 26:18 - 26:27
    sesquilinearity, a seldomly used word, and
    they're commonly defined classes of
  • 26:27 - 26:36
    operators in terms of how the adjoint that
    is defined there acts and how some other
  • 26:36 - 26:40
    operators for example where the adjoint is
    the inverse which is a generalization from
  • 26:40 - 26:46
    the fact that for rotation operators in
    normal Euclidean space, the transpose is
  • 26:46 - 26:54
    the inverse. And now that we have
    operators we can define expectation values
  • 26:54 - 26:59
    just by some formula. For now, we don't
    know what expectation values are, but we
  • 26:59 - 27:04
    can assume, it has something to do with
    the measurement values of the operator
  • 27:04 - 27:10
    because: why else would I tell you about
    it. And later on we will show that this is
  • 27:10 - 27:15
    actually the expectation value of the
    quantity if we prepare a system always in
  • 27:15 - 27:21
    the same fashion and then do measurements
    on it, we get random results each time,
  • 27:21 - 27:29
    but the expectation value will be this
    combination. And now again: a bit of
  • 27:29 - 27:35
    mathematics: eigenvalue problems. Well
    known: You can diagonalize a matrix and
  • 27:35 - 27:40
    you can diagonalize linear operators. You
    have some equation A psi equals lambda
  • 27:40 - 27:47
    psi, where lambda is just a scalar. And if
    such an equation holds for some vector Psi
  • 27:47 - 27:53
    then it's an eigenvector and if we scale
    the vector linearly, this will again be an
  • 27:53 - 28:01
    eigenvector. And what can happen is that
    to one eigenvalue there are several
  • 28:01 - 28:05
    eigenvectors, not only one ray of
    eigenvectors, but a higher dimensional
  • 28:05 - 28:12
    subspace. And important to know is that
    so-called Hermitian operators, that is
  • 28:12 - 28:18
    those that equal their adjoint, which
    again means that the eigenvalues equal the
  • 28:18 - 28:24
    complex conjugate of the eigenvalues have
    a real eigenvalues. Because if a complex
  • 28:24 - 28:33
    number equals its complex conjugate, then
    it's a real number. And the nice thing
  • 28:33 - 28:40
    about those diagonalized matrices and all
    is: we can develop any vector in terms of
  • 28:40 - 28:47
    the eigenbasis of the operator, again just
    like in linear algebra where when you
  • 28:47 - 28:51
    diagonalize a matrix, you get a new basis
    for your vector space and now you can
  • 28:51 - 28:58
    express all vectors in that new basis. And
    if the operator is Hermitian the
  • 28:58 - 29:05
    eigenvectors have a nice property, namely
    they are orthogonal if the eigenvalues are
  • 29:05 - 29:11
    different. And this is good because this
    guarantees us that we can choose an
  • 29:11 - 29:17
    orthonormal, that is a basis in the vector
    space where to basis vectors always have
  • 29:17 - 29:24
    vanishing scalar product are orthogonal
    and are normal, that is: we scale them to
  • 29:24 - 29:29
    length one, because we want our
    probability interpretation, and in our
  • 29:29 - 29:37
    probability interpretation we need to have
    normalized vectors. So now we have that
  • 29:37 - 29:42
    and now we want to know: How does this
    strange function Psi, that describes the
  • 29:42 - 29:50
    state of the system, evolve in time. And
    for this we can have several requirements
  • 29:50 - 29:56
    that it must fulfill. So again we are
    close to software engineering and one
  • 29:56 - 30:01
    requirement is, that if it is a sharp wave
    packet, so if we have a localized state
  • 30:01 - 30:07
    that is not smeared around the whole
    space, then it should follow the classical
  • 30:07 - 30:13
    equation of motion because we want that
    our new theory contains our old theory.
