Quantum State Evolution of a Free Particle at a Defined Position

[bluesunday]

The question is: what is the quantum state of a free particle t time after its detection at the position r₀ in t=0?

I know I have to use the evolution operator with the hamiltonian of a free particle. My actual problem is more stupid than that: I don't really know how to express the STATE of a free particle of well defined position (r₀ in t=0, in this case)

 

[bluesunday]

If we know exactly the position of the particle at t=0, does this mean that at this time its state is an eigenstate of position, with eigenvalue r₀? How can I put that into:

[URL]http://upload.wikimedia.org/math/1/0/3/10317da44bf13fbd709cd642c5143b9f.png[/URL]

Should I just skip Dirac notation and work with wavefuncions? I don't know... Please help.

 

[G01]

I'd work in the momentum basis for this problem.

The wave function for a free particle at a definite position is a delta function.

Fourier transform this wavefunction into momentum space and then time evolve. It should be straightforward in the momentum basis.

If you need your answer in position space, then you can always Fourier transform back.