Quantum Mechanics - how did Feynman derive this postulate?

Question

Feynman proposed a path integral formulation of quantum mechanics based upon the principle of least action.

His postulates were:
1. The probability for any fundamental event is given by the absolute square of a complex amplitude.
2. The amplitude for some event is given by adding together all the histories which include that event.
3. The amplitude a certain history contributes is proportional to
exp((i / hCross) * action of that history )

My question is this - How did Feynman come up with his third postulate. Did he derive it? If so, how?

Yes, the exponential is a phase term for incorporating the action principle into quantum mechanics.
My question is, why this form for the phase? In other words, where did it actually come from, or get derived from?

Answer 1

Just to be more explicit as to how this derives from conventional quantum mechanics - since that's what you seem to be asking - consider a system with one degree of freedom and a Hamiltonian:

H = P^2/(2M) + V(q)

The time evolution operator which moves the system forward infinitesmally in time (dt) is:

exp(-iH dt) = exp(-i [P^2/(2M) + V(q)] dt)

In general any operator O can be expressed as a matrix in q values . The part exp(-iV(q)dt) is already diagonal and it doesn't take too much imagination to wonder what that first (kinetic) term would look like expressed as a matrix. It turns out to be very similar to the kinetic term in the Lagrangian (there's some philosophy required since the integral in the following is technically divergent):


= \integral e^(ipq') exp(-ip^2dt/(2M)) e^(-ipq) dp
= \integral e^(ip(q'-q)-ip^2dt/(2M)) dp
= exp(iM(q'-q)^2/(2dt))
= exp(i(M/2)((q'-q)/dt))^2 dt)

In other words, the amplitude for going from any state |q> at one time, to another state |q'> an infinitesmal time in the future is:

exp(i [(M/2)(dq/dt)^2 - V(q)]dt = exp(iL dt)

The amplitude for a finite time interval is just the matrix product or convolution of each of the above matrix elements for going from t to t+dt which is precisely what a path integral is.

Answer 2

The phase term in Feyman's third postulate is based on the classical equations of motion called the Lagrangian, which is just a very general mathematical description.

The Lagrangian is used here because that's how a classical description is converted to a quantum mechanical description, or quantized. You first start out with the classical equations of motion, then you add the quantization. Here Feyman have chosen a particular way to conduct the quantization, via the path integral and then summing all the contributions of the action history.

Why does that work? Well, that's why it took a brilliant physicist like Feyman to figure this out. It's not obvious and it shows that there are alternate interpretations of what's happening in the quantum world. So here Feyman is saying (mathematically with the 3 postulates) that the behavior of any quantum system is based on all possible classical behaviors of this system added together, and that how much each classical behavior contributes to the quantum behavior depends on the phase of the action history. And interestly enough, those action histories that include bizzare classical behaviors are mostly "out of phase" and do not contribut much, if at all, to the final behavior of the quantum system.

Answer 3

Action principles have a long history in physics. Feynman was probably trying to find a way of bringing the action into quantum mechanics.

Answer 4

it quite is amazing reasoning, yet no. The universe became a particle, not a wave. additionally, considering that while does collapsing a wave function make a particle explode into an infinitely increasing mass of what we now call the universe? greater importantly, physicists do not fairly understand the way such issues paintings. have you ever heard of String theory? that ought to be the reason of the extraordinary habit of small debris. Strings are reported to vibrate interior the 10th length, that's a factor of the time-honored universe. Your theory of the universe as a particle appearing as a wave function might in hassle-free terms be precise if there became something even bigger than the strings contained interior of it, appearing as strings in a length bigger than this finished universe. the start of the time-honored universe is contained interior the 1st 5 dimensions. Strings on the 10th length might have given beginning to this universe, yet some being bigger than the 1st dimensions ought to have been latest on the time of the huge bang!

Answer 5

Could be in his book 'Surely You're Joking Mr. Feynman'

Answer 6

With difficulty!!!!