  • 30:13 - 30:18
    And the time evolution must conserve the
    total probability of finding the particle
  • 30:18 - 30:22
    because otherwise we couldn't do
    probability interpretation of our wave
  • 30:22 - 30:29
    function, if the total probability of the
    particle wouldn't remain one. Further we
  • 30:29 - 30:35
    wish the equation to be first order in
    time and to be linear because for example
  • 30:35 - 30:42
    the Maxwell equations are linear and show
    nice interference effects, so we want that
  • 30:42 - 30:46
    because then simply a sum of solutions is
    again a solution, it's a good property to
  • 30:46 - 30:52
    have and if it works that way: Why not?
    And the third and the fourth requirement
  • 30:52 - 31:01
    together already give us more or less the
    form of the Schroedinger equation. Because
  • 31:01 - 31:05
    linearity just says that the right-hand
    side of some linear operator applied to
  • 31:05 - 31:12
    Psi and the first order in time just means
    that there must be a single time
  • 31:12 - 31:20
    derivative in the equation on the left-
    hand side. And this i*hbar: we just wanted
  • 31:20 - 31:24
    that there, no particular reason we could
    have done this differently, but it's
  • 31:24 - 31:32
    convention. Now with this equation we can
    look: What must happen for the probability
  • 31:32 - 31:42
    to be conserved and by a simple
    calculation we can show that it must be a
  • 31:42 - 31:48
    Hermitian operator. And there is even more
    than this global argument. There's local
  • 31:48 - 31:52
    conservation of probability, that is, a
    particle can't simply vanish here and
  • 31:52 - 31:59
    appear there, but it must flow from one
    point to the other with local operations.
  • 31:59 - 32:05
    This can be shown when you consider this
    in more detail. Now we know how this
  • 32:05 - 32:09
    equation of motion looks like, but we
    don't know what this mysterious object H
  • 32:09 - 32:16
    might be. And this mysterious object H is
    the operator of the energy of the system
  • 32:16 - 32:22
    which is known from classical mechanics as
    the Hamilton function and which we here
  • 32:22 - 32:27
    upgrade to the Hamilton operator by using
    the formula for the classical Hamilton
  • 32:27 - 32:33
    function and inserting our p and our
    operators. And we can also extend this to
  • 32:33 - 32:40
    a magnetic field. And by doing so we can
    show that our theory is more or less
  • 32:40 - 32:46
    consistent with Newtonian mechanics. We
    can show the Ehrenfest theorem, that's the
  • 32:46 - 32:55
    first equation. And then those equations
    are almost Newton's equation of motion for
  • 32:55 - 33:06
    the centers of mass of the particle
    because this is the expectation value of
  • 33:06 - 33:11
    the momentum, this is the expectation
    value of the position of the particle.
  • 33:11 - 33:16
    This just looks exactly like the classical
    equation. The velocity is the momentum
  • 33:16 - 33:23
    divided by the mass. But this is weird:
    Here we average over the force, so the
  • 33:23 - 33:30
    gradient of the potential is the force, we
    average over the force and do not take the
  • 33:30 - 33:35
    force at the center position, so we can't
    in general solve this equation. But again
  • 33:35 - 33:39
    if we have a sharply defined wave packet
    we recover the classical equations of
  • 33:39 - 33:45
    motion, which is nice. So we have shown
    our new theory does indeed explain why our
  • 33:45 - 33:51
    old theory worked. We only still have to
    explain why the centers of mass of massive
  • 33:51 - 33:56
    particles are usually well localized and
    that's a question we're still having
  • 33:56 - 34:08
    trouble with today. But since it otherwise
    works: don't worry too much about it. And
  • 34:08 - 34:12
    now you probably want to know how to solve
    the Schroedinger equation. Or you don't
  • 34:12 - 34:18
    want to know anything more about quantum
    mechanics. And to do this we make a so-
  • 34:18 - 34:25
    called separation ansatz, where we say, we
    have a form stable part of our wave
  • 34:25 - 34:31
    function multiplied by some time dependent
    part. And if we do this we can write down
  • 34:31 - 34:36
    the general solution for the Schrodinger
    equation. Because we already know that the
  • 34:36 - 34:41
    one equation that we get is an eigenvalue
    equation or an eigenvector equation for
  • 34:41 - 34:46
    the energy eigenvalues, that is the
    eigenvalues of the Hamilton operator. And
  • 34:46 - 34:51
    we know that we can develop any function
    in terms of those and so the general
  • 34:51 - 34:58
    solution must be of the form shown here.
