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CHM 532 Notes on Wavefunctions and the Schrödinger Equation

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CHM 532 Notes on Wavefunctions and the Schrödinger Equation CHM 532 Notes on Wavefunctions and the Schr¨odinger Equation In class we have discussed a thought experiment1 that contrasts the behavior of classical particles, classical waves and quantum particles. The thought experiment consists of some method of generating the particles or waves (e.g. a gun), a barrier with two slits and a detector. The details of this thought experiment can be found in The Feynman Lectures on Physics Volume 3, Chapter 1 (Addison-Wesley, Reading MA, 1965) and are not discussed in these notes. Here, we discuss some of the implications of the thought experiment. 1 Wave Packets 1.2 1 0.8 f(x) 0.6 0.4 0.2 0 -6 -4 -2 0 2 4 6 x Before discussing the implications of the double-slit experiment, we first review an im- portant property of wave packets. Recall that for classical wave motion in an ideal string, 1A thought experiment does not imply that the experimental results would be different if the experiment were actually performed. Rather, the thought experiment is a simplification of real experiments. We have every confidence that if the double-slit apparatus could actually be constructed for electrons, the results would be identical to those discussed in class 1 a wave packet is some localized disturbance in the string. Because f(x − ct) for any twice differentiable f(x) is a solution to the classical wave equation, at t = 0 there are a large set of possible wave packets. For simplicity we choose a Gaussian wave packet that has the form 1 2 2 f(x) = √ e−x /2(∆x) (1) 2π∆x and is plotted in the figure for the case that ∆x = 1. The parameter ∆x is often called the standard deviation of the Gaussian, and the standard deviation is a measure of the width of the wave packet. The Gaussian function has the properties that follow: Z ∞ f(x) dx = 1 (2) −∞ Z ∆x f(x) dx ∼= .67 (3) −∆x Z ∞ xf(x) dx = 0 (4) −∞ and Z ∞ x2f(x) dx = (∆x)2. (5) −∞ Equation (2) is a normalization condition (the total area under the Gaussian curve is unity), and Eq. (3) implies that approximately two-thirds of the total area under the Gaussian lies in the range −∆x ≤ x ≤ ∆x. Equation (4) expresses that the average of x with respect to f(x) is zero, and Eq. (5) says that the average of x2 is (∆x)2 so that hx2i − hxi2 = (∆x)2 (6) the standard expression for the standard deviation in probability and statistics. We now imagine that we create a Gaussian wave packet in a string (by pulling the string in some way), and we ask into what distribution of sinusoidal wavelengths is the Gaussian wave packet composed. As we have learned, the distribution of wavelengths is given by g(k) the Fourier transform of the Gaussian2 1 Z ∞ g(k) = √ f(x)e−ikx dx (7) 2π −∞ ∞ 1 Z 1 2 2 = √ √ e−x /2(∆x) e−ikx dx. (8) 2π ∞ 2π∆x 2In evaluating the Fourier integral of a Gaussian, we use the important result ∞ Z 2 2 e−ax +bxdx = (π/a)1/2eb /4a. −∞ 2 1 2 2 = √ e−k (∆x) /2. (9) 2π From Eq. (9) we see that the Fourier transform of the Gaussian wave packet is a Gaussian distributions of wavelengths [remember that the wave vector k is related to the wavelength λ by k = 2π/λ]. Writing 1 ∆k = (10) ∆x Eq. (9) becomes 1 2 2 g(k) = √ e−k /2(∆k) . (11) 2π We then find that for a Gaussian wave packet having width (standard deviation) ∆x, the distribution of wavelengths is also a Gaussian of width 1/∆x; i.e. the widths of the wave packet and its Fourier transform are not independent. The result that we have proved for a Gaussian wave packet is general for all wave packets. The width of the packet is always inversely related to the width of the distribution of wavelengths. In fact, it can be proved in general that 1 ∆x∆k ≥ (12) 2 where the widths for both the wave packet and its Fourier transform are defined as in Eq. (6). 2 The Notion of Probability We next need to have some elementary notion about what is meant by the probability of an event. The usual treatment of probability can be more mathematically formal than needed in the study of quantum mechanics. Rather than giving definitions of probability (usually defined using set theory), it is perhaps more useful to explain the basic notions that we need in terms of a simple example. Let us consider a paper bag that contains 25 red marbles and 75 green marbles. If the bag is shaken so that the marbles are thoroughly mixed and one marble is drawn at random from the sack, the probability of obtaining a red marble is 25/100 and the probability of obtaining a green marble is 75/100. We interpret this probability to mean that if we consider a large collection of identically prepared bags each containing 25 red marbles and 75 green marbles, and we draw one marble at random from each sack, a red marble can be expected to be found 25% of the time and a green marble can be expected to be found 75% of the time. In a crude way, we calculate the probability by dividing the number of possible outcomes for a given event by the total number of possible outcomes. In the next section we must modify this description for the case that the possible outcomes form a continuum. 