Solution to 2001 Problem 46

We assume that the energies and momenta involved are nonrelativistic. The de Broglie wavelength is defined as
\begin{align*}\lambda := \frac{h}{p} = \frac{h}{\sqrt{2 m K}}\end{align*}
where K is the kinetic energy of the particle. Initially, K = E, but after the particle enters the region in which it has a potential energy V, the kinetic energy becomes K = E - V. Therefore, the initial de Broglie wavelength must be multiplied by a factor of
\begin{align*}\sqrt{\frac{E}{E-V}} = \sqrt{\frac{1}{1 - V/E}}\end{align*}
in order to obtain the new de Broglie wavelength. Therefore, the correct answer is (E).

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