    And those states of specific energy have a
  • 34:58 - 35:02
    simple evolution because their form is
    constant and only their phase changes and
  • 35:02 - 35:09
    depends on the energy. And now this thing
    with the measurement in quantum mechanics
  • 35:09 - 35:13
    is bad. You probably know Schroedinger's
    cat and the point is: there you don't know
  • 35:13 - 35:16
    whether the cat is dead or alive while you
    don't look inside the box. While you don't
  • 35:16 - 35:19
    look inside the box as long as you don't
    measure it's in a superposition or
  • 35:19 - 35:24
    something. So you measure. You measure
    your cat and then it's dead. It isn't dead
  • 35:24 - 35:29
    before only by measuring it you kill it.
    And that's really not nice to kill cats.
  • 35:29 - 35:37
    We like cats. The important part here is,
    the TL;DR, quantum measurement is
  • 35:37 - 35:43
    probabilistic and inherently changes the
    system state. So I'll skip the multi
  • 35:43 - 35:53
    particle things. We can't describe
    multiple particles. And just show the
  • 35:53 - 35:59
    axioms of quantum mechanics shortly. Don't
    don't read them too detailed, but this is
  • 35:59 - 36:04
    just a summary of what we've discussed so
    far. And the thing about the multiple
  • 36:04 - 36:10
    particles is the axiom 7 which says that
    the sign of the wave function must change
  • 36:10 - 36:15
    if we exchange the coordinates of
    identical fermions. And this makes atom
  • 36:15 - 36:21
    stable by the way. Without this atoms as
    we know them would not exist. And finally
  • 36:21 - 36:27
    there is a notational convention in
    quantum mechanics called Bra-Ket-notation.
  • 36:27 - 36:35
    And in Bra-Ket-notation you label states
    by their eigenvalues and just think about
  • 36:35 - 36:42
    such a Ket as an abstract vector such as x
    with a vector arrow over it or a fat set x
  • 36:42 - 36:49
    is an abstract vector and we can either
    represent it by its coordinates x1 x2 x3,
  • 36:49 - 36:53
    or we can work with the abstract vector
    and this Ket is such an abstract vector
  • 36:53 - 37:02
    for the L2 function Psi of r. And then we
    can also define the adjoint of this which
  • 37:02 - 37:08
    gives us, if we multiply the adjoint and a
    function, the scalar product. So this is a
  • 37:08 - 37:15
    really nice and compact notation for many
    physics problems. And the last equation
  • 37:15 - 37:21
    there just looks like component wise, like
    working with components of matrices, which
  • 37:21 - 37:32
    is because it's nothing else. This is just
    matrix calculus in new clothes. Now for
  • 37:32 - 37:44
    the applications. The first one is quite
    funny. There's a slight missing. Okay. Uh
  • 37:44 - 37:49
    the first one is a quantum eraser at home.
