In another round of the 'quantum wars' on the OU fora we have got round to discusssing orbitals and what they mean. It wasn't till I started thinking about it that I realised the usual picture of an electron orbiting the nucleus in an orbit of fixed radius like a mini-solar system, and that if it loses energy it 'jumps' from one to another is totally misleading to say the least. The latter view is definitely true of the Bohr model and I guess most people carry this over to their thinking when (if) they study quantum physics further.
However it is just not correct, When you solve Schrodinger's equation for the hydrogen atom what, one ends up with for each energy level is a 3 dimensional probability density function which gives the probability of finding an electron in a given region. This means that the electron in a given orbital can be anywhere in principle allowed by the particular orbital (better called probability density function) so there is a small but finite probability that it could be 1 m, 10m and so on away from the nucleus. Of course it will be closer to the average of the probabiity density function.
The crucial point is however that when the electron either gains or loses energy that loss or gain in energy is fixed (quantised). However the electron does not 'jump' from one orbital to another what happens is that the probability density function changes in accordance with the appropriate function for that energy level.
Thus the misleading picture given by the Bohr model is totally inadequate to do justice to the picture presented by the solution to Schrodinger's equation.
Matter's aren't helped by the depiction of the energy level diagrams with their pictures of arrows going from one energy level to another giving the impression that the particle is actually jumping from one energy level to another. But those 'jumps' are changes in energy not position.
Finally it is important to remember that those pictures of orbitals shown in chemistry or physics textbooks are pictures of probability density functions. The usual convention when drawing the boundary surfaces is to draw the boundary marking off the region where the electron is likely to be found 95% of the time they do not represent the fixed distance of the electron from the nucleus of the atom. 95% of the time the electron will be inside the so called 'shell' but there is also a 5% probability that it will be outside the shell. Here are some pretty pictures of hydrogen atom probability density functions for you you to drool over.
https://search.yahoo.com/search?ei=utf-8&fr=ytff1-yff22&p=Pictures%20of%20hydrogen%20atom%20orbitals%20&type=
And here is the first experimental observation of the probability density function of the hydrogen atom
http://io9.com/the-first-image-ever-of-a-hydrogen-atoms-orbital-struc-509684901
Sunday 24 August 2014
Monday 4 August 2014
Yet More Quantum madness the so called Cheshire cat
For those like me, who despair at the increasing tendency to the irrationality of the world, even in science. The recently published claim that it has been possible to separate out the spin component of a quantum object from it's spatial component can only cause one to wonder whether the world has gone mad.
For those who aren't aware this BBC post is typical of the hype
http://www.bbc.co.uk/news/science-environment-28543990
and the original paper is here
http://www.nature.com/ncomms/2014/140729/ncomms5492/full/ncomms5492.html
So the claim at first appears to be by a special arrangement of beam splitters it is possible to disembody a particles spin from it's location. So that in one path only the location of the particle can be determined on another part it's magnetic moment which seems to have magically been disentangled from the particle. A literal interpretation of this is just not credible.
First of all spin has no spatial properties so it can't move anywhere. Spin is a vector which is allowed quantum mechanically to take a number of fixed directions usually up or down. it is a property of a quantum object. So if the authors of the paper are claiming what the hype says they are then that would require a rewriting of the whole of quantum mechanics of spin and as far as I can tell the authors are not seeking to challenge the standard quantum mechanical concept of spin.
So what exactly have the authors shown? The key to interpreting the experiment is the concept of quantum superposition there are essentially two interpretations
The first beloved of the popular literature, and it would seem journalists who want to peddle the alleged mysterious aspects of quantum mechanics, is called the ontic (short for ontological) interpretation.
The ontic view is that quantum superpositions are real namely that a cat really is in limbo between being alive or dead or in the case of interferometer experiments each particle really does travel down both paths simultaneously or in the two slit experiment passes through both slits simultaneously.
