A free particle of mass m moves in one dimension. The initial wave function of the particle is ψ(x, 0). ​

Question

A free particle of mass m moves in one dimension. The initial wave function of the particle is \(\psi\)(x, 0).

(a) Show that after a sufficiently long time t the wave function of the particle spreads to reach a unique limiting form given by

\(\psi(x, t) = \sqrt{m/ht} \exp (-i\pi/4) \exp(imx^2/2ht)\phi(mx/ht)\),

where \(\phi\) is the Fourier transform of the initial wave function:

\(\phi(k) = (2\pi)^{-1/2} \int \psi(x, 0) \exp(-ikx)dx\)

(b) Give a plausible physical interpretation of the limiting value of \(|\psi(x, t)|^2\).

Answer 1

(a) The Schrodinger equation is

and so after a long time t(\(t \to \infty\)),

(b)

Because p(k) is the Fourier transform of $(z, 0), we have

which shows the conservation of total probability. For the limiting case of t \(\to \infty\), we have 

\(|\psi(x, t)|^2 \to 0.|\phi(0)|^2 = 0\),

which indicates that the wave function of the particle will diffuse infinitely.