Archive for April, 2012

More evidence that quantum mechanics is nonlocal?

Posted in Uncategorized on April 28, 2012 by quantummoxie

In a string of recent posts I have developed a gedankenexperiment that involves neutral pions in an optical cavity decaying to photons.  I finally figured out how to handle the photons in the cavity courtesy of a book on quantum optics by Mandel and Wolf. So let’s imagine we have a single neutral pion at rest inside an optical cavity.  99% of the time, a neutral pion will decay to two photons.  In order to simplify the math a bit, let’s work with a simple 1-D cavity with mirrors on each end acting as the “walls” of the potential well (this is one of the examples discussed in Mandel and Wolf). As Mandel and Wolf describe, the normal modes for light in an empty optical cavity are found by solving the Helmholtz equation and setting appropriate boundary conditions. In the 1-D case, the energy eigenvalues for the light are given as

E_{n}=\frac{hcn}{2L}

where L is the length of the “box,” i.e. the width of the cavity.  Now, according to special relativity, the four-momentum must be conserved for the decay process.  As such, we have

\left(\begin{array}{c}m_{\pi} \\ 0 \\ 0 \\ 0 \end{array}\right)=\left(\begin{array}{c}E_{\gamma} \\ -E_{\gamma} \\ 0 \\ 0 \end{array}\right)+\left(\begin{array}{c}E_{\gamma} \\ E_{\gamma} \\ 0 \\ 0 \end{array}\right).

This actually tells us right away that the energies of the emitted photons should be about 67.5 MeV since the rest pass a neutral pion is roughly 135 MeV.  But wait a second.  As we saw earlier, since they’ll be in a box, these photons will be quantized.  How do we interpret this, then?  Is 67.5 MeV equal to E for a particular value of n, or does it simply tell us that the ratio of n/L is limited?  Either way, something non-local is going on because, at the instant of decay, the photons must immediately know that they are in a box.

Now here’s another peculiar thing.  The most logical way to interpret the 67.5 MeV is to say that it is one of the eigenvalues for the photon in the cavity.  Thus, if we plug it in for E_{n}, we find that n/L is equal to some number.  This number is positive and real, but – crucially – not necessarily an integer.  But we know that n has to be an integer.  There are only two possible conclusions that we can reach from this:

  1. L cannot be any value, i.e. paradoxically, the size of the box is somehow predetermined by the fact that it contains a neutral pion initially, which is utterly absurd, or
  2. if the box is the wrong size, the neutral pion simply cannot decay via this decay mode.  But, once again, this implies that the pion somehow knows that it is in a box to begin with!  In fact it also implies that the pion knows something about the size of the box!

While the second option squares better with our ability to arbitrarily choose a box size, it does, nevertheless, imply that there is something non-local going on.  The only explanation I can think of is that the pion is somehow entangled with the box.  But then, if it is, how did it get that way?  Suppose I were able to isolate a neutral pion in a vacuum and then bring two halves of a spherical cavity together around it.  How does it become entangled with the box if it never comes in contact with the box (or anything else, for that matter)?  The only explanation I can think of is that the neutral pion itself is a non-local object.

“Pion stars,” spherical cavities, and more on the nature of mass

Posted in Uncategorized on April 21, 2012 by quantummoxie

In a recent post on information and the nature of mass, I presented a gedankenexperiment that involved a spherical, thin-shelled, perfectly internally reflective optical cavity that initially contains neutral pions that then decay to photons.  Using special relativistic arguments I demonstrated how the total mass of the system should remain unchanged in the process.  So, for instance, if the cavity were large enough such that some massive object could orbit (due to gravity), the object’s orbit should not change after the photons decay.  This experiment has continued to gnaw away at my brain and I began to think of the quantum mechanical aspects of the problem.

Let’s idealize this a bit in order to simplify things and start by only considering a single neutral pion (initially at rest) in the cavity.  Quantum mechanically we can treat the problem as a spherical “particle in a box” problem.  Since the neutral pion is initially non-relativistic since it is at rest, the three-dimensional Schrödinger equation can be validly applied in order to find the energy eigenstates of the associated pion.  This is a fairly standard procedure and, in fact, is often included in elementary quantum mechanics texts as a problem.  One simply solves the radial Schrödinger equation and tacks on the spherical harmonics.  Solutions generally take the form of Bessel functions (note, however, that the probability densities depend only on r since the distributions are spherically symmetric).

But what happens when the pion decays to the photons?  Photons are highly relativistic and applying the Schrödinger equation here is a bit suspect (though note that it can be used to find approximate solutions, e.g. see Slater’s classic Quantum Theory of Matter).  We might try to apply the Klein-Gordon equation in the limit that m\to\infty. But it isn’t clear that this would give the correct answer either. Iwo Bialynicki-Birula has written several papers on photon wavefunctions (a nice summary paper is here).  Tom Moore also does a nice job “explaining” the two-slit experiment with wavefunctions in his book Six Ideas That Shaped Physics, Unit Q.  In the latter, Moore uses the free-particle approximation and really just treats the stream of photons as a fluctuating semi-classical E&M field.  I think for the present problem a semi-classical approach such as this might be a bit oversimple.

At any rate, this raises a number of intriguing questions.  Presumably, conservation of energy would require that there be some kind of correspondence between the energy eigenstates of the pion and that of the photon pair (aside from the well-known phase correspondence for such a decay that is due to the conservation of charge conjugation, i.e. (-1)^{l+s}=(-1)^{n}).  How would such a restriction affect the solutions to the photon eigenvalue problem?  In the gedankenexperiment for mass, we invoked special relativity in order to explain that the mass remains unchanged.  Since this also tells us that the energy of the system is related to the mass, does this quantization of energy levels necessarily quantize the mass of the system?  Could this tell us a little something about quantum gravity in the process?

