Electrons and Matter Waves — Question 9

Question 9

Matter wavelength changes under a Galilean change of reference frame

A nonrelativistic particle has laboratory momentum $p=mv>0$ and wavelength $\lambda=h/(mv)$. An observer moves along +x at speed $u$; momentum in that frame is $p\prime=m(v-u)$. Define wavelength using the magnitude of momentum.

1. Find the wavelength and propagation direction for $u=v/2$ and $u=2v$.

2. Interpret the plane-wave state in the frame $u=v$.

Question 9 - Solution

Matter wavelength changes under a Galilean change of reference frame

(a) Method and calculation

$$u=v/2:\quad p\prime=mv/2,\quad\boxed{\lambda\prime=2\lambda,\quad\text{direction }+x}.$$$$u=2v:\quad p\prime=-mv,\quad\boxed{\lambda\prime=\lambda,\quad\text{direction }-x}.$$

(b) Interpretation and further calculation

$$u=v:\quad\boxed{p\prime=0,\quad k\prime=0}.$$$$\psi\prime(x\prime)\propto e^{ik\prime x\prime}=1,\qquad\lambda\prime\to\infty.$$

Physical check

This is a delocalized momentum eigenstate, not an assertion that a material object has infinite size. A physical localized packet has a range of momenta.