Question

Consider a one-dimensional square-well potential

V(x) = 0, x < 0,

V(x) = -V₀, 0 < 2 < a,

V(x) = 0, x > a,

where V₀ is positive. If a particle with mass m is incident from the left with nonrelativistic kinetic energy E, what is its probability for transmission through the potential? For what values of E will this probability be unity?

Answer 1

Let the wave function be

The constants R, S, A, B are to be determined from the boundary conditions that \(\psi\)(x) and \(\psi'\)x are both continuous at x = 0 and x = a, which) give

and the probability for transmission is

Resonance transmission occurs when k' = nr, i.e., when the kinetic energy E of the incident particle is

E = n²π²h²/2ma² - V₀.

The probability for transmission, P, then becomes unity.