[The wave image below depicts a wave packet, the wave aspect of a particle; but unlike the physical particle this wave packet is not in physical space; it is a vibration in an abstract mathematical space known as Hilbert Space. The only physical significance of this vibration is that it is related to the probability of finding the physical particle within a given volume of physical space.]
Quantum mechanics is the physics of microscopic phenomena. Newtonian physics which is known as classical physics describes all natural phenomena in the scales observable by humans, namely the macroscopic world. But when we enter the world of atoms Newtonian mechanics breaks down. Atomic phenomena cannot be explained or understood in terms of laws of physics postulated by Isaac Newton. Physicists of the 20th century had to find new laws and postulates that could explain and predict atomic phenomena. This new physics which applies to small scales is known as Quantum Physics, or Quantum Mechanics. It has been shown that all the laws of Newton and classical physics can be derived from the laws of quantum mechanics. In fact, the laws of quantum mechanics are the fundamental laws from which Newtonian physics is derived as an approximation or a special case. In other words, our world is essentially quantum mechanical and not Newtonian, though we perceive only the Newtonian aspect of it at human scales: World is Newtonian when you look at it, but it is quantum mechanical when we are not looking at it. But this itself is a prediction of quantum mechanics: According to the principles of quantum mechanics the world at human scale must appear classical and Newtonian.
What distinguishes quantum mechanics from classical mechanics is the wave-particle duality. It is possible to explain all the strange phenomena of atomic realm by reference to the dual nature of elementary particles. Here I state and restate the fundamental features of quantum mechanics all of which are based on the wave-particle quality. The fundamental constituents of nature have a particle aspect and a wave aspect, and above all their particles aspect is entangled to their wave aspect which is the source of all the strangeness of quantum phenomena. Below I present the deep physical and philosophical implications of foundations. If it is too technical at certain points it is because their omission would damage our purpose. (FE=Fundamental Entanglement)
1) There is a fundamental entanglement between certain physical variables. The most important of these are position, momentum, energy, and time: Position x is fundamentally entangled with momentum p. Energy E is fundamentally entangled with time t.
2) The product of these fundamentally entangled variables is always of the dimension of classical action which has the dimension of angular momentum. [xp] = [Et]
3) Due to this FE there is always an uncertainty relation between any pair of fundamentally entangled variables whose products have the dimension of action. Thus, the uncertainty relations involve Planck’s constant h.
4) This FE is expressed in De Broglie equation which is also the expression of wave-particle duality:
Pλ = h
De Broglie’s equations tie the particle aspect to the wave aspect, the product of which has the dimension of action again: P stands for the momentum of the particle; landa stands for the associated wavelength, and h is Planck’s constant. De Broglie’s equation is also equivalent to the two following equations:
Here P stands for the momentum of the particle; k known as wave-number stands for a wave aspect related to wavelength; E is energy, and omega is the angular frequency. In both these formulas the left hand side is related to the corporeal aspect, and the right hand side is related to the wave aspect. The relation, the FE, is mediated by Planck’s constant.
5) Considering the classical relations for the phase of a wave, kx-ωt, if we replace the wave number and the angular frequency with their quantum mechanical counterparts we arrive at the following which relates the phase to action:
6) Beginning with only the De Broglie relation we arrive at quantum mechanics when we consider the wave phase above to be the phase of an abstract wave which is obtained by replacing the above formulate with the one in classical waves: kx-ωt =(px-Et)/ɦ
In the classical case we generally write wave as following:
Now insert the phase relation above to obtain the equivalent wave formula:
Ψ(x,t)=ei(kx-ωt) = ei(px-Et)/ɦ
The above which is obtained from imposing the De Broglie relation on classical waves is nothing but the quantum mechanical wave function, the solution to the Schrodinger equation which plays the role of Newton laws in the microscopic realm.
If here we define the action S to be: S = S(x,t) = px-Et, then we can write the wave function as following:
This is the most general form of quantum mechanical wave function as the solution to Schrodinger equation. As a matter of fact, this wave is the solution to Hamilton-Jacobi equation in classical physics. If we differentiate the above wave function with respect to time, and replacing E with the Hamiltonian H, then we derive the following non-linear partial differential equation:
This is none other than the famous Hamilton-Jacobi equation from which we can derive the Schrodinger equation:
This is the wave equation for quantum particles. The peculiar fact of wave-particle duality is that the particle aspect is related to a wavelength of a non-physical wave. The wave aspect of phenomena, which is related to vibrations, is not a physical wave; it is a wave in an abstract Hilbert Space, and hence not observable. In other words, the observable aspect of phenomena is associated with the vibrations of a non-observable wave. This is the wave-particle duality that is behind all strangeness of quantum mechanical world. We can express the De Broglie formula of wave-particle duality in a more philosophical way:
The manifest is associated with the vibrations of the unmanifest.
This expression, though still different from saying that “the manifest is the vibration of the unmanifest,” is a restatement of Advaita Vedanta metaphysics. But it is also the very definition of string theory.
More details about this subject can be found in my book Nondual Perspectives on Quantum Physics.