Roger's Equations

In "The Lagrangian - A Qualitative Analysis" and "The Hamiltonian - A Qualitative Analysis", I have discussed the principle of least action and how it can be used to find the equations of motion for a system.

In "The Hamiltonian - Quantum Mechanics", I explain why this idea breaks down because of the uncertainty principle. I also explain how the wave-function is used to find results in a system where the position and momentum are not exactly known, but are in fact expressed as probability distribution.

Next I would like to talk about Richard Feynman's work on how the principle of least action works in quantum mechanics. First however I'd like to touch upon "calculus of variations".

Calculus of Variations

Integrals are nothing more than glorified summations. You can call it the area under a curve if you wish, but that is just another way of saying the same thing. It's no accident the symbol ∫ looks like a stretched out S. Sometimes in the confusion of calculus it's easy to forget that when you do an integral, what you're really doing is adding stuff up.

I've been talking about the principle of least action. As with most words in Physics, the word action has a very specific meaning, it is defined to be:

Where S is action, t is time, and L is the Lagrangian and defined as:

L = T - V

where T is kinetic energy and V is potential energy

so

Action - the sum of the differences between the Kinetic and Potential Energies at each instantaneous moment in time for a selected length of time.

and

Principle of Least Action - In nature, the motion through space over time is such that the action is minimized, meaning:

S= ∫ T-V dt = minimum

where S is action, T is kinetic energy, V is potential energy, and t is time.

In calculus we use derivatives to help us find the point that is the minimum (local or global) for a particular curve.

In order to minimize action though, we are looking for a curve that is a minimum, not just a point. Just as there are an infinite number of points on a curve, there are an infinite number of curves (paths) between two points, and we want to find the minimum action path. To find that minimum, you have to use Calculus of Variations. The Euler-Lagrange equations are simply the conditions, in terms of the Lagrangian, that produces minimum action for the system.

where L is the Lagrangian, t is time, and q and qdot are generalized position and velocity variables.

The Hamiltonian

The Lagrangian is a function of Kinetic and Potential Energies that is expressed in terms of Position and Velocity. The Hamiltonian is a function of Kinetic and Potential Energies that is expressed in terms of Position and Momentum. The Hamiltonian, which is the total energy of a system, is derived from the Lagrangian:

where p is momentum, and T+V, the total energy, is the Hamiltonian.

The Hamiltonian of a system can be derived from the Lagrangian of a system with the equation:

where q and qdot are generalized position and velocity variables, p is generalized momentum variables, t is time, and L is the Lagrangian.

Quantum Mechanics

The problem with Quantum Mechanics is you can't have exact position and momentum at the same time. This is descibed by the uncertainty principle:

where Δx is the standard deviation of position, Δp is the standard deviation of momentum, and hbar = h/2π where h is Planck's constant.

Since Kinetic Energy is momentum dependent energy, and Potential Energy is position dependent enegy, it is impossible to calculate the Lagrangian of a system in the classical way (with an exact T and an exact V at the same time). However, since there is a relation between the standard deviations (uncertainties) of position and momentum, systems can be described by wavefunctions. Wavefunctions contain the possible combinations of position and momentum and their relative likelyhoods. One can find the likelyhood of a "state" of a system by operating on the wavefunction by projecting that wavefunction on a state and getting the corresponding probability amplitude, which is squared (modulus squared) to produce the probability that the wavefunction can be found in that state. Thus we have a description of a system in which we can calculate physical likelyhoods while preserving the uncertainty between position and momentum as described in the uncertainty principle.

Feynman - Many Paths

Just as the Uncertainty Principle forbids exact position and momentum, it also forbids exact paths between two points in time. How could it not? After all, before when we were talking about action we said that it was the sum of all the instantaneous (T-V) over a given length of time. Since we can't know T and V exactly at the same time (since T is momentum dependent and V is position dependent), we certainly can't find a sum of them exactly.

Just as there are many (infinite) possible states of a system, there are also an infinite number of paths between two points in time that represent the minimum action. Just as each state has a particular probability associated with it, so too does each path have a probability associated with it.

Richard Feynam was the first to realize this idea of "many paths". Feynman realized that in order to describe a system, one must consider all the probabible paths it could take. Obviously some paths are more likely than others. Inclusion of as many of the paths as possible, specifically the paths that are most probable, leads to a greater accuracy in the calculation of the system. This is the origin of Feynman Diagrams, they are a way of tracking the different paths of a system, generally the most probable and thus most important to accurately calculating the system.

What does it all mean?

"How does an electron know which path to take?" will often be answered by "It takes every path". You'll read that as an ad hoc explanation to the many paths idea , and it sounds reasonable enough, but I never liked it. It implies more certainty than I think there really is. After all, the question itself seems to imply a known starting point and ending point, which is impossible due to the uncertainty principle. An electron can only have a likely starting point and a likely end point.

Special thanks to the following websites:

http://en.wikipedia.org/wiki/Feynman
http://www.mathpages.com/home/kmath320/kmath320.htm
http://en.wikipedia.org/wiki/Principle_of_least_action
http://www.eftaylor.com/leastaction.html