How Does Path Integral Formalism Derive the Quantum Propagator?

[squigglywolf]

Path Integral Formalism

Reading through Shankar atm, up to page 232/233.
Reference to pages if interested.
http://books.google.co.nz/books?id=2zypV5EbKuIC&printsec=frontcover&source=gbs_vpt_reviews#v=onepage&q=232&f=false(sorry I am too noob at latex to type all the formulas out..)

It's looking at getting the propagator from the path integral method with potentials of the form $$\mbox{V = a + bx + cx^2 +d}\dot{x} \mbox{+ex}\dot{x}$$

I think I understand the steps until 8.6.6 but can't understand why it is assumed L is a quadratic polynomial? Cant V(x) depend on say x^3 and higher?

Also equation 8.6.12, in evaluating the path integral it assumes it has no memory of Xcl, so it can only depend on t, thus it gets replaced by A(t). What exactly does this mean? Is this because we are now taking all paths between 0 --> 0? i.e. the paths no longer have any knowledge of what the classical endpoints are (x --> x' ) ?
This whole section 8.6 has kinda confused me.

 

[atyy]

Terms higher than quadratic can be included. He's just doing it for the simple cases.

I think the reasoning is that the integral is over all paths consistent with the endpoints, so the integral should be some number that changes depending on what the endpoints are. The start point is fixed by Shankar's convention. So only the end point will come into play. The classical path has been subtracted out, so the end point always has position zero, which means that all that's left is what t is at the end point.

 

[squigglywolf]

Question about the propagator.

Am I right in interpreting it in the equation ψ(x,t) = ∫U(x,t;x',t')ψ(x',t') dx' as the transition amplitudes of |x'> to |x> (where these are the position basis vectors) ?

Isn't U(x,t;x',t') in ψ(x,t) = ∫U(x,t;x',t')ψ(x',t') dx' kinda saying that, there is a certain amount of ψ at (x',t') that will be found at x at a later time, t, and then sum over these contributions from all x' ?

I'm finding it hard to see how the path integrals provides a propagator which can be interpreted like this. I am guessing that points connected by a classical path have a high transition amplitude, the reason for this being that there are many paths possible close by which coherently add together. Similarly points which ARENT connected classically, say, (x',t') and (x+ε,t), do not permit as many paths that will add together coherently since the action has no stationary point between a path directly connecting them. Is the way I see this correct?

 

[atyy]

It's just saying that states at two different times are unit vectors in a Hilbert space, and should be related by some "rotation". The rotation with time (ie. time evolution) is specified by the Schroedinger equation (or path integral).

U(x,t;x',t') is the matrix element in the position basis of the time evolution operator (see Shankar's Eq 1.8.40 & 5.1.10-12).

 

[squigglywolf]

Ah thanks for the references about propagator, forgot about those. In particular the paragraph after 5.1.13 pretty much confirms what I thought, didn't explain it the best myself.

Still can't quite grasp how PI gives a proper propagator, it seems that the endpoints (x,t) are fixed to coincide with the classical path. Especially in the way it is worked out on page 232, with y(0) = y(t) = 0, it looks as though you only get a transition amplitude for a single pair of points, but not for any neighbouring points like (x',t') --> (x+ε,t'), unless their is a distinct 'classical' path for each pair. Hopefuly my question makes sense. I think I'm missing something fundamental in the mathematics of working out the propagator this way.

 

[atyy]

Each quantum state |ψ> is a vector with a coordinate representation <x|ψ>=ψ(x), called the wave function. You can think of x as indexing the elements of the coordinate representation of the vector.

Similarly, the time evolution operator |U(t')><U(t')| is a "matrix" with a coordinate representation <x'|U(t')><U(t')|x>=U(x',t',x), called the coordinate representation propagator. You can think of x,x' as indexing the elements of the matrix, so <x'|U(t')><U(t')|x> is also called a matrix element of the coordinate representation of the time evolution operator.

|U(t')><U(t')|x> is the time evolution operator acting on an initial state with definite position x, so |U(t')><U(t')|x> will be the state at time t'. In other words, if |ψ(0)> = |x>, then |ψ(t')> = |U(t')><U(t')|x>. So for fixed initial state |x>, <x'|ψ(t')> = <x'|U(t')><U(t')|x> is the wave function at time t'. That's one way of interpreting the particular matrix element of the coordinate representation of the time evolution operator.

For any particular pair of start (x,0) and end points (x',t') corresponding to any particular matrix element of the position representation of U, there is a classical trajectory x(t). (Hamilton's principle)

In general, we have to know all matrix elements of an operator to specify it. So while we hold x,x' fixed to calculate one matrix element, we have to do that for all pairs x,x' to specify the operator.