CHM 532 Notes on Wavefunctions and the Schrödinger Equation

Transcription

1 CHM 532 Notes on Wavefunctions and the Schrödinger Equation In class we have discussed a thought experiment 1 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 f(x) x Before discussing the implications of the double-slit experiment, we first review an important property of wave packets. Recall that for classical wave motion in an ideal string, 1 A 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

2 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 f(x) = 1 2π x e x2 /2( x) 2 (1) 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: and x x f(x) dx = 1 (2) f(x) dx =.67 (3) xf(x) dx = 0 (4) x 2 f(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 x 2 is ( x) 2 so that x 2 x 2 = ( 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 Gaussian 2 g(k) = 1 2π f(x)e ikx dx (7) = 1 1 e x2 /2( x) 2 e ikx dx. (8) 2π 2π x 2 In evaluating the Fourier integral of a Gaussian, we use the important result e ax2 +bx dx = (π/a) 1/2 e b2 /4a. 2

3 = 1 2π e k2 ( x) 2 /2. (9) 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 k = 1 (10) x Eq. (9) becomes g(k) = 1 2π e k2 /2( k) 2. (11) 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 x k 1 (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

4 3 An Implication of the Double-slit Experiment From the double-slit experiment performed on quantum particles 3, 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 doubleslit 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 P = b a P (x, t) dx = b a Ψ (x, t)ψ(x, t) dx (14) 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 Principle 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 p = hk = h (15) λ where h = 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 3 All 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

5 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 x p h (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 uncertainty 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 ability 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. The uncertainty principle tells us that we cannot simultaneously know exact values of the position and momentum of a particle at the same time, but the uncertainty principle does not imply that we can know nothing about each variable. The wavefunction used to calculate the probability in Eq. (13) is mathematically analogous to a classical wave. Like a classical wave, the wavefunction can be decomposed into a weighted mixture of sinusoidal waves by calculating its Fourier transform. From the de Broglie relation [Eq. (15)], we know each sinusoidal wave of definite wavelength corresponds to a particular momentum. Consequently, the Fourier transform of the wavefunction used in Eq. (13) gives a new kind of wavefunction expressed in terms of the momentum of the particle rather than the position. We can then define the momentum space wavefunction by Φ(p, t) = 1 Ψ(x, t)e ipx/ h dx. (17) 2π h The momentum space wavefunction is the Fourier transform of the position space wavefunction where we take the momentum p to be the Fourier transform variable rather than the wave vector k. The factor of 1/ h that appears in Eq. (17) is a result of the variable transformation from k to p. It is left as an exercise to show that δ(x x ) = 1 e ip(x x )/ h dp (18) 2π h 4 A more formal and general derivation of the uncertainty principle is given later in the semester 5

6 and Ψ(x, t) = 1 2π h Φ(p, t)e ipx/ h dp. (19) The interpretation of the momentum space wavefunction is completely analogous to the usual coordinate space wavefunction. We interpret P (p, t)dp = Φ (p, t)φ(p, t)dp = Φ(p, t) 2 dp (20) to be the probability that a measurement of the momentum of the particle at time t lies in the range p to p + dp. 5 The Schrödinger Equation Just like classical waves obey the classical wave equation, the wavefunctions that provide information about the probability of measuring the position of particles obey a wave-like equation that we call the Schrödinger equation. The Schrödinger equation provides one of two equivalent formulations of the laws of non-relativistic quantum mechanics that were developed in the mid 1920 s (the other was developed by Heisenberg and is often called matrix mechanics). The Schrödinger equation is a law of nature, and like other laws of nature (Newton s laws, Couloumb s law,...) the Schrödinger equation cannot be derived. Its ultimate justification lies in the extent to which the Schrödinger equation agrees with experiment. We currently believe the Schrödinger equation when properly interpreted describes all physical phenomena provided the particle velocities do not approach the speed of light (where a relativistic treatment is necessary). Although the Schrödinger equation cannot be derived, we can give arguments that help us understand the particular form of the equation. The thinking that we discuss here is similar to that used originally by Schrödinger when he guessed properly the equation that quantum particles obey. We make the following observations: 1. Owing to the interference effects observed in the double-slit experiment, the function Ψ(x, t) = Ae i(kx ωt) (21) must be a solution to the equation [Eq. (21) explains classical interference the mathematical form of which is identical to the interference probability distribution observed for quantum particles]. 2. From the Davisson-Germer experiment, we have the de Broglie relation [Eq. (15)]. 3. We assume the Planck-Einstein relation E = hν = hω. (22) 6

