GSO001 Problem 17

The time-independent Schrödinger equation:

-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)

In Region I ( $x < 0$ , $V(x) = 0$ ):

\frac{d^2\psi_1(x)}{dx^2} + k_1^2 \psi_1(x) = 0, \quad k_1 = \frac{\sqrt{2mE}}{\hbar}

General solution (incident wave + reflected wave):

\psi_1(x) = A e^{ik_1 x} + B e^{-ik_1 x}

In Region II ( $x \ge 0$ , $V(x) = V_0$ ):

\frac{d^2\psi_2(x)}{dx^2} + k_2^2 \psi_2(x) = 0, \quad k_2 = \frac{\sqrt{2m(E-V_0)}}{\hbar}

Only transmitted wave:

\psi_2(x) = C e^{ik_2 x}

Applying boundary conditions at $x = 0$ (continuity of $\psi$ and $\psi'$ ):

Eliminating $C$ :

k_1(A - B) = k_2(A + B)

\frac{B}{A} = \frac{k_1 - k_2}{k_1 + k_2}

The reflection coefficient:

R = \left| \frac{B}{A} \right|^2 = \left( \frac{k_1 - k_2}{k_1 + k_2} \right)^2 = \left( \frac{\sqrt{E} - \sqrt{E-V_0}}{\sqrt{E} + \sqrt{E-V_0}} \right)^2

Substituting numerical values:

R = \left( \frac{17 - 8}{17 + 8} \right)^2 = \left( \frac{9}{25} \right)^2 = \frac{81}{625}

This is already an irreducible fraction, so $A = 81$ , $B = 625$ .

$A + B = 81 + 625 = 706$

Reflection and Transmission of Matter Waves at a 1D Step Potential