Propagator

Date:: 11 Aug 2019

Dependency

What is the problem ?

In Quantum Mechanics there is no actual path of a particle but only the probability of transition from one state \(a\) to another state \(b\)
What is the probability \(P_{a\to b}\) from \((t_a,x_a)\) to \((t_b,x_b)\) ?

Derivation

Step 1 Feynman says

\[\begin{split}\begin{align*} & P_{a\to b} = |K_{a\to b}|^2\\ & where\\ & \text{Propagator }K_{a\to b} = \sum\limits_{\text{every path }x(t)\text{ from }a \to b} {1\over A} e^{{i\over \hbar} S[x(t)]} \\ & A: \text{ Normalization Factor} \end{align*}\end{split}\]

This looks really like Boltzmann Factor

Makes me question whether Quantum Mechanics is actually a result of Classic probability

Step 2 Approximation for small time interval \(\epsilon\)

\[\begin{align*} & K_{a\to b} \approx {1\over A} \exp \left[ {i\over \hbar} \cdot \epsilon \cdot L({x_b+x_a\over 2}, {x_b-x_a\over \epsilon}, {t_b+t_a\over 2}) \right] \end{align*}\]

Step 3 Wave Functions are simply Propagators with common starting point \((t_0,x_0)\)

\[\begin{split}\begin{align*} & \psi(x_i,t_i) \equiv K_{0\to i} \\ & \psi(x_j,t_j) = \int_{-\infty}^{\infty} dx_i \; \psi(x_i,t_i) K_{i \to j} \end{align*}\end{split}\]

Doubt 1

The integration over \(dx_i\) has dimension of Length, but in Step 1 we are only summing over all possible paths, which should be dimensionless

Derivation Part 2

Step 4 Combine Step 2 and Step 3 to get Schrodinger Equation

\[\begin{split}\begin{align*} & \psi(x,t+\epsilon) ={1\over A} \int_{-\infty}^{\infty} dy \; \psi(y,t) \exp \left\{ {i\over \hbar} \epsilon \left[ {1\over 2} m({x-y\over \epsilon})^2-V({x+y \over 2},t) \right] \right\} \\ & \text{After some Taylor Expansions...}\\ & \psi(x,t) + \epsilon {\partial \psi \over \partial t} = \psi(x,t) - {\hbar \epsilon \over 2im} {\partial^2 \psi \over \partial x^2} - {i\epsilon \over \hbar}V(x,t) \psi \\ &\boxed{ i\hbar {\partial \psi \over \partial t} = - {\hbar^2 \over 2m} {\partial^2 \psi \over \partial x^2} + V\psi \;\;\;\; \text{Schrodinger Equation}}\\ \end{align*}\end{split}\]