7 4. We expect the wave-like equation for quantum particles to have, at most, a single derivative in time. With a differential equation that is first order in time, we have only one boundary condition in time. We expect that whatever we can say about a system at one time should be sufficient to predict what we can say about a system at later times. It is unphysical to expect that we need information about the system at two or more different times. This restriction in the order of the time derivative is often called causality. Newton s second law has a second time derivative, but we satisfy the two required boundary conditions by specifying the coordinates and momenta of the particles at one time. Such a specification is forbidden by the uncertainty principle in the quantum domain, and we restrict ourselves to a single time derivative. Using the restrictions given by these assumptions, we now ask what equation would give Eq. (21) as a solution. We first differentiate Eq. (21) with respect to time and use the Planck-Einstein relation or Ψ t = t Aei(kx ωt) = iωψ = i Ē h Ψ (23) h Ψ = EΨ = (T + V )Ψ (24) i t where we have decomposed the total energy E into its kinetic energy T and potential energy V. We next differentiate Eq. (21) twice with respect to the coordinate x and use the de Broglie relation 2 Ψ x 2 = k2 Ψ = p2 2 Ψ. (25) h Then T Ψ = p2 2m Ψ = h2 2 Ψ 2m x. (26) 2 Using Eq. (24), we then obtain the Schrödinger equation in one dimension h i In three dimensions Eq. (27) generalizes to h i where we have used the short-hand notation Ψ t = h2 2 Ψ + V Ψ. (27) 2m x2 Ψ t = h2 2m 2 Ψ + V Ψ (28) 2 f(x, y, z) = 2 f x f y f z 2 (29) ( 2 is often called the Laplacian). Equations (27) and (28) are often called the timedependent Schrödinger equation to distinguish them from another equation called the Schrödinger equation that is given in Section 7 (the other equation is the one called the Schrödinger equation in our textbook and the one we learn to solve this semester). 7

8 6 The Equation of Continuity and the Time-dependent Schrödinger Equation There is an important equation that describes the flow of particles through space. This equation, called the equation of continuity, is an expression of the conservation of the number of particles. In words, the equation of continuity says that if one examines some volume of space with particles entering or leaving, the net flow of particle current out of the volume of space must equal the time rate of change of the integrated density of particles in the volume of space. The equation of continuity is a common condition used in fluid dynamics and to study the flow of charges in electricity and magnetism. To derive the equation of continuity, we need a definition and a theorem. Definition: Let v(x, y, z) be a vector function of x, y and z. is defined by v = v x x + v y y + v z z Then the divergence of v (30) The Divergence Theorem: Let an area S enclose a volume of space τ, and let J be any vector function defined in τ. Then S J ˆn ds = τ J dτ (31) where ˆn is a unit vector normal (perpendicular) to the surface at ds. Although the proof of the divergence theorem is not difficult, we do not provide the proof here, but rather use the divergence theorem to derive the equation of continuity. Figure 1 represents a cylindrical volume of space and the arrows represent current flow entering the left face of the region of space and leaving through the right face. To simplify the development, we assume there is no flow of particles perpendicular to the flow indicated in the figure. We let J be the current density; i.e. the number of particles per unit area per unit time in the chosen volume of space. The usual particle current I is obtained by integrating the current density over the surface area I = J ˆn ds (32) S The negative sign in Eq. (32) accounts for the negative contribution to the current coming from the right hand face (where J ˆn is positive) and the positive contribution coming from the left face (where J ˆn is negative). If the signs are confusing, notice that ˆn points in a direction opposite to J on the left and in the same direction as J on the right. The current is also equal to the time derivative of the integrated particle density I = t τ ρ dτ. (33) 8

