Coherent States for the Hydrogen Atom

Abstract

The long-standing problem of finding coherent states for the (bound state portion of the) hydrogen atom is positively resolved. The states in question: (i) are normalized and parameterized continuously, (ii) admit a resolution of unity with a positive measure, and (iii) enjoy the property that the temporal evolution of any coherent state by the hydrogen atom Hamiltonian remains a coherent state for all time. Harmonic-oscillator coherent states In modern terms, the distinguished set of states Schrodinger introduced for the harmonic oscillator 1 are now commonly known as coherent states 2 and are given, for all z 2 I C, by jzi j e (za y \\Gammaz a) j0i = e \\Gamma 1 2 jzj 2 1 X n=0 z n p n! jni; (1) where [a; a y ] = 1 as usual. Here jni; n = 0; 1; 2; : : :, denote normalized eigenstates of the number operator N , N jni = njni, which may be identified with the eigenstates of the harmonic oscillator Hamiltonian H 0 = !N = !a y a (¯h = 1). As such it follows that ...