    Because if you encode the "which way"
  • 37:49 - 37:56
    information into a double slit experiment
    you lose your interference pattern. And we
  • 37:56 - 38:01
    do this by using a vertical and horizontal
    polarisation filter. And you know from
  • 38:01 - 38:16
    classical physics then it won't make an
    interference pattern. And if we then add a
  • 38:16 - 38:26
    diagonal polarization filter then the
    interference pattern will appear again. So
  • 38:26 - 38:31
    now, just so you've seen it, the harmonic
    oscillator can be exactly solved in
  • 38:31 - 38:36
    quantum mechanics. If you can solve the
    harmonic oscillator in any kind of physics
  • 38:36 - 38:42
    then you're good, then you'll get through
    the axioms when you study physics. So the
  • 38:42 - 38:48
    harmonic oscillator is solved by
    introducing so-called creation and
  • 38:48 - 38:54
    destroyer operators and then we can
    determine the ground state function, in a
  • 38:54 - 38:58
    much simpler manner than if we had to
    solve the Schrödinger equation explicitly
  • 38:58 - 39:06
    for all those cases. And we can determine
    the ground state function, so the function
  • 39:06 - 39:11
    of lowest energy. This can all be done and
    then from it by applying the creation
  • 39:11 - 39:19
    operator create the highest eigenstate of
    the system and get all of them. Then
  • 39:19 - 39:23
    there's this effect of tunnelling that
    you've probably heard about and this just
  • 39:23 - 39:28
    means that in quantum mechanics a
    potential barrier that is too high for the
  • 39:28 - 39:32
    particle to penetrate does not mean that
    the particle doesn't penetrate at all but
  • 39:32 - 39:37
    that the probability of finding the
    particle inside the barrier decays
  • 39:37 - 39:43
    exponentially. And this can for example be
    understood in terms of this uncertainty
  • 39:43 - 39:48
    relation because if we try to compress the
    particle to the smaller part of the
  • 39:48 - 39:52
    boundary layer then its momentum has to be
    high so it can reach farther in because
  • 39:52 - 39:59
    then it has more energy. And there's this
    myth that tunnelling makes particles
  • 39:59 - 40:05
    traveling to travel instantaneously from A
    to B and even some real physicists believe
  • 40:05 - 40:13
    it. But sorry it's not true. The particle
    states is extended anyway and to defining
  • 40:13 - 40:17
    what how fast the particle travels is
    actually not a well-defined thing in deep
  • 40:17 - 40:23
    quantum regimes, and also the shooting
    equations is not relativistic. So there is
  • 40:23 - 40:28
    nothing, really nothing stopping your
    particle from flying around with 30 times
  • 40:28 - 40:34
    the speed of light. It's just not in the
    theory. Another important consequence of
  • 40:34 - 40:39
    quantum mechanics is so-called
    entanglement and this is a really weird
  • 40:39 - 40:44
    one, because it shows that the universe
    that we live in is in a way non-local,
  • 40:44 - 40:52
    inherently non-local. Because we can
    create some states for some internal
  • 40:52 - 40:57
    degrees of freedom of two atoms and move
    them apart then measure the one system and
  • 40:57 - 41:01
    the measurement result of the one system
    will determine the measurement result of
  • 41:01 - 41:08
    the other system, no matter how far
    removed they are from each other. And this
  • 41:08 - 41:12
    was first discovered in a paper by
    Einstein Podolski and Rosen and they
  • 41:12 - 41:18
    thought it was an argument that quantum
    mechanics is absurd. This can't be true,
  • 41:18 - 41:23
    but sorry it is true. So this works and
    this kind of state that we've written
  • 41:23 - 41:33
    there that is such an entangled state of
    two particles. But important to remark is
  • 41:33 - 41:37
    that there are no hidden variables, that
    means the measurement result is not
  • 41:37 - 41:42
    determined beforehand. It is only when we
    measure that is actually known what the
  • 41:42 - 41:48
    result will be. This is utterly weird but
    one can prove this experimentally. Those
  • 41:48 - 41:52
    are Bell tests. There's a Bell-inequality
    that's the limit for theories where they
  • 41:52 - 41:57
    are hidden variables and it's by real
    experiments they violate the inequality
  • 41:57 - 42:03
    and thereby show that there are no hidden
    variables. And there's a myth surrounding
  • 42:03 - 42:07
    entanglement namely that you can transfer
    information with it between two sides
  • 42:07 - 42:13
    instantaneously. But again there's nothing
    hindering you in non relativistic quantum
  • 42:13 - 42:18
    mechanics to distribute information
    arbitrarily fast. It doesn't have a speed
  • 42:18 - 42:24
    limit but you can't also count communicate
    with those entangled pairs of particles.