9 Figure 1: Equating Eqs. (32) and (33) and using the divergence theorem ρ dτ = t τ S J ˆn ds = τ J dτ (34) or J + ρ = 0. (35) t Equation (35) is called the equation of continuity and is a differential expression for particle conservation. We now show that the time-dependent Schrödinger equation satisfies the equation of continuity. By demonstrating that the Schrödinger equation satisfies the equation of continuity, we verify our equation makes physical sense and we provide a quantum mechanical expression for the probability current density J. We require an expression for J in terms of our wavefunctions for use later in the course. The quantum expression for the particle density is given by ρ = Ψ Ψ (36) the absolute square of the wavefunction. We differentiate ρ with respect to time to obtain ρ t = Ψ Ψ Ψ + Ψ t t. (37) 9

10 The time derivatives of the wavefunctions are given by the Schrödinger equation Ψ t = 1 ] [ h2 i h 2m 2 Ψ + V Ψ (38) and Then Now Ψ t ρ t = = 1 ] [ h2 i h 2m 2 Ψ + V Ψ. (39) h 2mi [( 2 Ψ )Ψ Ψ 2 Ψ]. (40) [Ψ Ψ ( Ψ )Ψ] = Ψ 2 Ψ ( 2 Ψ )Ψ (41) so that the equation of continuity [Eq. (35)] is satisfied if we identify J = h 2mi [Ψ Ψ ( Ψ )Ψ]. (42) It is possible to show that if a second derivative in time had appeared in our quantum mechanical laws, it would have been impossible to construct a probability current density that could satisfy the equation of continuity. 5 It is also impossible to satisfy the equation of continuity if the potential energy function is allowed to be complex. This latter statement is proved in a homework problem. 7 The Time-independent Schrödinger Equation We now attempt to solve the time-dependent Schrödinger equation using the separation of variables method. As with the classical equation for wave motion, the separation of variables method gives solutions that are valid only for specific time-dependent boundary conditions. Learning the physical nature of the boundary conditions that lead to the separation of variables solution is an important goal of CHM 532. We assume a solution to the time-dependent Schrödinger equation of the form Ψ( r, t) = ψ( r)φ(t) (43) and substitute this product wavefunction into Eq. (28) to obtain ( ψ( r) h ) ) φ = φ(t) ( h2 i t 2m 2 ψ + V ψ. (44) 5 See D. Bohm, Quantum Theory, (Dover Publications, New York, 1951), Chapter 4 10

11 As usual in the separation of variables method, we move all terms depending on time only to one side of the equation and all terms dependent only on coordinates to the other side of the equation with the result h 1 dφ i φ dt = 1 ) ( h2 ψ 2m 2 ψ + V ψ = E. (45) Because the left side of the equation depends on time only and the right side of the equation depends on coordinates only, each side must equal a separation constant. Because the separation constant has units of energy, we call the separation constant E. The differential equation for the temporal part of the total wavefunction can be solved immediately giving The right side of the equation becomes φ(t) = e iet/ h. (46) h2 2m 2 ψ + V ψ = Eψ (47) and is often called the time-independent Schrödinger equation. When there is no ambiguity, Eq. (47) is often just called the Schrödinger equation. Our textbook refers to Eq. (47) as just the Schrödinger equation, and we spend a good portion of the remainder of the semester solving Eq. (47) for a variety of problems and learning to interpret its solutions. The solutions to the time-dependent Schrödinger equation that we obtain using separation of variables take the form Ψ( r, t) = ψ( r)e iet/ h, (48) and we emphasize once again that these separation of variables solutions are physical solutions only for particular time-dependent boundary conditions. There is one property of these separation of variables solutions that we can understand quickly. The particle density associated with Eq. (48) is ρ( r, t) = Ψ ( r, t)ψ( r, t) = ψ ( r)ψ( r) (49) and is independent of time. Consequently, the solutions to the time-independent Schrödinger equation and the separations of variables solutions are often called stationary states. 11