The solution to Schrodinger's equation is a real wave and something physical (even though it doesn't actually affect anything (for example exert a force on a particle as an electric fiield does), For an N body problem, it is some sort of field in 3N + 1 configuration space as opposed to our 3+1 dimensional space time. Also when a measurement is made the solution to Schrodinger's eqaution collapses into one of the states. This is usually called wave-packet collapse but it really should be called the solution to Schrodinger's equation collapse.
On the ontic view, it is only by measurement that a system jumps from it's state of limbo of possible states into a definite state. Extrapolating (slightly unfairly I admit) this means the moon really isn't there when one looks at it. It would seem from the language used by the authors talking of the wavefunction travelling down both paths simultaneously they subscribe to the ontic view. On this view in this experiment, it really does seem that the spin part of a qauntum objects solution to Schrodinger's equation can be detached from it's spatial part. But just what disembodied spin means hasn't really been explained and if it really was disembodied then all the text books on quantum mechanics would have to be rewritten.
Conversely from an epistemic (statistical) point of view, the solution to Schrodinger's eqaution is a probability amplitude who's modulus squared gives when suitably normalised gives rise to a probability density function. The Born interpretation, on this view the superposition of states represents the lack of knowledge of an observer about a given situation and nothing physical. As the solution to Schrodinger's equation is primarily a statistical quantity, applying it to work out the behaviour of one particle is just as meaningful, as trying to predict the roll of a single dice or a spin of a roulette wheel. When the solution to Schrodinger's equation 'collapses' all that happens is that one of the possibilities is realised. Albeit in order to get the correct probabilities for a given situation one has to use complex numbers which accounts (mathematically at least) for all the interference effects associated with many quantum phenomenon.
I have discussed how this works for two state systems in previous posts
http://chrisfmathsphysicsmusic.blogspot.co.uk/search?q=Two+state+
Where I also point out that the Dirac notation can also be applied to classical systems. But in order to explain many quantum phenomenon, one has to extend our notion of probability to include complex numbers.
On this epistemic\statistical view then when a beam of particles passes through a beam splitter, a single particle, does not travel through both paths simultaneously, but the solution to Schrodinger's equation represents the probability amplitude that it will traverse one path or the other. On this view in the Cheshire Cat experiment, if the neutron passes one path it's location, will be measured on the other it's magnetic moment will be measured. There is no disembodiment of the spin part of the neutron's solution to Schrodinger's equation from it's spatial part. That realtively prosaic account of course won't attract attention and so would be ignored by most people.
The epistemic view and the ontic view, as they are interpreting the same mathematical object, namely the Solution to Schrodinger's equation cannot be empirically distinguished from each other. However given the more modest claims of the epistemic view vis a vis our notions of quantum wierdness I know which one I prefer. All the epistemic (statistical view) is asking is that we extend our notions of probability to include complex numbers. In contrast the ontic view would claim that wavepacket collapse is real, that spin's can be dismbodied from their constituent particles and that act's of measurement do create reality. Of course some people would find the epistemic\statistical view unsatisfactory, accusations of instrumentalism etc but if there are no hidden variables then there can be nothing behind the statistics.
I really do have to ask why is there such a hunger for irrational explanations of quantum phenomenon. I really do despair some times.
For those who aren't aware this BBC post is typical of the hype
http://www.bbc.co.uk/news/science-environment-28543990
and the original paper is here
http://www.nature.com/ncomms/2014/140729/ncomms5492/full/ncomms5492.html
So the claim at first appears to be by a special arrangement of beam splitters it is possible to disembody a particles spin from it's location. So that in one path only the location of the particle can be determined on another part it's magnetic moment which seems to have magically been disentangled from the particle. A literal interpretation of this is just not credible.
First of all spin has no spatial properties so it can't move anywhere. Spin is a vector which is allowed quantum mechanically to take a number of fixed directions usually up or down. it is a property of a quantum object. So if the authors of the paper are claiming what the hype says they are then that would require a rewriting of the whole of quantum mechanics of spin and as far as I can tell the authors are not seeking to challenge the standard quantum mechanical concept of spin.
