# Is it wrong to talk about wave functions of macroscopic bodies?

Does a real macroscopic body, like table, human or a cup permits description as a wave function? When is it possible and when not?

For example in the "Statistical Physics, Part I" by Landau & Lifshitz it is argued that such systems must be described via the density matrix (chapter I, about statistical matrix). As far as I got it, roughly speaking, macroscopic bodies are so sensible to external interaction that they never can be counted as systems, one have to include everything else to form a system. Is my interpretation right?

When is it wrong to talk about wave functions of bodies that surround us?

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Those degrees of freedom of a quantum system that are described by a pure partial state must be very well shielded from unwanted interactions with the environment, otherwise they will be decoherered to a mixed state in a moment. This shielding can be done for a few degrees of freedom (like a superconducting current) but not for position and momentum of macroscopic bodies. Therefore these dof are always described by density matrices.

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and what is partial state? – Yrogirg Aug 14 '12 at 7:05
@Yrogirg: It is the part of a state of a macroscopic object that is considered as relevant. For example, in quantum optics, a beam is a complicated object but for many experiments one only looks at the partial state in the direction of the beam. Thus the position/momentum degrees of freedom are ignored. As a result, the description simplifies, and one may look at pure polarizartion or angular momentum states embedded in the more complex system. – Arnold Neumaier Aug 14 '12 at 8:53

It's not wrong, but you have to consider configuration spaces with exponentially large dimensionality. For N nonrelativistic particles, it's 3N dimensional not counting spin. This is beyond our ability. So, we take partial traces and "collapse" the wavefunction.

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It would still be wrong because you cannot write a wavefunction for an object that is entangled with something else, in this case the environment. – dbrane Oct 4 '12 at 22:28

Long wires are real macroscopic bodies, kilometers of superconducting wires are used at the LHC of CERN and the currents can be described by quantum mechanical equations.

Crystals also can be described by quantum mechanical equations, and can be quite large, maybe not as large as a table. Superfluids too are in the realm of macroscopic quantum mechanics.

The difference with a random object, like a table, is that the individual wave functions of the microcosm of molecules and atoms that compose them are incoherent with each other. Coherence means that all the phases of the probability wave functions of the ~10^23 molecules per mole composing them are lost statistically, in contrast with the examples of coherence above. That is why we use the density matrix to describe the behavior of such systems.

So the random bodies that surround us cannot be described by one wave function in the sense of a solution of one quantum mechanical equation, except when careful conditions are met as in the examples above.

Edit in response to comment:

"Coherence means that all ... " Could you please elaborate more on this, maybe with the help of math?

Any wave solution will have a constant angle phi as a phase with another wave solution.

These phases are what define interference and beat patterns in waves.Coherence means that the phases are known.

The square of the quantum mechanical wave solution is the probability to find the particle at that (x,y,z,t) and the interferences patterns when the phases are fixed are also probability functions.

And you say that superconducting wires are described by usual QM,

Not usual QM, it is a special solution within the quantum mechanical theory, from the link:

Since the discovery of superconductivity, great efforts have been devoted to finding out how and why it works. During the 1950s, theoretical condensed matter physicists arrived at a solid understanding of "conventional" superconductivity, through a pair of remarkable and important theories: the phenomenological Ginzburg-Landau theory (1950) and the microscopic BCS theory (1957).[12][13] Generalizations of these theories form the basis for understanding the closely related phenomenon of superfluidity, because they fall into the Lambda transition universality class, but the extent to which similar generalizations can be applied to unconventional superconductors as well is still controversial. The four-dimensional extension of the Ginzburg-Landau theory, the Coleman-Weinberg model, is important in quantum field theory and cosmology. Superfluidity of helium and superconductivity both are macroscopic quantum phenomena.

and the link has further references.

hence their wave functions belongs to a kind of tensor product of state spaces of constituting free atoms.

If you read up on superconductivity you will see that it is not what you assume.

The theory describes superconductivity as a microscopic effect caused by a condensation of Cooper pairs into a boson-like state. The theory is also used in nuclear physics to describe the pairing interaction between nucleons in an atomic nucleus.

But what happens when the temperature rises?

the cooper pairs break up with higher temperatures and incoherence reigns.

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"Coherence means that all ... " Could you please elaborate more on this, maybe with the help of math? And you say that superconducting wires are described by usual QM, hence their wave functions belongs to a kind of tensor product of state spaces of constituting free atoms. But what happens when the temperature rises? Why does this product space stops being an eligible state space describing the wire? – Yrogirg Aug 13 '12 at 16:05

Almost always, one cannot write a wavefunction for a macroscopic object even in principle, because if something is macroscopic, it means that it is usually entangled strongly with the environment (i.e., decohered by it, as others have pointed out). By definition of entanglement, if two systems are entangled, then the combined system cannot be written as the product of wavefunctions of each system, i.e., one cannot associate any wavefunction with the individual systems.

The exception is of course for very carefully prepared systems at low temperatures where you try to minimize the decohering influence of the environment. Perhaps a table near absolute zero.

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Does a real macroscopic body, like table, human or a cup permits description as a wave function?

Well, apart from the usual discussions always triggered by this question there's something even more fundamental:

Most people (except for a very small number of intelligent ones) don't even realize how much those "Macroscopic bodies" are just a metaphysical fiction of our own speech enhanced primate brains.

We have the ability to reduce enormously complex conglomerates of molecules into very short sequences of audio information like "table", "human" or "cup" Without such a huge data reduction factor of the visual information our eyes receive into just a few bytes, our brains wouldn't be able to do all the Object Oriented Processing it does. However, it comes at a cost of a build-in disability leading to faulty reasoning:

• If you talk about the wave-function of a human being, do you include or exclude his/hers:

Glasses? Cloths? Teeth-fillings? The stomachs-filling? The internal bacterial ecosystem? The internal air and other gasses? Pace-makers? Transplanted organs? Which of the 27 components of the daily vitamin pill? The energy of the radio waves propagating in the body? and so on, and so on.

There are just as many "macroscopic bodies" as there are opinions on this.

• In the hypothetical case of an interference experiment, which definition determines the interference?

Or can we have the human getting interfered away leaving only his Teeth-fillings? And if so, can we then also interfere a human away but leaving only one of his eyes and one of his legs? But we never considered this to be a macroscopic body because we don't have a special word for people missing one eye and one leg. Can we also have a random 63% of all the human body's atoms interfered away while the remaining 37% becomes a bloody mess? Why would such a messy interference be more or less likely as the other cases which we can describe because we have words like eye, leg and Teeth-filling? Does physics depend on our vocabulary?

• Does it even make sense at all to talk about the interference of macroscopic objects?

You can guess my opinion here.

Hans.

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good parameters at

Phys. Rev. Lett. 106, 220401 (2011)

Quantification of Macroscopic Quantum Superpositions within Phase Space

http://prl.aps.org/abstract/PRL/v106/i22/e220401

http://arxiv.org/pdf/1106.0062v2.pdf

the experimental actual limit is around 430 atoms.

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Where exactly is that information in the article? – Yrogirg Nov 9 '12 at 11:23

The wavefunction describes the quantum state of isolated systems in pure states. If the system is not isolated or is not in a pure state, wavefuntion theory cannot be used.

A table in a room is not an isolated system. A human is a typical example of dissipative system (interchanging both energy and matter with surrounds). Their quantum states are given by density matrices as correctly noticed by Landau & Lifshitz.

Macroscopic bodies can be counted as systems, but they are not isolated systems, the density matrix of any system (open, closed or isolated) only depends of the variables of that system, not of the variables of the surrounds or of other systems.

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