# What is a state in physics?

What is a state in physics? While reading physics, I have heard many a times a "___" system is in "____" state but the definition of a state was never provided (and googling brings me totally unrelated topic of solid state physics), but was loosely told that it has every information of the system you desire to know. On reading further, I have found people talking of Thermodynamic state, Lagrangian, Hamiltonian, wave-function etc etc which I think are different from one another. So in general I want to know what do we mean by state in physics and is there a unique way to describe it?

• en.wikipedia.org/wiki/State
– user83548
Dec 11 '15 at 18:51
• exactly my question, there are so many definition for it, why shouldn't be there only one state. Dec 11 '15 at 18:55
• There are different notions of state because in physics, one often uses different mathematical models to describe systems in different contexts (e.g. different length scales), and in these different models, the state of a system is described by a different kind of mathematical object. Dec 11 '15 at 18:57

The definition of a state of a system, in physics, strongly depends on the area of physics one is dealing with and it comes as one of the initial definitions once such underlying theory has to be set up. In particular one has:

1. classical mechanics: a state of a system is a point $m\in TQ$ (or equivalently $T^*Q)$ in the tangent bundle of the configuration space (or the phase space, respectively). Such state is identified on a local chart with a set of coordinates $(q_i, \dot{q}_j)\in\mathbb{R}^N$ representing positions and velocities of all the particles at a given time $t$. Such description is equivalent to require the uniqueness of the solution of the Newton's equations once initial conditions are specified.

2. thermodynamics: a state is a set of extensive variables $(X_1,X_2,\ldots,X_N)$ that uniquely specify the value of the entropy function as $S(X_1,X_2,\ldots,X_N)\in\mathbb{R}$. Such variables represent the macroscopic extensive parameters (as volume, number of particles, total energy and so on and so forth) from which one can derive the corresponding associated intensive variables taking derivatives of the entropy as, for instance, $p=T(\partial S/\partial V)$ and similars.

3. quantum mechanics: a state is any element $|\psi\rangle\in\mathcal{H}$ of a Hilbert space together with a collection of self-adjoint operators $(A_1,\ldots,A_n, H)$. Special role is played by the Hamiltonian $H$, whose action mirrors classical mechanics giving the evolution in time of the state $|\psi(t)\rangle$. A collection of states (i. e. an ensamble) is instead described by a density matrix $\rho$ such that the expectation value of any operator on the ensamble can be defined as $\langle O \rangle = \textrm{tr}(\rho O)$.

4. field theories: very subtle as the definition of a state strongly depends on the theory at hand (quantum gravity, loop quantum gravity, string theory, QFT all have slightly different definitions of states).

EDIT: as per the suggestions in the comments below, more complex states and descriptions may and do arise, therefore the above is supposed to only be taken as general walkthrough.

• I would argue, that your definition of thermodynamical state is a bit restrictive. One often works in ensembles with thermodynamic potentials, that do not necessarily only depend on extensive quantities (e.g. the canonical ensemble with $F(T, V, N)$). Plus there is a typo in an equation: $\partial_V S = p / T$. It also wouldn't hurt to mention the density matrix in quantum mechanics (as it is more general than a wave function). But this is nitpicking, so +1 anyway. Dec 11 '15 at 19:51
• Good answer. Just some remarks/questions: for point 2. I do not think you need to give such a specific role to the entropy function; it's a state function like the energy for instance. For point 3. is it really any element of Hilbert space? Shouldn't there be a set of observables associated to it somehow? I mean, if you take the Hilbert space of spin states of a spin half particle, good luck to get any information on the position of the particle. Dec 11 '15 at 20:05
• Yes, I agree with your remarks and I have edited my answer accordingly. Dec 11 '15 at 20:31
• For mechanical systems perhaps it would be better to take the coordinates and the impuls instead of coordinates and velocities? Dec 12 '15 at 6:46
• That is possible only when the Hessian of the Lagrangian is invertible. If so, yes. Dec 12 '15 at 7:25

Our physics prof once put it informally that way:

A state is a set of variables describing a system which does not include anything about its history.

The set of variables (position, velocity vector) describes the state of a point mass in classical mechanics, while the path how the point mass got from point $A$ to point $B$ is not a state.

• That's nice and crisp. Dec 11 '15 at 20:58

Informally speaking, a complete description of a physical system is referred to as its state. Completeness of the state of a system means that it provides all the possible information about the system, i.e. everything that can be possibly known about the system has to be contained in the specification of its state.

Every physical theory is ultimately based on the following three fundamental postulates:

• The postulate which defines the way we describe a state of a system.
• The postulate which specify what kind of information about observables, i.e. measurable properties of the system, is contained in the description of its state.
• And the postulate which provides us with a law that governs the time evolution of the system and allows us to predict its future state given the current one.

And in view of these fundamental postulates the meaning of completeness of the description provided by the state of a system is that all possible information about observables should be contained in the specification of the state and it should also be possible to use it to obtain all possible information about observables at any time in the future.

To make the definition of a state more formal and less vague we have to at least distinguish between classical and quantum theories because concrete manifestations of the above mentioned postulates for these two families of physical theories differ significantly. For instance, the meaning of the "all possible information about observables" phrase in quantum theories is quite unconventional from the classical point of view. And the rigorous definitions сan be given only for a particular physical theory since different mathematical objects are used to represent the state of a system in different theories as discussed in details in the answer given by Gennaro Tedesco.

• Good answer, but I would add that a state does not necessarily need to contain every bit of information that can possibly be known about the system. It just needs to contain all information that is relevant for the particular model being used. For example, a thermodynamic state contains only pressure, temperature, and number of particles, and completely ignores the individual motions of the particles, even though those could be known (in some thought-experimental system). Dec 12 '15 at 18:25
• @DavidZ, true, but overall the question of what properties do we "attach" to a system is more about the system itself then about its state. And in that respect the notion of a system is even more metaphysical (at least when taken in general) than that of its state. Again, there will be at least a big difference between quantum and classical theories... Dec 12 '15 at 18:51
• This is probably my favor answer, but I would like to ask about "physics laws give us time evolution of states", how about general relativity? Jan 5 '16 at 0:25

State in physics is a usefully ambiguous term, which is used in different ways in different fields; it's probably best understood in opposition to dynamics: state is static, and says nothing about motion; whereas dynamics tells you how one state evolves into another.

For example, in classical picture a state would be both the position and the momentum of a particle; knowing all the states of all the particles in the universe gives a snap-shot of the universe, or the state of the universe; but knowing all this does not tell you the state at some future moment - for this one also needs to know the dynamics - that is, the equations of motion; or simply how one state changes into another.

Another example, would be QM; there a state encodes the quantum system at hand, and (in the Schrodinger picture), are time-independent; the dynamics would then be given by Schrodingers equation which says how the state - the wave or potential - evolves.

(There is here, though the crucially complicating factor of observables, and acts of measurement).

However, it's also worth noting that there is another picture, the Heisenberg picture, where states do not evolve but observables do - this picture is more useful for the move into relativistic QM and/or QFT.

Roughly, you describe state in physics as a series of particular values assigned to the different magnitudes that you can measure of the system, i.e. a value for the energy, pressure, temperature, ... or any magnitude that you're interested in.

So the state is a way of describing which properties has the system that you're going to study.

• I'm not sure if this really covers it, because e.g. a quantum state is not a measurable property. In quantum mechanics the state actually contains more information than what you can ever measure. Dec 12 '15 at 18:26