*Exploring Quantum Physics – Week 2, Lecture 3*

Developed in Feynman’s 1948 paper: Space-Time Approach to Non-Relativistic Quantum Mechanics

## Summary

For a particle traveling from a coordinate

The probability of going to is the sum of the probability (or weight) of taking a particular path, over all possible paths:

## Trajectories

- Consider a particle localized at at time
*t*=0, that is - Evolve according to Schrödinger:

- What “part” will propagate to ?

i.e. what is the overlap with , - How will it get there? (In Q.M. it follows all possible trajectories)

## Evolution Operator

*Where is the evolution operator*

Plug into Schrödinger:

i.e. does not depend on the initial condition of

Verify:

*Asside: define the exponential of an operator, with it’s Taylor series:
*

## Propagator

a.k.a. the transition amplitude

Exponential of an operator is messy (infinite sum, see above) so simplify using:

Combining the above:

Reapplying this *N* times gives:

Let becomes small, and using a first order approximation:

**Propagator becomes a product:**

*Asside: difference between *

– Represents an “abstract” state of the system, coordinate-independent

– Represents in a coordinate system.

*Asside: the Dirac delta-function*

- Extension to a “continuum” basis

*where C normalizes:*

- Fourier transform:

## Breaking down the propagator equation

*because is an eigen state of the coordinate system*

## Putting it all together

*where is the Lagrangian*