# Wavefunction of a particle

einai
What's the wave function in coordinate space &Psi;x0(x') of a particle (in 1-D) located at a certain position x0? What about the wave function &Phi;x0(p') in momentum space? Now, consider the totality of these wave functions for different values of x0. Do they form an orthonormal set?

The only thing I know is that if I know &Psi;x0(x'), I can Fourier transform it to &Phi;x0(p')? But what's &Psi;x0(x')? I'm really confused.

Ambitwistor
Originally posted by einai
What's the wave function in coordinate space &Psi;x0(x') of a particle (in 1-D) located at a certain position x0? What about the wave function &Phi;x0(p') in momentum space? Now, consider the totality of these wave functions for different values of x0. Do they form an orthonormal set?

The wavefunction of a 1-D particle localized at a position x=x0 is delta(x-x0), where delta is the Dirac delta function. In fact, the Dirac delta "function" is not really a function at all, but a "distribution". You can sort of think of it as a spike in the limit as the spike becomes infinitely tall and thin. In momentum space, the wavefunction is just constant: equal probability to have all momenta. You cannot really speak of these as orthonormal functions in the Hilbert space of states, because the Dirac delta isn't a function, and the constant function isn't normalizable.