Question

Let the solution to the one-dimensional free-particle time-dependent Schrodinger equation of definite wavelength \(\lambda\) be \(\psi\)(x, t) as described by some observer 0 in a frame with coordinates (x, t). Now consider the same particle as described by wave function \(\psi'\) (x', t') according to observer 0’ with coordinates (x', t') related to (x, t) by the Galilean transformation

x’ = x - vt,

t’ = t.

(a) Do \(\psi\)(x, t), \(\psi'\)(x', t') describe waves of the same wavelength?

(b) What is the relationship between \(\psi\)(x, t) and \(\psi'\)(x', t') if both satisfy the Schrodinger equation in their respective coordinates?

Answer 1

(a) The one-dimensional time-dependent Schrodinger equation for a free particle

\(ih\partial _t\psi(x, t) = (-h^2/2m)\partial_x^2\psi(x, t)\)

has a solution corresponding to a definite wavelength \(\lambda\)

As the particle momentum p is different in the two reference frames, the wavelength \(\lambda\) is also different.

(b) Applying the Galilean transformation and making use of the Schrodinger equation in the (x', t') frame we find

Considering

making use of Eq. (1) and the definitions of k and w, we see that

This is just the Schrodinger equation that \(\psi\)(x, t) satisfies. Hence, accurate to a phase factor, we have the relation