Wavefunction of a particle

  • Thread starter einai
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What's the wave function in coordinate space Ψx0(x') of a particle (in 1-D) located at a certain position x0? What about the wave function Φx0(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 Ψx0(x'), I can Fourier transform it to Φx0(p')? But what's Ψx0(x')? I'm really confused.

Thanks in advance! :smile:
 
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Originally posted by einai
What's the wave function in coordinate space Ψx0(x') of a particle (in 1-D) located at a certain position x0? What about the wave function Φx0(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.
 

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