  • 42:24 - 42:28
    You can just create correlated noise at
    two ends which is what quantum
  • 42:28 - 42:36
    cryptography is using. So now because this
    is the hackers congress, some short
  • 42:36 - 42:42
    remarks and probably unintelligible due to
    their strong compression about quantum
  • 42:42 - 42:47
    information, a qubit, the fundamental unit
    of quantum information, is a system with
  • 42:47 - 42:55
    two states zero and one. So just like a
    bit. But now we allow arbitrary super
  • 42:55 - 42:58
    positions of those states because that is
    what quantum mechanics allows. We can
  • 42:58 - 43:03
    always superimpose states and quantum
    computers are really bad for most
  • 43:03 - 43:09
    computing tasks because they they have to,
    even if they build quantum computers they
  • 43:09 - 43:14
    will never be as capable as the state-of-
    the-art silicon electrical computer. So
  • 43:14 - 43:18
    don't fear for your jobs because of
    quantum computers. But the problem is they
  • 43:18 - 43:24
    can compute some things faster. For
    example factoring primes and working with
  • 43:24 - 43:30
    some elliptic curve algorithms and so on
    and determining discrete logarithm so our
  • 43:30 - 43:35
    public key crypto would be destroyed by
    them. And this all works by using the
  • 43:35 - 43:41
    superposition to construct some kind of
    weird parallelism. So it's actually I
  • 43:41 - 43:47
    think nobody really can imagine how it
    works but we can compute it which is often
  • 43:47 - 43:52
    the case in quantum mechanics. And then
    there's quantum cryptography and that
  • 43:52 - 43:56
    fundamentally solves the same problem as a
    Diffie-Hellman key exchange. We can
  • 43:56 - 44:01
    generate the shared key and we can check
    by the statistics of our measured values
  • 44:01 - 44:07
    that there is no eavesdropper, which is
    cool actually. But it's also quite useless
  • 44:07 - 44:10
    because we can't detect a man in the
    middle. How should the quantum particle
  • 44:10 - 44:15
    knows of the other side is the one with
    that we want to talk to. We still need
  • 44:15 - 44:19
    some shared secret or public key
    infrastructure whatever. So it doesn't
  • 44:19 - 44:27
    solve the problem that we don't have
    solved. And then the fun fact about this
  • 44:27 - 44:31
    is that all the commercial implementations
    of quantum cryptography were susceptible
  • 44:31 - 44:35
    to side channel text, for example you
    could just shine the light with a fiber
  • 44:35 - 44:41
    that was used, read out the polarization
    filter state that they used and then you
  • 44:41 - 44:51
    could mimic the other side. So that's not
    good either. So finally some references
  • 44:51 - 44:55
    for further study. The first one is really
    difficult. Only try this a few. Forget the
  • 44:55 - 45:01
    other two but the second one. Sorry that
    they're in German. The first and the last
  • 45:01 - 45:05
    are also available in translation but the
    second one has a really really nice and
  • 45:05 - 45:10
    accessible introduction in the last few
    pages so it's just 20 pages and it's
  • 45:10 - 45:15
    really good and understandable. So if you
    can get your hands on the books and are
  • 45:15 - 45:22
    really interested, read it. So thank you
    for the attention and I'll be answering
  • 45:22 - 45:28
    your questions next.
    [Filler, please remove in amara]
  • 45:28 - 45:33
    *Applause*
    [Filler, please remove in amara]
  • 45:33 - 45:41
    Herald: Thank you Sebastian. Do we have
    questions? And don't be afraid to sound
  • 45:41 - 45:45
    naive or anything. I'm sure if you didn't
    understand something many other people
  • 45:45 - 45:49
    would thank you for a good question.
    Sebastian:As to understanding things and
  • 45:49 - 45:53
    quantum mechanics, Fineman said" You can't
    understand quantum mechanics, you can just
  • 45:53 - 45:57
    accept that there there's nothing to
    understand. That's just too weird."
