wavemech.htm(Ó R. Egerton)

Y = A cos (kx - w t) ................ (1)

where A is the amplitude of the wave, k = 2p /l is known as the wavenumber (l = wavelength) and w = 2p f is an angular frequency, measured in radians/sec (f being the frequency in Hz). The phase velocity of the wave is defined to be v_p = f l = w /k .

For mechanical waves, such as sound or water waves, Y (Greek letter psi) would represent the physical displacement of atoms at position x and time t . In the case of an electromagnetic wave, it would represent electric or magnetic field. In the case of matter waves, Y in Eq.(1) is known as a wavefunction and must satisfy a wave equation known as the Schrodinger equation.

Unfortunately, Eq.(1) cannot serve as a wave-mechanical description of a single particle, such as an electron (or a single photon of electromagnetic radiation) since the wavefunction has no localization; the cosine function continues oscillating over an infinite range of x . However, if we add two cosine waves, with slightly different wavenumbers k₁ and k₂ and angular frequencies (w ₁ and w ₂), the result is an amplitude-modulated wave; see Fig.4.11. This is like a standing wave, with nodes and antinodes, except that each "bunch" of waves moves forward in the x-direction with a group velocity v_g given by

v_g = (w ₂ - w ₁) / (k₂ - k₁) = D w / D k ................ (2)

D p D x > h/2p ................ (3)

Here D p represents the uncertainty in our knowledge of the momentum (p=h/l =hk/2p ) of a particle and D x is the corresponding uncertainty in our knowledge of its position. This relationship follows directly from the wave description of a particle. For example, the length of each wave packet in Fig. 4.11 can be shown to be D x = 2p /D k, so that the product D k D x = 2p , equivalent to D p D x = h , which is in agreement with Eq.(3).

As an example of this principle in operation, consider the Bohr model of the hydrogen atom in its ground state. Our uncertainty in knowledge of the position of the orbiting electron might be taken as the diameter of its orbit: D x = 2 a₀. Since at any one instant the electron might have an x-component of velocity anywhere between v₁ and -v₁, we can say that D v = 2 v₁, giving D p D x = (mD v) D x = 2m(h/[2p ma₀])(2 a₀) = 2h/p , again in agreement with Eq.(3).

The most famous example of Eq.(3) is the Heisenberg microscope, a Gedanken (thought) experiment where we try to image a single particle by focussing the electromagnetic radiation which it scatters. If the optical system consists of a single lens which subtends a semiangle q at the object (as in Fig. 4.26). the image resolution is limited (by diffraction at the lens edges) to D x = 0.6l / sinq . However, a photon initially travelling along the optic axis) which is scattered through an angle q transfers a horizontal momentum of magnitude p_x = p sinq = (h/l ) sin q to the particle. This momentum might be in the +x or -x direction, so the uncertainty in our knowledge of the momentum of the particle (after it has scattered the photon) is D p = 2(h/l ) sinq , leading to D p D x = [2(h/l ) sinq ][0.6l /sinq ] = 1.2 h , again validating Eq.(3). If we choose a small wavelength l in an attempt to minimize D x, we 'disturb' the particle more because the short-wavelength particle transfers a larger momentum. Therefore Eq.(3) is often interpreted by saying that by making a measurement on an object, we necessarily change it; the wavelike properties of the object deny us the option of performing measurements on an 'isolated' system.

Because the product D pD x often comes out a factor of 5 or 6 larger than the lower limit prescribed by Eq.(3), the Heisenberg uncertainty principle is sometimes written as D p D x » h .

D E D t > h/2p ................ (4)

where D E is the uncertainty in our knowledge of the energy of a particle, if its measurement is carried out over a time interval D t . Eq.(4) allows us to estimate the width of the emission lines emitted by atoms, for example in a gas-discharge lamp. The lifetime in the excited state is commonly of the order of 10^-8 seconds, so D E cannot be less than approximately 10^-34/10^-8 = 10^-26 J, or about 10^-7 eV. For visible light, the photon energy is two or three eV, so the fractional width D l /l must be at least three parts in 10^8 . If the pressure in the lamp is increased, collisions between gas atoms tend to reduce the lifetime D t and the spectral lines become broader.

Given two alternative descriptions of the same object (such as an electron) how do we know which is applicable in a given situation ? Answers to this question begin to emerge when we investigate simple situations such as diffraction of electrons by a double slit, sometimes called the Young's slit experiment. This experiment can be performed fairly easily with visible light (i.e. photons) and, with some technical difficulty, with electrons. The result is a series of parallel fringes, recorded on a plane behind the double slit; see Fig. 4.28. Using a wave picture for the incident radiation (electrons, photons, ... ) maxima in intensity should occur where there is constructive interference, corresponding to a path difference which is equal to an integral number of wavelengths: n l = D sinq , where D is the separation of the two narrow slits. The experimental results are in agreement with this prediction, showing that the wave picture correctly describes the overall effect.

The usual answer to this question is that the probability of arrival of a particle (at a particular location on the detector plane) is determined by the resultant intensity of two interfering matter waves, one (Y ₁) representing an electron passing through the upper slit, the other (Y ₂) representing an electron passing through the lower slit. This is consistent with Max Born's probabilistic interpretation of the electron wavefunction Y . Strictly speaking, Y must be represented as a vector (or a complex number with real and imaginary parts), indicating that it contains phase as well as amplitude information. However, if we measure just the length of the vector, represented by | Y | , we obtain a scalar quantity whose square is proportional to the probability of finding the electron at a particular point in time and space.

The wavelike properties of an electron also show up when the particle is confined within a small region of space. Usually this confinement is by electrostatic forces, which can be visualised in terms of the corresponding electrostatic potential energy. The simplest case to consider is a one-dimensional square potential well, in which the potential energy rises abruptly to infinity at x=0 and x=L ; see Fig. 5.8c. Since an electron of finite kinetic energy (and wavenumber k) could not exist in the regions x<0 and x>L , its wavefunction must be zero at these boundaries. This places constraints on the wavefunction Y , if represented by a simple sine or cosine function as in Eq.(1). An integral number of half-wavelengths must fit inside the well (Fig. 5.11a) or n l /2 = L , where the integer n is a quantum number. Fig. 5.11b shows the square of the magnitude of the wavefunction and indicates that the probability of finding an electron inside the well varies with its position x , in a way which depends on the value of n .

E = (1/2) m v^2 = (p^2)/2m = (1/2m)(h/l )^2 = n^2 h^2 /(8 m L^2) .............. (5)

A more realistic situation is represented in Fig. 5.12, where the potential energy rises to a value U outside the region 0<x<L . We assume E<U, since the electron is confined by the potential. Even so, the Schrodinger equation for Y has solutions outside the well, although these decay rapidly (exponentially) away from the boundaries x=0 and x=L (Fig. 5.13). The wavefunction decays by a factor of e (2.718) over a distance called the penetration depth, whose value depends on U but is typically a fraction of a nanometer.

The conduction electrons in a metal are confined by an electrostatic potential not unlike that of Fig. 5.12. They are contained by a potential barrier at each surface whose height is U = j , the work function. The electrons can escape if given energy in excess of j , for example by interaction with photons (photoelectric effect). However, the electrons can also "leak out" if a second solid is brought up very close to the first, with the gap between them less than about 1 nm (several times the penetration depth). This behaviour is called tunnelling and is a direct result of the wavelike nature of electrons.