Asked by: Robyn Scott

The basic picture of the Newtonian mechanics can be described as follows. There are particles, with specified positions and velocities, interacting with each other by means of forces. There are several kinds of forces in Nature. These forces can act between two particles, and their strength and direction depend on the positions and the velocities of the particles. Second Newton's Law connects between the forces acting on a particle and the resulting acceleration. Knowledge of the positions and the velocities of all the relevant particles at a specific moment of time allows to predict the positions and the velocities at any other time.

The laws which govern the behavior of the sub-atomic particles are completely different. It is impossible to assign a specific position and velocity to a particle. Each particle can be in a superposition of different states, which means that in some sense it is located at the same time in a whole region of space and has a whole range of velocities. If you measure the position (or the velocity) of the particle, you just get one of the values from that range, in random (possibly with different probabilities for each value). However, this is NOT because the particle actually HAD that position and you just hadn't known that, but the particle really HAD a whole range of positions the moment before the measurement. This is something strange and beautiful.

The ability of the particle to be in several different states simultaneously results in a well-known wave-particle duality: the sub-atomic particles (electrons, neutrons and other) can behave like waves and show interference. Suppose we have a particle source aimed towards a wall with two slits where the particles can pass, and a detecting screen beyond this wall. First we allow the particles to pass only through one of the slits, and then only through the second one. In a third experiment, the particles can pass though both the slits. When looking at the results, the results of the last experiment seem to be completely unrelated to the results of the first two. This happens because when particles are allowed to pass through both slits it's not that some of them pass through the first slit and some of them through the second one, but in some sense each particle passes through both of them. On the detecting screen we see a picture identical to one which is obtained from interference of waves.

The theory which is able to describe the sub-atomic particles is the Quantum Mechanics. In Quantum Mechanics, a system (sometimes a single particle) can be described by a wave function (or by a vector in a multi-dimensional space). The information contained in the wave function is just the weight of each possible state in the current state of the system (actually there is something more: the phase of each state, which I will not discuss here). The wave function allows to calculate the possible results (and their probabilities) of any measurement which can be performed on the system. The development of the wave function in time is described by Schroedinger's Equation (which is analogical to Newton's Laws), given the initial wave function and the so-called Hamiltonian operator of the system (which takes into account the interactions between the particles).

Newtonian Mechanics turns out to be a private case of Quantum Mechanics. In some situations, the behavior of the sub-atomic particles can be described well enough by Newton's Laws, but the more general theory is the Quantum Mechanics. To see the beauty and understand the basics of Quantum Mechanics, I would recommend reading about it in 'Feynman's Lectures on Physics' by R. P. Feynman or in 'The Principles of Quantum Mechanics' by P. A. M. Dirac.

Answered by: Yevgeny Kats, Physics Grad Student, Bar-Ilan University, Israel

'As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.'**Albert Einstein**

(*1879-1955*)

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