  • 45:57 - 46:02
    Herald: Ok,we've found some questions. So
    microphone one please.
  • 46:02 - 46:09
    Q: Can you explain that, if you measure a
    system, it looks like you changed the
  • 46:09 - 46:15
    state of the system. How is it defined
    where the system starts? No. How is it
  • 46:15 - 46:20
    defined when the system ends and the
    measurement system begins. Or in other
  • 46:20 - 46:24
    words why does the universe have the
    state? Is there somewhere out there who
  • 46:24 - 46:29
    measures the universe?
    A: No. There's at least the beginning of a
  • 46:29 - 46:35
    solution by now which is called
    Decoherence and which says that this
  • 46:35 - 46:40
    measurement structure that we observe is
    not inherent in quantum mechanics but
  • 46:40 - 46:44
    comes from the interaction with the
    environment. And we don't care for the
  • 46:44 - 46:48
    states of the environment. And if we do
    this, the technical term is traced out the
  • 46:48 - 46:53
    states of the environment. Then the
    remaining state of the measurement
  • 46:53 - 47:00
    apparatus and the system we're interested
    in will be just classically a randomized
  • 47:00 - 47:06
    state. So it's it's rather a consequence
    of the complex dynamics of a system state
  • 47:06 - 47:11
    and environment in quantum mechanics. But
    this is really the burning question. We
  • 47:11 - 47:16
    don't really know. We have this we know
    decoherence make some makes it nice and
  • 47:16 - 47:21
    looks good. But it also doesn't answer the
    question finally. And this is what all
  • 47:21 - 47:25
    those discussions about interpretations of
    quantum mechanics are about. How shall we
  • 47:25 - 47:29
    make sense of this weird measurement
    process.
  • 47:29 - 47:37
    Herald: Okay. Microphone 4 back please.
    Q: Could you comment on your point in the
  • 47:37 - 47:44
    theory section. I don't understand what
    you were trying to do. Did you want to
  • 47:44 - 47:49
    show that you cannot understand really
    quantum mechanics without the mathematics
  • 47:49 - 47:52
    or?
    A: Well, yes you can't understand quantum
  • 47:52 - 47:56
    mechanics without the mathematics and my
    point to show was that mathematics, or at
  • 47:56 - 48:02
    least my hope to show was that mathematics
    is halfways accessible. Probably not
  • 48:02 - 48:08
    understandable after just exposure of a
    short talk but just to give an
  • 48:08 - 48:13
    introduction where to look
    Q: OK. So you are trying to combat the
  • 48:13 - 48:18
    esoterics and say they they don't really
    understand the theory because they don't
  • 48:18 - 48:29
    understand the mathematics I understand
    the mathematics. I'm just interested. What
  • 48:29 - 48:34
    were you were trying to say.
    A: I was just trying to present the
  • 48:34 - 48:39
    theory. That was my aim.
    Q:Okay. Thank you. Okay.
  • 48:39 - 48:46
    Herald: Microphone two please.
    Q: I know the answer to this question is
  • 48:46 - 48:49
    that
    Herald: Can you go a little bit closer to
  • 48:49 - 48:53
    the microphone maybe move it up please.
    Q: So I know the answer to this question
  • 48:53 - 49:00
    is that atoms behave randomly but could
    you provide an argument why they behave
  • 49:00 - 49:07
    randomly and it is not the case that we
    don't have a model that's. So, are atoms
  • 49:07 - 49:12
    behaving randomly? Or is it the case that
    we don't have a model accurate enough to
  • 49:12 - 49:18
    predict the way they behave.
    A: Radioactive decay is just as random as
  • 49:18 - 49:24
    quantum measurement and since it. If if we
    were to look at the whole story and look
  • 49:24 - 49:28
    at the coherent evolution of the whole
    system we would have to include the
  • 49:28 - 49:34
    environment and the problem is that the
    state space that we have to consider grows
  • 49:34 - 49:38
    exponentially. That's the point of quantum
    mechanics. If I have two particles I have
  • 49:38 - 49:43
    a two dimensional space. I have 10
    particles I have a 1024 dimensional space
  • 49:43 - 49:47
    and that's only talking about non
    interacting particles. So things explode
  • 49:47 - 49:52
    in quantum mechanics and large systems.