So what exactly have the authors shown? The key to interpreting the experiment is the concept of quantum superposition there are essentially two interpretations
The first beloved of the popular literature, and it would seem journalists who want to peddle the alleged mysterious aspects of quantum mechanics, is called the ontic (short for ontological) interpretation.
The ontic view is that quantum superpositions are real namely that a cat really is in limbo between being alive or dead or in the case of interferometer experiments each particle really does travel down both paths simultaneously or in the two slit experiment passes through both slits simultaneously.
The solution to Schrodinger's equation is a real wave and something physical (even though it doesn't actually affect anything (for example exert a force on a particle as an electric fiield does), For an N body problem, it is some sort of field in 3N + 1 configuration space as opposed to our 3+1 dimensional space time. Also when a measurement is made the solution to Schrodinger's eqaution collapses into one of the states. This is usually called wave-packet collapse but it really should be called the solution to Schrodinger's equation collapse.
On the ontic view, it is only by measurement that a system jumps from it's state of limbo of possible states into a definite state. Extrapolating (slightly unfairly I admit) this means the moon really isn't there when one looks at it. It would seem from the language used by the authors talking of the wavefunction travelling down both paths simultaneously they subscribe to the ontic view. On this view in this experiment, it really does seem that the spin part of a qauntum objects solution to Schrodinger's equation can be detached from it's spatial part. But just what disembodied spin means hasn't really been explained and if it really was disembodied then all the text books on quantum mechanics would have to be rewritten.
Conversely from an epistemic (statistical) point of view, the solution to Schrodinger's eqaution is a probability amplitude who's modulus squared gives when suitably normalised gives rise to a probability density function. The Born interpretation, on this view the superposition of states represents the lack of knowledge of an observer about a given situation and nothing physical. As the solution to Schrodinger's equation is primarily a statistical quantity, applying it to work out the behaviour of one particle is just as meaningful, as trying to predict the roll of a single dice or a spin of a roulette wheel. When the solution to Schrodinger's equation 'collapses' all that happens is that one of the possibilities is realised. Albeit in order to get the correct probabilities for a given situation one has to use complex numbers which accounts (mathematically at least) for all the interference effects associated with many quantum phenomenon.
I have discussed how this works for two state systems in previous posts
http://chrisfmathsphysicsmusic.blogspot.co.uk/search?q=Two+state+
Where I also point out that the Dirac notation can also be applied to classical systems. But in order to explain many quantum phenomenon, one has to extend our notion of probability to include complex numbers.
On this epistemic\statistical view then when a beam of particles passes through a beam splitter, a single particle, does not travel through both paths simultaneously, but the solution to Schrodinger's equation represents the probability amplitude that it will traverse one path or the other. On this view in the Cheshire Cat experiment, if the neutron passes one path it's location, will be measured on the other it's magnetic moment will be measured. There is no disembodiment of the spin part of the neutron's solution to Schrodinger's equation from it's spatial part. That realtively prosaic account of course won't attract attention and so would be ignored by most people.
The epistemic view and the ontic view, as they are interpreting the same mathematical object, namely the Solution to Schrodinger's equation cannot be empirically distinguished from each other. However given the more modest claims of the epistemic view vis a vis our notions of quantum wierdness I know which one I prefer. All the epistemic (statistical view) is asking is that we extend our notions of probability to include complex numbers. In contrast the ontic view would claim that wavepacket collapse is real, that spin's can be dismbodied from their constituent particles and that act's of measurement do create reality. Of course some people would find the epistemic\statistical view unsatisfactory, accusations of instrumentalism etc but if there are no hidden variables then there can be nothing behind the statistics.
I really do have to ask why is there such a hunger for irrational explanations of quantum phenomenon. I really do despair some times.
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