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Introduction to Quantum Mechanics
Consider the quantum mechanical analog to the classical problem of a ball (mass $m$ ) bouncing elastically on the floor. ${ }^{13}$
(a) What is the potential energy, as a function of height $x$ above the floor? (For negative $x$, the potential is infinite $-$ the ball can't get there at all.)
(b) Solve the Schrödinger equation for this potential, expressing your answer in terms of the appropriate Airy function (note that $B i(z)$ blows up for large $z$, and must therefore be rejected). Don't bother to normalize $\psi(x)$.
(c) Using $g=9.80 \mathrm{~m} / \mathrm{s}^{2}$ and $m=0.100 \mathrm{~kg}$, find the first four allowed energies, in joules, correct to three significant digits. Hint: See Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, Dover, New York
(1970), page 478; the notation is defined on page 450 .
(d) What is the ground state energy, in $\mathrm{eV}$, of an electron in this gravitational field? How high off the ground is this electron, on the average? Hiut: Use the virial theorem to determine $\langle x\rangle$.