The Quantum Mechanics Solver 26 uniquely illustrates the application of quantum mechanical concepts to various fields of modern physics. It aims at encouraging the reader to apply quantum mechanics to research problems in fields such as molecular physics, condensed matter physics or laser physics. Advanced undergraduates and graduate students will find a rich and challenging source of material for further exploration. This book consists of a series of problems concerning present-day experimental or theoretical questions on quantum mechanics | 256 25 Quantum Reflection of Atoms from a Surface Fig. . A hydrogen atom incident on a liquid helium bath. The oscillations of the surface of the liquid are studied in Sect. v0 z - a. Z 3 Express a in terms of C6 and n. . Experimentally one finds a x 10 i 2eV A3. At what distance from the surface does gravity become larger than the Van der Waals force In what follows we shall neglect the gravitational force. . Show that the eigenstates of the Hamiltonian which describes the motion of the H atom are of the form kj. where k represents a plane running wave propagating in the plane Oxy . parallel to the surface of the liquid He and where o is an eigenstate of the Hamiltonian which describes the motion along the z axis RZ kL a . - K Z . y LxLy . We want to evaluate the number of bound states of the motion along the z axis in the potential Vo Z -03 if Z zmin Z3 Vo Z X if Z Zmin . We shall use the WKB approximation. a Justify the shape of this potential. b What is the continuity condition for the wave function at Z zmin c Show that the quantization condition for a motion with turning points zmin and b is i k Z dZ n J n J Zmin 4 with n integer 0. Excitations on the Surface of Liquid Helium 257 d Infer the order of magnitude of the number of bound states as a function of zmin and a. What is the domain of validity of the result e The parameter zmin for the surface of liquid He is of the order of 2 A. How many bound states does one expect for the motion along the z axis f Experimentally one finds that there is a single bound state H-liquid He whose energy is E0 x 10 5 eV. Compare this result with the WKB prediction. This unique bound state in the z-axis motion will be denoted in the rest of the chapter. Excitations on the Surface of Liquid Helium The general dispersion relation for waves propagating on the surface of a liquid is -2 gq ---q3 with q q q Po where xq and q qx are respectively the frequency and the wave vector of
