Friday, March 28, 2014

The Wave Function 01

— 1. The Wave Function —
The purpose of this section is to introduce the wave function of Quantum Mechanics and explain its physical relevance and interpretation.

— 1.1. The Schroedinger Equation —
Classical Dynamics' goal is to derive the equation of motion, $ {x(t)}$, of a particle of mass $ {m}$. After finding $ {x(t)}$ all other dynamical quantities of interest can be computed from $ {x(t)}$.
Of course that the problem is how does one finds $ {x(t)}$? In classical mechanics this problem is solved by applying Newton's Second Axiom

$ \displaystyle F=\frac{dp}{dt} $
For conservative systems it is $ {F=-\dfrac{\partial V}{\partial x}}$ (previously we've used $ {U}$ to denote the potential energy but will now use $ {V}$ to accord to Griffith's notation).
Hence for classical mechanics one has

$ \displaystyle m\frac{d^2 x}{dt^2}=-\dfrac{\partial V}{\partial x} $
as the equation that determines $ {x(t)}$ (with the help of the suitable initial conditions).
Even though Griffith's only states the Newtonian formalism approach of Classical Dynamics we already know by Classical Physics that apart from Newtonian formalism one also has the Lagrangian formalism and Hamiltonian formalism as suitable alternatives (and most of time more appropriate alternatives) to Newtonian formalism as ways to derive the equation of motion.
As for Quantum Mechanics one has to resort the Schrodinger Equation in order to derive the equation of motion the specifies the Physical state of the particle in study.

— 1.2. The Statistical Interpretation —
Of course now the question is how one should interpret the wave function. Firstly its name itself should sound strange. A particle is something that is localized while a wave is something that occupies an extense region of space.
According to Born the wave function of a particle is related to the probability of it occupying a region of space.
The proper relationship is $ {|\Psi(x,t)|^2dx}$ is the density probability of finding the particle between $ {x}$ and $ {x+dx}$.
This interpretation of the wave function naturally introduces an indeterminacy to Quantum Theory, since one cannot predict with certainty the position of a particle when it is measured and only its probability.
The conundrum that now presents itself to us is: after measuring the position of a particle we know exactly where it is. But what about what happens before the act of measurement? Where was the particle before our instruments interacted with it and revealed is position to us?
These questions have three possible answers:

  1. The realist position: A realist is a physicist that believes that the particle was at the position where it was measured. If this position is true it implies that Quantum Mechanics is an incomplete theory since it can't predict the exact position of a particle but only the probability of finding it in a given position.
  2. The orthodox position: An orthodox quantum physicist is someone that believes that the particle had no definite position before being measured and that it is the act of measurement that forces the particle to occupy a position.
  3. The agnostic position: An agnostic physicist is a physicist that thinks that he doesn't know the answer to this question and so refuses to answer it.
Until 1964 advocating one these three positions was acceptable. But on that year John Stewart Bell proved a theorem, On the Einstein Podolsky Rosen paradox, that showed that if the particle has a definite position before the act of measurement then it makes an observable difference on the results of some experiments (in due time we'll explain what we mean by this).
Hence the agnostic position was no longer a respectable stance to have and it was up to experiment to show if Nature was a realist or if Nature was an orthodox.
Nevertheless the disagreements of what exactly is the position of a particle when it isn't bening measured, all three groups of physicists agreed to what would be measured immediately after the first measurement of the particle's position. If at first one has $ {x}$ then the second measurement has to be $ {x}$ too.
In conclusion the wave function can evolve by two ways:

  1. It evolves without any kind of discontinuity (unless the potential happens to be unbounded at a point) under the Schrodinger Equation.
  2. It collapses suddenly to a single value due to the act of measurement.
The interested reader can also take a look at the following book from Bell: Speakable and Unspeakable in Quantum Mechanics


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