828 Wave Packet

Wave Packet

Helge Kragh

A wave packet is a concentrated train of (quantum) waves of various wavelengthsor momenta with the property that the packet is confined within a small region ofspace. Such a packet can be constructed by adding a very large number of waves sochosen that their sum interferes destructively everywhere except in a small region. Ifharmonic waves of different momenta are superposed, the packet can be expressedin the form ψ(x) = ∫ A(k)eikxdk where k = p/� and A(k) is the amplitude corre-sponding to the wave number k.

Although speculative attempts to identify atoms with systems of standing wavescan be found back in the nineteenth century, in a quantum context it was Schrodingerwho invented wave packets and related them to atomic particles. In his secondcommunication on � wave mechanics Schrodinger discussed the possibility ofconstructing a wave group or packet equivalent to a pointlike particle, such as anelectron, and in a subsequent paper of 1926 he provided a more elaborate discus-sion in which he introduced the � superposition principle. Analyzing the case ofa one-dimensional harmonic oscillator, Schrodinger constructed for the first time awave packet as an exact solution of the � Schrodinger equation. Making use of thesuperposition principle, he constructed a wave packet of the form ψ =∑ anψn/n!,where a is a large number, ψn are the eigenstates, and 0 � n � ∞. The result-ing wave packet, he showed, remains compact as time goes on and it has an energywhich is exactly the same as the one of the classical oscillator. Schrodinger’s wavepacket was a “minimum uncertainty wave packet,” the first example of what laterbecame known as “� coherent states.” He believed that this result would be validalso for electrons moving in atomic orbits and, if so, that it indicated that perhapselectrons and other particles are wave packets. At the end of his paper he foresawthat it was only a matter of time until “the representation by wave mechanics of thehydrogen atom” � Bohr’s atom model would be achieved.

However, in letters to Schrodinger from June 1926, Lorentz demonstrated thata permanent wave packet cannot be constructed for an atomic electron and thatSchrodinger’s success with the harmonic oscillator was accidental. “In the presentform of your theory you will be unable to construct wave packets that can repre-sent electrons moving in very high Bohr orbits,” Lorentz wrote. It is unknown howSchrodinger reacted, but most likely Lorentz’ critique contributed to a change in hisontology: by the fall of 1926 Schrodinger concluded that his original belief in theprimacy of waves was not an integral part of wave mechanics.

Some of Lorentz’s objections were independently made by Heisenberg in hisfamous paper of 1927 in which he introduced the � Heisenberg uncertainty prin-ciple, which he derived by means of arguments based on wave packets. Accordingto Heisenberg, “Schrodinger’s reasoning is only viable for the case of the harmonic

Wave Packet 829

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oscillator treated by him; in all other cases a wave packet spreads out in the courseof time over the whole immediate neighborhood of the atom.” He observed that thepeculiar properties of the wave packet Schrodinger had found was a consequence ofthe fact that the energy levels of the harmonic oscillator are equally spaced (namely,given by En = (n + 1/2)�ω). Moreover, Heisenberg found that the size of theprobability wave packet – ψ ψ* rather than ψ – representing a freely moving parti-cle would increase indefinitely with the time.

Wave packets were not only important in the chain of arguments that led Heisen-berg to his uncertainty relations, they also played a crucial role in Bohr’s physicalinterpretation of quantum theory and his formulation of the � complementarity prin-ciple in the fall of 1927 where he used wave packets to represent both � light quantaand � electrons. The problem with the wave packet picture illustrated to Bohr that“the contrast between the wave theory superposition principle and the assumptionof the individuality of particles” was irremediable. At that time, Schrodinger hadabandoned his wave ontology and no longer thought of electrons as constituted bywave packets.

The papers by Schrodinger and Heisenberg were discussed by several physicistsin 1927–1928, including George Darwin, Earle Kennard and Arthur Ruark, who allrecognized that electrons cannot be represented just as wave packets. Or, as Kennardexpressed it, “the electron must always be assigned a greater degree of reality thanthat of a wave packet.”

As indicated by the title of Schrodinger’s paper of 1926, “The Continuous Tran-sition from Micro- to Macromechanics,” his aim was to understand the behaviour ofmacroscopic bodies from quantum principles. Although wave packets would not doas representations of subatomic particles, in 1927 Paul Ehrenfest showed that therewere no corresponding problems with spreading wave packets (Fig. 1) in the case ofmacroscopic bodies. As an example he calculated the time it would take for a par-ticle of mass m and represented by a probability wave packet of width Δ to spread

Fig. 1 Example of a wave packet. Source: Wikimedia Commons

830 Wave-Particle Duality: Some History

out to double its initial size. His result was T ∼= Δ√m/�. Because of the smallness

of � Planck’s constant (� = 1.05× 10−34 Js) this means that the doubling time isnearly infinite for a macroscopic particle. For a particle of linear size Δ = 0.001 cmand mass m = 1 g, the doubling time is about 10,000 times the age of the universe.

Another important work, relating to Schrodinger’s and Ehrenfest’s, was dueto Peter Debye, who showed that � wave packet, simulating mass and chargepoints, can be constructed also without using the special expansion coefficient thatSchrodinger had used in his treatment of the harmonic oscillator. Debye discussedin 1927 the behaviour of wave packets of one degree of freedom for any kind offorce, and found that their maxima move in accordance with the classical laws. Hiswork was one of many that aimed at showing the correspondence-like connectionbetween quantum mechanics and classical physics.

Primary Literature

1. E. Schrodinger: Der stetige Ubergang von der Mikro- zur Makromechanik. Die Naturwis-senschaften 14, 664–666 (1926)

2. W. Heisenberg: Uber den anschaulichen Inhalt der quantentheoretischen Kinematik undMechanik. Zeitschrift fur Physik 43, 172–198 (1927)

3. K. Przibram ed.: Briefe zur Wellenmechanik (Springer, Vienna 1963)

Secondary Literature

4. M. Jammer: The Philosophy of Quantum Mechanics (Wiley, New York 1974)5. F. Steiner: Schrodinger’s Discovery of Coherent States. Physica B 151, 323–326 (1988)6. H. Kragh, B. Carazza: Classical Behavior of Macroscopic Bodies from Quantum Principles:

Early Discussions. Archive for History of Exact Sciences 55, 43–56 (2000)

Wave-Particle Duality: Some History

Bruce R. Wheaton

Our modern understanding of light is the result of dispute since the scientific revo-lution of the seventeenth century. The roots of that contention, however, precede thecontributions of Aristotle, and I daresay the final story has yet to be written.

Following Plato and his student Aristotle, what we see in our lives are “sec-ondary” qualities that originate from an unseen world of “primary” events. In theirview whatever the primary causes of sound should seem similar to the water, andof matter to the rocks we encounter in life. The earlier philosophers tended to find