No. "In classical mechanics, observables are simply dynamical variables such as position, momentum, and functions of position and momentum...The measurement (or observation) of a classical observable

is a well-defined physical operation that, when performed on the system, yields a single real number, the so-called value of the observable...a quantum observable is mathematically represented by a linear Hermitian operator on an infinite dimensional separable Hilbert space and there is one such operator for each quantum observable such as the position, the momentum, the energy, and so on."

Debnath, L., & Mikusiński, P. (2005). Hilbert spaces with applications. (3rd Ed.) Academic press.

or from Shankar's classic intro textbook on QM:

"The independent variables x and p of classical mechanics are represented by Hermitian operators X an P with...matrix elements in the.." and so on (it's a pain not to be able to use LaTex here)

QM is a statistical mechanics, whether it is "irreducibly statistical" (per MP Sillverman) or whether some relative state interpretation is correct, the actual mathematical components "dynamic" only in the quite literally mathematical sense. Also, relativistic formulations of SE exist, but as the incompatibility between relativity and QM is one of the central unresolved issues in modern physics, it is even less readily related to the manner in which equations relate to observables/measurements in classical physics than in QM.

Short answer: it's not wrong, it's just deterministically probabilistic (yes, the paradoxical formulation was deliberate; that's QM).

@Andrew, Thank you for the clear answer.

Dear Andrew,

I know the mainstream conclusions of the QM. However, I always feel that the probability or the statistical solution must be an equilibrium result, otherwise there is no a meaningful probability density!

As you said, my consideration is just the reason why the QM and Relativity can not be easily put into a same framework! The Dynamics or the Force can be transformed into a geometrical curvature of space in a general relativity theory, so if we can derive a Force from QM and transfer it into a curved space, then we can treat the Time of SE as a 4th dimension and reversely get a second derivative of time when we transform SE back to a flat space. It is just a guess, I don't know!

Andrew,

Agree with you.

We know the quantum mechanics gives us probability! That is a very natural result because we only can give statistical information for a system which is consist of 10^23 particles at least and moves very quickly! The only method we use for such a system is statistics! No macroscopic mechanics for quantum mechanics and the thermodynamics!

Basic quantum mechanics is for a single particle. Here

one tries to locate a particle in space. Schrodinger Equation

describes non relativistic quantum mechanics whereas Dirac

equation describes relativistic quantum mechanics. Klein-Gordon

equation describes relativistic quantum mechanics for spin

zero particles. Let us now discuss the non relativistic case

for simplicity.

Using the method of separation of variables Schrodinger

equation can be reduced to a time independent equation.

Solution of the time part is oscillatory. The solution of space

part gives probability amplitude. This is a complex function

of space coordinates only. Thus it cannot give probability

by itself. The mos square of wave function psi^* psi gives

probability as a function of space coordinate.

You have made a very valid point. Where is the force ?

Actually we specify potential in schrodinger equation

not force. Potential is an integral of force and force is

a space derivative of potential with a minus sign.

Dear Professor Brahmachari:

I think it is important to clarify some of the meaning behind terms and descriptions you use, as it is easy to (for example) to misunderstand how the term "space" is used in quantum physics and this is not simplified by ambiguous use of terminology. For example, while of course you are correct that the Schrödinger equation should not be interpreted as a probability, neither should the mod square of psi, i.e., |ψ(x)|^2. But describing this as the probability amplitude is, I have found, not particularly useful for scientists unfamiliar with QM whereas "probability distribution", "probability distribution function", or even simply p.d.f. is far more easily understood even by physicists who don't use QM, QFT, QCD, etc. Even more importantly, the retention of the term "particle" in modern physics despite its new use to describe something fundamentally different from anything that existed in classical physics is nothing if not misleading (at least to the majority, who are familiar with the terms in the classical sense but not as they are used in quantum physics. This is particularly true when one is describing a "particle' in terms of spatial coordinates that are neither localized nor residing in some Euclidean nor even observed (or rather, the "observable" exists but unlike classical physics is a mathematical operation not a measurement).