    And therefore I would go so far as to say
  • 49:52 - 49:57
    that it's objectively impossible to
    determine a radioactive decay although
  • 49:57 - 50:04
    there are things there is I think one
    experimentally confirmed method of letting
  • 50:04 - 50:11
    an atom decay on purpose. This involves a
    meta stable states of nuclear and then you
  • 50:11 - 50:16
    can do something like spontaneous emission
    in a laser. you shine a strong gamma
  • 50:16 - 50:22
    source by it and this shortens the
    lifespan of the nucleus. But other than
  • 50:22 - 50:25
    that.
    Q: completely hypothetical case. If you
  • 50:25 - 50:30
    know all the starting conditions and what
    happens afterwards,wouldn't it be able
  • 50:30 - 50:37
    where you could say it's deterministic? I
    mean I'm playing with heavy words here.
  • 50:37 - 50:44
    But is it just that we we say it's random
    it's because it's very very complex right.
  • 50:44 - 50:48
    That's what I'm understanding.
    Herald: Maybe think about that question
  • 50:48 - 50:53
    one more time and we have a signal Angel
    in between and then you can come back.
  • 50:53 - 50:58
    Signal Angel do we have questions on the
    Internet?
  • 50:58 - 51:05
    Q: There's one question from the Internet
    which is the ground state of a BEH-2 has
  • 51:05 - 51:12
    been just calculated using a quantum
    eigensolver. So is there still some use of
  • 51:12 - 51:17
    quantum computing in quantum mechanics?
    A:Yes definitely. One of the main
  • 51:17 - 51:22
    motivations for inventing quantum
    computers was quantum simulators. Uh
  • 51:22 - 51:27
    Feynman and then invented this kind of
    quantum computing and he showed that with
  • 51:27 - 51:32
    digital quantum computer you can
    efficiently simulate quantum system. While
  • 51:32 - 51:36
    you can't simulate quantum systems with a
    classical computer because of this problem
  • 51:36 - 51:42
    of the exploding dimensions of the hilbert
    space that you have to consider. And for
  • 51:42 - 51:46
    this quantum computers are really really
    useful and will be used once they work,
  • 51:46 - 51:53
    which is the question when it will be.
    Perhaps never. Beyond two or three qubits
  • 51:53 - 51:59
    or 20 or 100 qubits but it's scalability
    for a real quantum computer. But quantum
  • 51:59 - 52:03
    simulation is a real thing and it's a good
    thing and we need it.
  • 52:03 - 52:08
    Herald: Okay. Then we have microphone one
    again.
  • 52:08 - 52:14
    Q:So very beginning, you said that the
    theory is a set of interdependent
  • 52:14 - 52:22
    propositions. Right? And then if a new
    hypothesis is made it can be confirmed by
  • 52:22 - 52:28
    an experiment.
    A: That can't be confirmed but, well it's
  • 52:28 - 52:34
    a philosophical question about the common
    stance that can be made probable but not
  • 52:34 - 52:37
    be confirmed because we can never
    absolutely be sure that there won't be
  • 52:37 - 52:41
    some new experiment that shows that the
    hypothesis is wrong.
  • 52:41 - 52:45
    Q: Yeah. Yeah. Because this line said that
    the experiment confirms.
  • 52:45 - 52:51
    A: Yeah, confirm in the sense that it
    doesn't disconfirm it. So it makes
  • 52:51 - 52:57
    probable that it's a good explanation of
    the reality and that's the point. Physics
  • 52:57 - 53:02
    is just models. We don't need to get
    nothing about the ontology that is about
  • 53:02 - 53:06
    the actual being of the world out of
    physics. We just get models to describe
  • 53:06 - 53:12
    the world but all what I say about this
    wave function and what we say about
  • 53:12 - 53:18
    elementary particles. We can't say they
    are in the sense that you and I are here
  • 53:18 - 53:23
    and exist because we can't see them we
    can't access them directly. We can only
  • 53:23 - 53:29
    use them as description tools. But this is
    my personal position on philosophy of
  • 53:29 - 53:33
    science. So there are people who disagree.