3 3 An Implication of the Double-slit Experiment From the double-slit experiment performed on quantum particles3, we can conclude that the distribution of particles observed at the detector obeys some of the same mathematical relations as found in the intensity distribution of classical waves. The results of the double- slit experiment do not imply that electrons (for example) are waves. Instead, we can say that the equations that govern the behavior of electrons must be similar in some way to classical wave equations. Because the interference observed in classical waves is described by taking the absolute square of a complex number, we can conclude that there exists some possibly complex function Ψ(x, t) such that P (x, t) dx = Ψ∗(x, t)Ψ(x, t) dx = |Ψ(x, t)|2 dx (13) where P (x, t)dx is the probability that the observation of a particle at time t gives a result between x and x + dx. The function Ψ(x, t) is called a wavefunction, and the probability of observing a position has been carefully defined using an infinitesimal interval. The reason we need the infinitesimal interval is there are a continuum of possible outcomes of a position measurement. Because the number of possible outcomes is infinite, the probability at a point is ill-defined; we can only define the probability over an interval. Another statement for probabilities over a continuum is Z b Z b P = P (x, t) dx = Ψ∗(x, t)Ψ(x, t) dx (14) a a represents the probability that a position measurement gives a result that lies in the interval a ≤ x ≤ b at time t. 4 The de Broglie Wavelength, the Uncertainty Princi- ple and Momentum Space Wavefunctions Having concluded that a wavefunction for particles exists, we need some understanding of the connection between the wavelengths associated with a wavefunction and particle properties. The relation between the wavelength and momentum of a particle was first proposed by de Broglie who wrote h p =hk ¯ = (15) λ whereh ¯ = h/2π with h Planck’s constant. In class we have given some of the ideas that de Broglie used to develop this relation, but what is more important is the experiment of Davisson and Germer who developed electron diffraction methods that verified the de Broglie 3All particles obey the laws of quantum mechanics. When we say quantum particles, we imply that we consider particles with sufficiently small mass that the effects of quantum mechanics can be observed. 4 relation. The wavelength associated with the momentum of a particle is often called the de Broglie wavelength. We assume Eq. (15) to be a verified experimental fact. An important consequence of the de Broglie relation is obtained from Eq. (12). We know from Fourier transform relations that the distribution of wavelengths and the degree of localization of a wave packet in space are inversely related. By substituting Eq. (15) into Eq. (12) we obtain h¯ ∆x∆p ≥ (16) 2 the Heisenberg Uncertainty Principle.4 The uncertainty principle is a direct consequence of our conclusion that quantum particles must obey some kind of wave equation. The uncer- tainty principle states that it is impossible to determine both the momentum and position of a particle simultaneously. Because the usual boundary conditions for Newton’s second law are the specification of the momentum and position of a particle at the same time, the boundary conditions for classical mechanics are excluded in the quantum domain. Our abil- ity to predict the future motion of objects in classical mechanics (often called determinism) is impossible in the quantum domain. Consequently, in quantum theory the most we can know about a physical system must be expressed in terms of probabilities. Equation (13) expresses information about the probability of finding a particle at some location in space. The location of particles is not the only physical information we might want to know about a system. For example, as in classical mechanics, we might also want to know something about the momenta of the particles.
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