One way to answer these questions in an ensemble manner (i.e. for many pions and many photons) is to consider the pions as a degenerate Fermi gas, i.e. to consider the internal pressure of a cavity as high enough that you essentially have a “pion star.”  After the pions have completely decayed to photons, the “star” is now modeled as a photon gas.  What is interesting about this approach is that it can be shown that the total energy of the pions in the cavity is dependent on the number of pions in the cavity but is not dependent on the volume, while the opposite is true for the total energy of the photons.  One could tie volume to the total energy of the fermions via the Fermi energy, but that is slightly circular reasoning in my book.  Either way, if we set the total energies equal to one another (for simplicity assume the temperature is the same in both the pion and photon cases and is very low, T\ll \epsilon /k), the number of pions is linearly proportional to the volume, which actually makes sense.

But that doesn’t really tell us much.  For our purposes here, the more fruitful approach might be to start with a single neutral pion that then decays to a pair of photons, \pi^{0}\to\gamma +\gamma, and see what that implies, first.  Then we can look at the ensemble characteristics to see how the situation differs.  In any case, I have not gone into this in much detail yet, but am working on it.  I think it would be an intriguing idea to pursue, though, particularly if we can tie together the quantum and relativistic mass considerations.  Any seriously legitimate contributions given in the responses to this post will, of course, lead to co-authorship (should you choose).  Any attempt to steal these ideas and beat me to publication will result in a serious ass-kicking (assuming the ideas are even worthy of publication).

Superpositions and mixed states

Posted in Uncategorized on April 9, 2012 by quantummoxie

One of the more widely misunderstood aspects of quantum mechanics is the difference between superpositional states and mixed states.  This is partly because we don’t do a good enough job defining our terms in physics (or rather agreeing to use the same consistent definition). This has particular relevance to my recent posts on decoherence and the Schrödinger cat paradox.  As Hongwan Liu pointed out on Quora, while the latter paradox was originally formulated in relation to superpositional states, it has evolved to actually be about mixed states.  So what are they?

A superpositional state is not actually as strange as it first appears (regardless of the interpretation of quantum mechanics that you happen to prefer). In theory, we could describe classical systems in terms of superpositional states just as easily if those systems were probabilistic. Consider a fair (unbiased) coin. Prior to flipping it (or, perhaps, after flipping it but prior to actually looking at it), we could describe its state as being

|\psi\rangle=\frac{1}{\sqrt{2}}\left(|\textrm{heads}\rangle+|\textrm{tails}\rangle\right).

In some interpretations of quantum mechanics, quantum mechanical objects in such states are considered to be simultaneously in such states. This is partly why certain interpretations of QM take electrons in an atom to be in a ‘cloud’ (i.e. simultaneously in all energy eigenstates at once) prior to any type of measurement. But it is to be emphasized that this is merely an interpretation. It is based on results of interferometer experiments. In a purely statistical interpretation of QM as well as in certain epistemic interpretations, the view of the states is closer to the example of the coin that I just gave. But whatever your view, the fact remains that the above is a superpositional state.

So what’s a mixed state? A mixed state requires a density matrix representation. Using the above example, we can form a density matrix for the state of the coin as

\begin{array}{lcl}\rho & = & |\psi\rangle\langle\psi| \\ & = & \frac{1}{2}|\textrm{heads}\rangle\langle\textrm{heads}|+\frac{1}{2}|\textrm{heads}\rangle\langle\textrm{tails}|\\ & & + \frac{1}{2}|\textrm{tails}\rangle\langle\textrm{heads}| + \frac{1}{2}|\textrm{tails}\rangle\langle\textrm{tails}| \\ & = & \left(\begin{array}{cc} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{array}\right)\end{array}.

Now we see that there do seem to be these strange situations in which the coin is both heads and tails simultaneously. These terms correspond to the off-diagonal elements of the density matrix and are truly quantum mechanical in that, while we can write such a matrix for a classical object like a coin, the meaning of the off-diagonal terms for non-quantum objects is not clear (in fact most people would simply say it has no meaning for classical objects, i.e. it makes no sense to even write down such a thing for a classical system). These off-diagonal terms are called coherences. The term ‘decoherence’ can be applied to this in two ways.

The interpretation-free way of defining decoherence is that it corresponds to any process that eliminates the coherences from a density matrix. Conversely, if we refer to decoherence as any process that irreversibly eliminates the coherences, we are, to some extent, biasing ourselves a bit towards interpretations that favor an environmental (or similar) explanation for the decay of such terms (assuming they decay to begin with). This is because many density operators are formed using the Schrödinger equation for density operators which naturally gives rise to exponential terms with time constants in them, i.e. unitary evolution produces terms with time-dependent exponentials. When these decay, the coherences go to zero and the diagonal terms become equal to classical probabilities (or, one might say the superpositional state turns into a mixed state). But I’m not entirely sure I agree that coherences are always time-dependent, exponentially decaying terms, in which case I don’t think the term ‘irreversible’ necessarily need apply (since there might be some other way to get rid of them). Or, if you were to insist that decoherence was an inherently irreversible process, then you would need a new term to describe a process by which the coherences vanish either reversibly or by a time independent process.

Either way, the superpositional state is a state that includes such coherences while a mixed state does not. Understanding the differences between them is more than merely a matter of learning QM. It gets at the heart of precisely what QM is and how it should (or can) be interpreted.