In quantum mechanics, we have many different postulates.
In classic mechanics, we have different laws.
As far as I know, physics's laws are temporarily correct until an anomaly. But what is the RIGOROUS difference between a "Postulate" versus a "Law"? They seem to be the same for me.
The QM postulates define the "rules" used in the mathematical formalism of Quantum Mechanics. From these postulates you can determine experimentally verifiable predictions, but the postulates themselves are not (or maybe even cannot) be experimentally verified. This can be seen by looking at the postulates (I am getting these from Shankar's Principle of Quantum Mechanics, Second Edition).
- The state of a particle is represented by a vector $|\psi(t)\rangle$ in a Hilbert space.
- The independent variables $x$ and $p$ of classical mechanics are represented by Hermitian operators $X$ and $P$ with the following matrix elements in the eigenbasis of $X$ $$\langle x|X|x'\rangle=x\delta(x-x')$$ $$\langle x|P|x'\rangle=-i\hbar\,\delta'(x-x')$$ The operators corresponding to dependent variables $\omega(x,p)$ are given Hermitian operators $$\Omega(X,P)=\omega(x\to X,p\to P)$$
- If the particle is in a state $|\psi\rangle$, measurement of the variable (corresponding to) $\Omega$ will yeild one of the eigenvalues $\omega$ with probability $P(\omega)\propto |\langle\omega|\psi\rangle|^2$. The state of the system will change from $|\psi\rangle$ to $|\omega\rangle$ as a result of the measurement.
- The state vector $|\psi(t)\rangle$ obeys the Schrodinger Equation $$i\hbar\frac{\text d}{\text dt}|\psi(t)\rangle=H|\psi(t)\rangle$$ where $H(X,P)=\mathcal H(x\to X,p\to P)$ is the quantum Hamiltonian operator and $\mathcal H$ is the Hamiltonian for the corresponding classical problem.
Notice how these postulates are all mathematical in nature. There isn't a way to actually test for a state vector, or to test what the matrix elements of an operator are. Therefore, these are postulates. They are assumed to be mathematically true, and we then see what else must be true because of these postulates. Then we can test the predictions made by the theory.
Note that the same exact thing is true for classical mechanics. The corresponding postulates are given in the same book:
- The state of the particle at any given time is specified by the two variables $x(t)$ and $p(t)$, i.e., as a point in a two dimensional phase space.
- Every dynamical variable $\omega$ is a function of $x$ and $p$: $\omega=\omega(x,p)$
- If the particle is in a state given by $x$ and $p$, the measurement of the variable $\omega$ will yield a value $\omega(x,p)$. The state will remain unaffected.
- The state variables change with time according to Hamilton's equations $$\dot x=\frac{\partial\mathcal H}{\partial p}$$ $$\dot p=-\frac{\partial\mathcal H}{\partial x}$$ These are mathematical postulates of classical mechanics. We can use them to derive experimentally testable "laws" linking the math to the physical world.
Therefore, the postulates are mathematical assumptions. Laws, (or rules, etc.) are then predictions from the postulates that that can be experimentally verified. Additionally, we usually give the "important" results the title of "law". For example, the time $t$ it takes for an object to fall a distance $h$ from rest close to the Earth is given by $t=\sqrt{2h/g}$, but we don't call this the "law of falling time", or something like that, even though we can experimentally confirm it.
As also mentioned in other posts, laws are not necessarily always true. They might only be true when a certain number of criteria are true. Yet when they are true, they are important and sort of constitute a "law" in the sense that it is something that must always be true if it can be true. For example, Newton's law of gravity is not always valid, yet it can explain many phenomena when it is valid. Additionally, some laws were "created" before the theoretical underpinnings were created (like Kepler's laws).
In (a possibly imprecise) summary, "postulates" are just something we assume to be mathematically true. If the predictions from the postulates turns out to be false, then we need to change one or more of the postulates. A "law" is something that can be derived from the postulates whose truth depends on how well it describes the physical world around us.
A Scientific Law is a statement that describes a phenomenon, and is backed up by repeated experimental evidence. They are developed from data.
A postulate is a statement made before a theoretical approach is outlined. The idea behind a postulate is that if it were the case that the postulate were not true, then the following theory would be invalidated.
In other words, a postulate is assumed to be true, and if it is not true in a given circumstance, then the theory that falls from it no longer works, while a Law is rigorously tested.
There is no exact authoritative difference, and the two words overlap considerably in meaning. A postulate is something assumed to be true, like an axiom. In principle a law should represent hard facts. However, historically we have found that so called laws have proved to be approximations- for example, Newton's law of gravitation, or Kepler's law of planetary motion- so the distinction between postulate and laws more a question of semantics than physical significance.
In my opinion the difference is historical. There are the laws of Newtonian mecanics: observations encapsulated into specific formulas. The four "laws" of electricity and magnetism were devised to explain electrical and magnetic phenomena. Maxwell used them in the same way as we use postulates in quantum mechanics, while also incorporating the laws of conservation of energy, momentum and angular momentum in addition to the quantum mechanics postulates and the quantum numbers conservation coming from observation that lead us to the standard model.
And lets not forget the principles, like Heisenberg's uncertainty which started from observations and ended being axiomatic ( in the commutators anticommutators of quantum field theory).
All are axiomatic for the given theory and models in the theory. They pick up the solutions of the general differential equations that have to do with physical observations. Some laws are more general and some confined to the specific theory, but they are necessary in order to have theories that are predictive for new experimental set ups.
