Chapter 12 The SchrÄodinger Wave Equation Topics The double-slit experiment. Representing particles by waves. Heisenberg's Uncer- tainty Principle. SchrÄodinger'swave equation. Stationary states. Interpretation of the wave-function. One dimensional solutions for a particle in an in¯nite square potential well. General features of solutions of SchrÄodinger'swave equation. The correspondence principle. This chapter is the centrepiece of our development of quantum mechanics and the `derivation' of the SchrÄodingerwave equation. Let us build up the equation, reviewing the clues provided in the pre- vious chapters. We have established that the wave- particle duality works in both directions: ² waves have particle properties, most vividly demonstrated by the photoelectric e®ect and the derivation of the Planck distribution. For photons, hº ¹h! E = hº = ¹h! p = = = ¹hk: c c ² particles have wave properties as illustrated by electron di®raction experiments. h p = ¹hk = : ¸ 1 The SchrÄodingerWave Equation 2 12.1 The Double-slit Experiment The famous double-slit experiment was ¯rst car- ried out by Thomas Young in Emmanuel College in 1802 { the importance of the experiment was that it demonstrated beyond any reasonable doubt the correctness of the wave theory of light. A pair of narrow slits separated by distance d is illuminated by a uniform plane wave and the di®racted rays observed on a screen at some distance R from the slits. A series of light and dark bands is observed, corresponding to constructive and destructive in- terference between the light rays from the two slits. The geometry of the experiment is shown in Figure 12.1. Let us revise the calculation of the observed inten- Figure 12.1. The geometry of Young's sity, but now using complex wave notation double-slit experiment. à = <[A ei(kx¡!t)] = < fA exp [i(kx ¡ !t)]g : where < means `take the real part of'. It will turn out that this is not just a mathematical conve- nience, but rather goes right to the heart of quan- tum mechanics. Complex numbers are the natural language of quantum mechanics. For convenience, we will drop the explicit use of <. From Figure 12.1, provided R À d, the paths travelled from the two slits to the observation point P are: d d r ¼ r ¡ sin θ and r ¼ r + sin θ: 1 2 2 2 We use Huyghens' construction according to which we can replace the slits by sources of light which are in phase and have the same amplitude. Therefore, the amplitude of the wave at P is à = A exp [i(kr1 ¡ !t)] + A exp [i(kr2 ¡ !t)] d d ¼ A exp i[k(r ¡ 2 sin θ) ¡ !t] + A exp i[k(r + 2 sin θ) ¡ !t] = A exp i(kr ¡ !t) fexp [ik(d=2) sin θ] + exp [¡ik(d=2) sin θ]g = 2A exp i(kr ¡ !t) cos[k(d=2) sin θ]: The intensity is proportional to the square of the The SchrÄodingerWave Equation 3 amplitude and so I / 4A2 cos2[k(d=2) sin θ] (12.1) (Figure 12.2). For small values of θ, sin θ ¼ θ and so the intensity distribution becomes µ ¶ kθd I / 4A2 cos2 : (12.2) 2 There is an important piece of mathematics in what Figure 12.2 Di®raction pattern from double we have just done. In the complex representation, slits. the intensity can be found by taking the square of the modulus of the complex function Ã, that is, I = Ãä = jÃj2; where ä is the complex conjugate of Ã. We do not need to take the real part at all. Let us repeat the double-slit experiment using a photon counting detector. In this case, the arrival The experiment with light was ¯rst performed by of each photon on the screen is detected separately. G.I. Taylor in 1909, and the electron di®raction When the light is of high intensity, we observe the experiment was performed by JÄonnsonin 1961 us- 2 ing 50 keV electrons and a slit separation of only usual cos (kθd=2) variation of light intensity on the 1 ¹m. screen (Figure 12.3(c)). As the light intensity is de- creased, we begin to see the pattern `shimmering' as we begin to see the individual photons arriving (Figure 12.3(b)). At the very lowest light intensi- ties, we observe individual photons arriving at the screen (Figure 12.3(a)). The extraordinary thing about this experiment is that the individual par- ticles of light know where to appear on the screen to produce the standard di®raction pattern, despite the fact that they only arrive one at a time. To put it another way, the physics of light quanta must be such that there is zero probability that the individ- ual photons will land on the screen at the nulls in the di®raction pattern. It is interesting to quote the words of Richard Feynman about the signif- icance of the double-slit experiment for quantum mechanics. `We choose to examine a phenomenon which is impossible, absolutely impossi- ble, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the Figure 12.3. Young's double-slit experiment only mystery.' carried out using a photon counting detector. The SchrÄodingerWave Equation 4 These results provide