Despite descriptions throughout the physics literature on the dynamics of quantum systems, these are mathematical dynamics. The statistical structure is incredibly successful as a theoretical framework, but what is the point about asking how force acts on a non-localized particle (a contradiction in terms, like a "living cat that is dead") in infinite dimensional Hilbert space? There are reasons to do so, but as far as explanations are concerned the only point I can see is to mislead.

Quantum mechanics is valid in subatomic range that is

10^-8 cms and less. In that domain wave particle duality

is in force. Thus even if I say it is the mechanics of a single

"particle" system you can always think of the particle as a

wave whose wavelength is given by de Broglie wavelength.

In fact if you notice the derivation of Schrodinger equation

from (first principles) you will find that the de Broglie

wavelength is built in into the dynamics.

Probability distribution function has the property

integral P(x) dx=1

range of integration is

-infty < x < infty.

This physically means that probability of finding the

particle some where in the interval -infty < x < infty

is always one. That is the particle must exist

somewhere. This is a property of non relativistic

quantum mechanics only. In relativity things change

drastically and one has to face complications due to

the presence of valence particles and sea particles.

Many thanks! The above contribution alone would be invaluable, but is doubly so when taken together with the one I initially objected to because (while in no way inaccurate and through no fault of your own) it could be misleading. Now we have the best of both worlds: the explanation using the language of modern physics and a kind of "translation" that doesn't sacrifice accuracy but does make more accessible (in my opinion) through the use of the more universal language of mathematics.

Andrew,

Here is a picture of wave function and probability distribution

for various states of the Harmonic oscillator. Notice from the

plot that the wave function can be negative but probability cannot.

In the ground state the maximum value of the probability is

at x=0. This is also the classical expectation. If you roll a ball

it will settle down at the bottom of the potential well. The first

excited state has two peaks in probability distribution and both

are shifted from the origin. Such a thing has no classical analog

and can happen exclusively in quantum mechanics. This means

that there are two positions (both are away from the origin) where

the particle can settle down in spite of the fact that these places

are not the ones where potential is minimum.

What "particle"? It isn't that the state ("excited state" or not) has no classical analogue, but that the state of quantum systems have no physical analogue (that is, they lack any one-to-one correspondence with some physical property, measurement, etc.). The state, as in classical physics, is described mathematically, but unlike classical physics for all intents and purposes it has no existence other than mathematically. Upon measurement , a state is "observed" by a mathematical function, not by measurement itself nor is that state ever in any classical sense observed but rather derived. Whatever the "particle's" ontological status, the closest we get to it is via a mathematical representation of a "wave" that becomes localized only once we fundamentally alter the "state" of the "particle". So, where in all of this jumble of particles that have none of the dynamics of particles, observables that are not observable properties and have no known physical analogues, and the use of probability distributions to describe what we only know of through probabilities, do we determine that "the particle can settle" anywhere without ever knowing anything about a particle other than that a system that has no properties of a particle is, when non-trivially disturbed, detected locally? My previous response was not a request for a elementary simplification of incredibly basic aspects of quantum physics, but for a more precise (or less misleading) use of terminology for those not used to the language of classical physics in quantum physics despite fundamental incongruities. Your previous response, in my opinion, covered this. So I don't understand the latest addition which adds more confusion than the first response I objected to (for the reasons I gave then).

No classical analog means no analog in classical mechanics.

That does not mean that they do not exist. They exist and that indeed is

the reason for saying quantum mechanics is the correct description

of atomic and subatomic scale physics.

This question also have a part on relativity effects in

quantum mechanics.

The Schrodinger equation is a non relativistic equation.

Consequently it does cannot describe particles moving

with relativistic velocities. However it can describe quite

accurately non relativistic phenomenon in Atomic Physics.

However, when one attempts to describe relativistic particles

such as the elementary particles (electron, muon etc.) then

one has to use the Dirac equation.

P.A.M. Dirac formulated the quantum theory of relativistic

particles in 1928, a few years after Schrodinger equation

was discovered in 1926.

Thanks for the above insightful answers to the question. I think the microscopic particles will follow the macroscopic laws when the particles are enlarged to the size of a macroscopic ball or our human beings shrinks to a particle size.