    Q:Thanks.
  • 53:33 - 53:40
    Herald: Microphone 2 please.
    Q: Or maybe superposition. By the way, so
  • 53:40 - 53:48
    on the matter of the collapsing of the
    wave function, so this was already reated
  • 53:48 - 53:53
    on the interpretation of Copenhagen and
    then as you mentioned it was expanded by
  • 53:53 - 53:59
    the concept of decoherence. And is this so
    the decoherence is including also in the
  • 53:59 - 54:03
    Ghirardi–Rimini–Weber interpretation or
    not?
  • 54:03 - 54:07
    A: Could decoherence be used in
    computation or?
  • 54:07 - 54:13
    Q: No so for the Ghirardi–Rimini–Weber
    interpretation of the collapsing of the
  • 54:13 - 54:16
    wave function.
    A:That's that's one that I don't know.
  • 54:16 - 54:24
    Okay I'm not so much into interpretations.
    I actually think that there's interesting
  • 54:24 - 54:30
    work down there but I think they're a bit
    irrelevant because in the end what I just
  • 54:30 - 54:34
    said I don't think you can derive
    ontological value from our physical
  • 54:34 - 54:41
    theories and in this belief, I think that
    the interpretations are in a sense void,
  • 54:41 - 54:45
    they just help us to rationalize what
    we're doing but they don't really add
  • 54:45 - 54:49
    something to the theory as long as they
    don't change what can be measured.
  • 54:49 - 54:58
    Q:Oh okay. Thanks.
    A: Sorry for being an extremist.
  • 54:58 - 55:03
    Q:Totally fine.
    Herald:Someone just left from microphone
  • 55:03 - 55:08
    one. I don't know if they want to come
    back. I don't see any more questions as to
  • 55:08 - 55:14
    signal angel have anything else. There is
    some more. Single Angel, do you have
  • 55:14 - 55:17
    something?
    Signal Angel: no.
  • 55:17 - 55:20
    Herald: Okay. Then we have microphone
    four.
  • 55:20 - 55:28
    Q: I want to ask a maybe an open question.
    I want to know are there probabilities of
  • 55:28 - 55:33
    quantum mechanics inherant part of nature
    or maybe in some future we'll have a
  • 55:33 - 55:37
    science that will determine all these
    values exactly?
  • 55:37 - 55:45
    A:Well um if decoherency theory is true,
    then quantum mechanics is absolutely
  • 55:45 - 55:54
    deterministic. But so let's say, Everett
    says that all those possible measurement
  • 55:54 - 55:59
    outcomes do happen and the whole state of
    the system is in a superposition and by
  • 55:59 - 56:04
    looking at our measurement device and
    seeing some value we in a way select one
  • 56:04 - 56:10
    strand of those superpositions and live in
    this of the many worlds and in this sense
  • 56:10 - 56:21
    everything happens deterministically, but
    we just can't access any other values. So
  • 56:21 - 56:28
    I think it's rather a matter of for now
    rather matter of philosophy than of
  • 56:28 - 56:33
    science.
    Q: I see. Thanks.
  • 56:33 - 56:39
    Herald: Anything else? I don't see any
    people lined up at microphones. So last
  • 56:39 - 56:47
    chance to round up now I think. Well then
    I think we're closing this and have a nice
  • 56:47 - 57:02
    applause again for Sebastian.
    Sebastian: Thank you. And I hope I didn't
  • 57:02 - 57:06
    create more fear of quantum mechanics than
    I dispersed.
Title:
35C3 - Quantum Mechanics
Description:

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Video Language:
English
Duration:
57:30

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