(5-8)    ①  We know that:     k  =   2π/λ   (definition of angular wavenumber)  and    p  =  h/λ   (de Broglie). Combining these, we get:  

                                                                                   p  = hk /2π.

②  This result suggests that a particle has some intrinsic momentum, by virtue of its waviness. We discovered, in section 5-7, that localising a particle requires superposing waves with a range of k values and this equation shows that this means superposing matter waves with a range of momenta. We have also established that the more localised the particle is, the bigger the range of k values – and therefore the bigger the range of momenta – required.

We have recognised that increasing the range of k also necessarily increases its mean value, <k>. Thus, the more localised a particle is, the greater the mean value of the momentum, <p>, must be. So, a particle that was fully localised, with no uncertainty in its position would require superposition of waves with an infinite range of momentum and, therefore, with an infinite mean momentum.