I have read recently that the motion of an electron of momentum p must be described by the means of a plane waves :[itex]\psi(\vec r,t)=Ae^{i(\vec k \cdot \vec r -wt)}=Ae^{i(\vec p\cdot \vec r -Et)/\hbar}[/itex](adsbygoogle = window.adsbygoogle || []).push({});

de Broglie hypothesis states that every particle of momentum p has a wavelength lamda.

I will split my question into three parts:

My first part concerns the plane wave by itself:

1) Why plane waves are written like this [itex]\psi(\vec r,t)=Ae^{i(\vec k \cdot \vec r -wt)}[/itex]

why not like this for example:[itex]\psi(\vec r,t)=Ae^{i(\vec k \cdot \vec r +wt)}[/itex]

2) Is it called a plane wave since in the far region they will approximately be like a plane?

3)what does the imaginary part of this wave means physically?

My second part concerns the wavefunction:

1) If the electron of momentum p is described by means of a plane wave, does this mean that we cant predict at all the position of the particle? since the plane waves has no sensible normalization ( [itex]|\psi(x)|^2=A[/itex] )in all the space.

2) If the answer of the above question is yes, then why in the cases of interference and diffraction the position of the electron can be predicted as there are bright,dark and intermediate fringes.Does the wavefunction change, in this case, from a plane wave to another wavefunction that have a sensible normalization?Can we relate this to wavepackets?

My third part concerns de Broglie hypothesis:

"Whenever the de Broglie wavelength of an object is in the range of, or exceeds its size, the wave nature of the object is detectable,hence it cannot be neglected.But if de Broglie wavelength is much too small compared to its size,the wave behavior of this object is undetectable".

Can somebody give an example that show me this?

THANKS!!

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# I Wave-particle duality and localization

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