waves and quantum mechanics

Waves and quantum mechanics –Schrödinger’s equation

If the world of quantum mechanics is so different from everything we have been taught before, how can we even begin to understand it? Miniscule electrons seem so remote from what we see in the world around us that we do not know what concepts from our everyday experience we could use to get started. There is, however, one lever from our existing intellectual toolkit that we can use to pry open this apparently impenetrable subject, and that lever is the idea of waves. If we just allow ourselves to suppose that electrons might be describable as waves, and follow the consequences of that radical idea, the subject can open up before us. Astonishingly, we will find we can then understand a large fraction of those aspects of our everyday experience that can only be explained by quantum mechanics, such as color and the properties of materials. We will also be able to engineer novel phenomena and devices for quite practical applications.

      On the face of it, proposing that we describe particles as waves is a strange intellectual leap in
the dark. There is apparently nothing in our everyday view of the world to suggest we should do so. Nevertheless, it was exactly such a proposal historically (de Broglie’s hypothesis) that opened up much of quantum mechanics. That proposal was made before there was direct experimental evidence of wave behavior of electrons. Once that hypothesis was embodied in the precise mathematical form of Schrödinger’s wave equation, quantum mechanics took off. Schrödinger’s equation remains to the present day one of the most useful relations in quantum mechanics. Its most basic application is to model simple particles that have mass, such as a single electron, though the extensions of it go much further than that. It is also a good example of quantum mechanics, exposing many of the more general concepts. We will use these concepts as we go on to more complicated systems, such as atoms, or to other quite different kinds of particles and applications, such as photons and the quantum mechanics of light. Understanding Schrödinger’s equation is therefore a very good way to start understanding quantum mechanics. In this text, we introduce the simplest version of Schrödinger’s
equation – the time-independent form – and explore some of the remarkable consequences of this wave view of matter.

 Rationalization of Schrödinger’s equation

Why do we have to propose wave behavior and Schrödinger equation for particles such as
electrons? After all, we are quite sure electrons are particles, because we know that they have
definite mass and charge. And we do not see directly any wave-like behavior of matter in our2.1 Rationalization of Schrödinger’s equation everyday experience. It is, however, now a simple and incontrovertible experimental fact that electrons can behave like waves, or at least in some way are “guided” by waves. We know this for the same reasons we know that light is a wave – we can see the interference and diffraction that are so characteristic of waves. At least in the laboratory, we see this behavior routinely. We can, for example, make a beam of electrons by applying a large electric field in a vacuum to a metal, pulling electrons out of the metal to create a monoenergetic electron beam (i.e., all with the same energy). We can then see the wave-like character of electrons by looking for the effects of diffraction and interference, especially the patterns that can result from waves interacting with particular kinds or shapes of objects. One common situation in the laboratory is, for example, to shine such a beam of electrons at a crystal in a vacuum. Davisson and Germer did exactly this in their famous experiment in 1927, diffracting electrons off a crystal of nickel. We can see the resulting diffraction if, for example, we let the scattered electrons land on a phosphor screen as in a television tube (cathode ray tube); we will see a pattern of dots on the screen. We would find that this diffraction pattern behaved rather similarly to the diffraction pattern we might get in some optical experiment; we could shine a monochromatic (i.e., single frequency) light beam at some periodic structure 1 whose periodicity was of a scale comparable to the wavelength of the waves (e.g., a diffraction grating). The fact that electrons behave both as particles (they have a specific mass and a
specific charge, for example) and as waves is known as a “wave-particle duality.”
The electrons in such wave diffraction experiments behave as if they have a wavelength

                                λ =h/p                                          (2.1)

where p is the electron momentum, and h is Planck’s constant
h ≅ 6.626 × 10 − 34 Joule ⋅ seconds . (This relation, Eq. (2.1), is known as de Broglie’s hypothesis). For example, the electron can behave as if it were a plane wave, with a “wavefunction” ψ propagating in the z direction, and of the form

                             ψ ∝ exp ( 2 π iz / λ ) .                     (2.2)

If it is a wave, or is behaving as such, we need a wave equation to describe the electron. We find empirically that the electron behaves like a simple scalar wave (i.e., not like a vector wave, such as electric field, E, but like a simple acoustic (sound) wave with a scalar amplitude; in acoustics the scalar amplitude could be the air pressure). We therefore propose that the electron wave obeys a scalar wave equation, and we choose the simplest one we know, the “Helmholtz” wave equation for a monochromatic wave. In one dimension, the Helmholtz equation is

                                  d ψ/dz 2 = − k 2 ψ                       (2.3)

This equation has solutions such as sin(kz), cos(kz), and exp(ikz) (and sin(-kz), cos(-kz), and
exp(-ikz)), that all describe the spatial variation in a simple wave. In three dimensions, we can
write this as

                                 ∇ 2 ψ = − k 2 ψ   (2.4)

where the symbol ∇ 2 (known variously as the Laplacian operator, “del squared”, and “nabla”,
and sometimes written Δ ) means

(2.5)
where x , y , and z are the usual Cartesian coordinates, all at right angles to one another. This has solutions such as sin(k.r), cos(k.r), and exp(ik.r) (and sin(-k.r), cos(-k.r), and exp(-ik.r)), where k and r are vectors. The wavevector magnitude, k , is defined as

                                  k = 2 π / λ                                  (2.6)

 we can rewrite our simple wave equation (Eq. (2.4))     as




which lead us