the clue as to how we should interpret the cos2(kθd=2) intensity distribution. This function describes the probability that a single par- ticle arrives at any particular location on the detec- tor. Therefore, the quantum mechanical wave func- tion Ã(x) will be used to describe a particle such as The Wave Function an electron with the property that the probability The wave function is de¯ned so that the of ¯nding the particle in the interval x to x + dx is probability of ¯nding the particle in the interval x to x + dx is p(x) dx = jÃj2 dx = Ãä dx; (12.3) p(x) dx = jÃj2 dx = Ãä dx which has the form of a probability density func- tion. This represents a huge change from classical me- chanics. According to classical physics we can spec- ify precisely the position and momentum of a par- ticle. This has been replaced by a probabilistic de- scription of phenomena at the atomic level. This di®erent perspective is forced upon us as a result of the need to describe particles by waves and vice versa. The consequences are very profound as we shall see. Our next task is to determine the di®erential equa- tion which determines the wave function à { that will turn out to be the SchrÄodingerwave equation. First, however, let us investigate how we go about representing particles by waves. 12.2 The Representation of Particles by Waves We need a means of localising the particle in space { in other words, of ¯nding a probability density which describes where the particle is located. This is done by developing the idea of a wave packet which we construct out of a superposition of waves. This leads to a consideration of the properties of Fourier series and Fourier transforms, which come only late in the mathematics course. We need only one key idea from these topics and that is that we can use sums of sine and cosine functions to repre- sent any function f(x) within some range x1 · x · x2. The reason this works is that, mathematically, The SchrÄodingerWave Equation 5 the sequences of sine waves sin k0x; sin 2k0x; sin 3k0x; : : : plus the series of cosine waves cos k0x; cos 2k0x; cos 3k0x; : : : form a complete, orthogonal set of functions, mean- ing that we can synthesise any pattern we like by selecting appropriately the constants A0;A1, A2, ::: and B1, B2, ::: in the series expansion X1 X1 Ã(x) = A0 + An cos nk0x + Bn sin nk0x: n=1 n=1 (12.4) This sum is the Fourier series for the function Ã(x) de¯ned in the interval x1 · x · x2. These ideas suggest a way in which we can rep- resent particles by waves. As an example, con- sider building up a gaussian distribution in the x- direction, which is to represent the probability den- sity distribution of the particle in that coordinate. Suppose the particle has momentum p, so that, according to the de Broglie hypothesis, the wave- length associated with it is ¸0 = h=p. The corre- sponding wavevector is 2¼ 2¼p k0 = = : (12.5) ¸0 h Figure 12.4. The distribution of terms The waves associated with k are sine or cosine 0 in the Fourier Series with ¢k = 0:05. waves, A sin k x or A cos k x, and they are not lo- 0 0 0 We are bearing in mind that in the calised at all { the extend all the way from ¡1 to limit of a large number of terms, the +1 and this is not very helpful. binomial series tends to a gaussian Now, let us start with A cos k0x and add some other distribution. cosine waves and, for illustrative purposes, let us add together a binomial sequence of terms, cen- . tred on k0, but separated from it by §n ¢k0, where . ....... .. p(k) . .. .. .. ¢k ¿ k . Figure 12.4 shows the binomial distri- . .. 0 0 . .. .. ................ ...... bution of amplitudes of cosine waves, which are the . ...... ...... ...... ...... 8 . ...... ...... coe±cients of the expansion of (1 + x) , that is, . ...... ...... ...... ...... ...... ...... ...... 8 2 3 4 . ...... .................... (1 + x) = 1 + 8x + 28x + 56x + 70x . .......... .......... .......... .......... 5 6 7 8 . .......... .......... + 56x + 28x + 8x + x ; . .......... .......... .......... .......... ............................ .............. .............. .............. where we have chosen k0 = 1 and ¢k0 = 0:05. .................................................................................................................................................................................................................................................................................................................................................................................................................... .......................... .... .... .... .......... 0 k0 k The SchrÄodingerWave Equation 6 The result of adding together this series of cosine waves is shown in Figure 12.5 by the oscillating line which has maximum at the origin, but tends rapidly to zero in a ¯nite distance rather than os- cillating to in¯nity.