Almost-periodic wave packets and wave packets of invariant shape

American Journal of Physics

Volumn 71, Issue 6, 2003, Pages 580-584

  Mentrup, Detlef ^a   Luban, Marshall ^b  

^a UNIVERSITY OF OSNABRÜCK   (Germany)

^b IOWA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038236355     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1555872     Document Type: Article

References (13)

1. Misleading remarks by two key figures in the development of quantum mechanics, W. Heisenberg and D. Bohm, that arose in reaction to Eqs. (8) and (9) are discussed in detail in Ref. 11.
2. A. S. Besicovitch, Almost Periodic Functions (Cambridge U.P., Cambridge, 1932).
3. H. Bohr, Fastperiodische Funktionen (Springer, Berlin, 1932).
4. E. Schrödinger, "Der stetige Übergang von der Mikro- zur Makromechanik," Naturwissenschaften 14, 664-666 (1926).
5. D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, NJ, 1950), pp. 306-308.
6. L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), 3rd ed., pp. 74-76.
7. W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001).
8. I. R. Senitzky, "Harmonic oscillator wave functions," Phys. Rev. 95, 1115-1116 (1954).
9. We will not concern ourselves with the delicate point that the equality written in Eq. (2) should really be replaced by the symbol ∼ so as to follow the notation of Refs. 2 and 3, because in some cases the series need not converge to f(t), but it is nevertheless "summable to f(t)."
10. C. M. Bender and S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
11. Schrödinger's result, Eq. (9), for the harmonic oscillator is actually of considerable historical interest as it supported his own proposed "electrodynamic" interpretation that the wave function describes the invariant shape of an extended particle. In the same article (Ref. 4), he expressed his conviction that wave packets of invariant shape could also be constructed for highly excited bound states of the hydrogen atom. This possibility was rejected by W. Heisenberg, "Über den anschaulichen Inhalt der quanten-theoretischen Kinematik und Mechanik," Z. Phys. 43, 172-198 (1927), especially pp: 183-185, on the basis that equal spacing of the energy levels was the crucial factor for achieving wave packets that do not spread with time, a condition satisfied for the harmonic oscillator, but not satisfied for hydrogen or any other atom: "...in all other cases, with the passage of time, a wave packet broadens with time." Likewise, in discussing Schrödinger's result of Eq. (9), Bohm (see Sec. 13.15 of Ref. 5) writes that, "Normally, we expect wave packets to spread out in time, but this particular packet does not," and he attributed this to the fact that the harmonic oscillator gives rise to strictly periodic wave packets which therefore cannot spread indefinitely with time. We disagree with both Heisenberg's and Bohm's remarks; the spreading of a wave packet occurs only if the initial wave packet incorporates continuum eigenstates. Moreover, the wave packet will be an almost-periodic function of the time and thus not broaden indefinitely if the inital wave packet does not include any contribution from continuum states, no matter what the spacing of the bound state energy levels.
12. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1998), 3rd ed., p. 40.
13. B. Mielnik, "Factorization method and new potentials with the oscillator spectrum," J. Math. Phys. 25, 3387-3389 (1984).