LECTURE 10. WAVE MECHANICS AND SCHROEDINGER’S EQUATION Topics - Concepts of a wave function - Introduction to Schroedinger’s equation for a free particle - Application of Schroedinger’s equation to a particle constrained in a box - Time independent Schroedinger’s equation - Ehrenfest Theorems and applications - Basic postulates - Eigenvalues and eigenfunctions - Degeneracy OBJECTIVES At the end of the lecture you should be able to 1) Explain the meaning of the wave function Ψ, and state its properties. 2) State and show steps leading to the time- dependent Schrödinger equation of a free particle 3) Write down the time-dependent Schrödinger equation for a restricted particle. 4) Verify the correct wave functions for free and confined particles. 5) Solve simple problem involving the wavelength Ψ. 6) Show that the energy of a particle confined to a potential well is quantized, hence draw the appropriate energy level diagram. 7) Define eigenvalues, eigenstates and eigenfunctions 8) Explain the concept of degeneracy. 9) For trapped particles, state the features that are predicted by quantum theory but not by classical theory. 10.1 Concepts of a wave function We have already discussed in the previous lecture the fact that a moving particle can be considered as a wave, with specific wavelength, which is inversely proportional to the magnitude of the momentum p. In Mechanics or Waves and Optics courses, you have learnt that in a medium the wave disturbance in the x-direction can be represented by the transverse displacement (y). Recall the wave equation 2 y 2 y c 2 (10.1) 2 2 t x where c is a constant with the dimensions of velocity. Equation (10.1) is a differential equation. Similarly in Wave Mechanics ( a semi-quantum theory constructed to solve problems involving electrons and other sub-atomic particles, for which classical laws were found to be deficient) we look for an equation that can be used for the wave s representing particles. For this purpose we introduce a new wave function (), which at every point in space and time can be written as (x,y,z,t). This wave function has the following characteristics: 1) , in general, is not real but a complex function, that it is, it has a real part and an imaginary part. (Recall what real and complex numbers are). This means that the wave function should be interpreted in terms of probabilities instead of certainties. Therefore the square of is interpreted as the probability (per unit volume) that the particle will be found at that point. The probability that a particle is somewhere (in volume element dV) is unity so that 2 dV 1 (10.2) 2) is not a measure of any simple physical entity such as displacement. 3) is continuous in space. That is, the wave function should not have breaks in it for the solution to be valid. We note that in Wave mechanics, physical information about the particles is contained in complex wave functions from which probability predictions can be got. 10.2 The Schrödinger Equation The Schrödinger theory of quantum mechanics specifies the laws of wave motion that the particles that comprise microscopic systems obey. (In Newtonian Mechanics the Newton’s second law, and in electromagnetic theory the Maxwell’s equations serve this role). This is achieved by stating, for each system, the equation that controls the behaviour of the wave function , and also by giving the relation between the behaviour of the wave function and the behaviour of the particle. Schrödinger’s theory is therefore an extension of the de Broglie’s postulate. It also bears close resemblance of Newton’s theory of motion of particles in macroscopic systems. It is worth noting that the de Broglie’s postulate only says that the motion of a microscopic particle is governed by the propagation of an associated wave. It never says how the wave propagates. It was Erwin Schrödinger’s (1926) who developed an equation that tells us the behaviour of any wave function of interest. We shall assume a free particle, i.e. one that is not acted on by an external force, and further, that the motion is one-dimensional. Now consider a particle of momentum P and energy E. Its wavelength is given by h and p frequency by E . We can define h 2 P Wave number k (10.3) And E Angular frequency 2v (10.4) 2 For a free particle E P , thus 2m 2mE k (10.5) Now if the particle travels in the x-direction, the wave will also travel in the same direction. The harmonic wave (correct wave function) meeting this condition is x,t e it kx for motion in the positive x-direction (10.6) and x,t e it kx for motion in the negative x-direction (10.7) Note that in arriving at the correct harmonic wave function we have invoked the superposition principle of quantum mechanics: that is, if 1 and 2 are possible wave functions, then so is 1 2 . We now want to find out the wave equation obeyed by these wave functions. We rewrite Eq. (10.6) by expressing and k in terms of P, that is, Et x i 2mE eit kx e (10.8) We differentiate Eqn. (10.8) once with respect to time (t) and twice with respect to x to obtain E i (10.9) t and 2 2mE 2 2 (10.10) x Comparing equations (10.9) and (10.10) we notice that (10.11) 2 2 2 x,t i 2 x,t 2m x t This is the wave equation for a free particle; and it is called The-time dependent Schrödinger Equation in one dimension. If the particle is restricted the presence of a force is represented by some given potential energy function (V(x,t)), equation (10.11) now takes the form 2 2 V x,t (10.12) i t 2m t 2 (We have used the fact that (i 2 1) Exercise Generalize equation (10.12) into the three dimensions form. The general Schrödinger Equation is the basic equation of quantum mechanics. It serves as the equation of motion for the wave function and it allows us to calculate the wave function at any later time given the initial conditions at some initial time. Of the solutions to the Schrödinger equation, the standing-wave solutions are of the most importance. Wave mechanics explains the quantization of energy in terms of standing waves. Just as the possible standing waves in an air column or in a stretched string (see Fig. 10.1) must have an integral number of wavelengths or half-wavelengths with the air column or stretched string so does the discrete set of frequencies and energies of a quantum-mechanical particle bound within a system arise. