# Is there a quantum state for a large system

My understanding of quantum mechanics is that the state of a system is represented by a vector in multidimensional complex vector space. Is there, in principal, a state vector that represents a large, classical object such as, say, a cheeseburger, at an instant in time? If so, what is the physical meaning of that "state"?

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–  Qmechanic Oct 26 '12 at 11:01
The context is the same but the goal of the question is different - the meaning of the state of a cheeseburger, rather than whether it is a wave function. –  Arnold Neumaier Oct 27 '12 at 9:43

Quantum states of macroscopic systems are routinely considered in statistical mechanics. They used to derive both the thermodynamic properties of macroscopic materials and the way they deform and respond to external forces.

However, these macroscopic quantum states are never described by state vectors (pure states) but always by density matrices (mixed states).

In practice, quantum derivations are restricted to simple and fairly homogeneous materials, because of the difficulty to work numerically with more complex states. But there is no limitation in theory of the size to which quantum mechanics applies; in particular, it would apply to a cheeseburger if one would model it as an $N$-particle system with $N$ of the order of $10^{25}$. For example, it is applied to derive the conditions of the hydrodynamic reactive flow in the interior of the sun. (Although the sun's apparent size is similar to that of a cheeseburger, its true size is much bigger.)

The state of a quantum system (no matter whether consisting of a single qubit or of $10^{25}$ atoms) describes all properties of a system that can possibly be measured.

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In principle yes, but the problem is that any system cannot be isolated from it's environment and interactions with the environment affect the wavefunction of your system. This is the principle behind decoherence. The larger the system the more quickly it interacts with the environment, and for something as large as a cheeseburger the interaction is so rapid we have no chance of observing any quantum behaviour.

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Why can't we isolate a burger from the environment? –  MBN Oct 26 '12 at 9:13
It's impossible to isolate anything from it's environment. Even in a peffect vacuum there are still CMB photons flashing around. I suppose a perfect vacuum at absolute zero might do it, but that's not very practical. Anyhow, and electron in a Young's slits experiment doesn't have many degrees of freedom so it's reasonably easy to stop interactions destroying the behaviour you're trying to observer. A cheesburger has so many degrees of freedom the probability of interaction is effectively 100% and it's impossible to prevent the interaction destroying the behaviour you're trying to observer –  John Rennie Oct 26 '12 at 9:45
Yes, but it is a point in principle, a burger in empty space, no matter how unrealistic this is, is isolated, and even so it will have classical (or very nearly classical) behaviour because of its many internal degrees of freedom. –  MBN Oct 26 '12 at 10:21
No, if I understand John correctly he is saying that a burger will in principle follow quantum behaviour regardless of the number of degrees of freedom but that in practice this cannot be observed because of the fast interaction with the environment (and this behaviour is still quantum actually, it just happens also to coincide with the classical physics). –  SMeznaric Oct 26 '12 at 18:47
I understood that, but the question is not about what is possible in practice. It is a principle question. A burger in empty space will by isolated, because there is nothing to isolate it from, will it show quantum behavior and what? –  MBN Oct 27 '12 at 8:56

You make two different answers: "Is there a quantum state for a large system?" and "Is there, in principal, a state vector that represents a large, classical object such as, say, a cheeseburger, at an instant in time?". The answers are "yes" and "no", respectively.

State vectors $|\Psi\rangle$ only represent pure states and therefore do not apply to non-isolated systems as cheeseburgers. The quantum state of such systems is given by a density matrix $\rho$.

The physical meaning of those states is the ordinary, they represent the properties of a given system in condensed form. Instead giving a list of properties you can compute any property of the cheeseburger using this state [*].

[*] In principle! Of course, the computation is very difficult and plagued with technical difficulties.

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