Question

why group waves associated with a single particle

Answer 1

Einstein introduced us to the particle properties of waves in 1905 (photoelectric effect). Compton
scattering of x-rays by electrons (which we skipped in Chapter 2) confirmed Einstein's theories.
You ought to ask "Is there a converse?" Do particles have wave properties?
De Broglie postulated wave properties of particles in his thesis in 1924, based partly on the
idea that if waves can behave like particles, then particles should be able to behave like waves.
Werner Heisenberg and a little later Erwin Schrödinger developed theories based on the wave
properties of particles.
In 1927, Davisson and Germer confirmed the wave properties of particles by diffracting
electrons from a nickel single crystal.
3.1 de Broglie Waves
Recall that a photon has energy E=hf, momentum p=hf/c=h/, and a wavelength =h/p.
De Broglie postulated that these equations also apply to particles. In particular, a particle of
mass m moving with velocity v has a de Broglie wavelength of
λ =
h
mv .
where m is the relativistic mass
m =
m
1- v / c . 0
2 2
In other words, it may be necessary to use the relativistic momentum in =h/mv=h/p.
In order for us to observe a particle's wave properties, the de Broglie wavelength must be comparable
to something the particle interacts with; e.g. the spacing of a slit or a double slit, or the spacing
between periodic arrays of atoms in crystals.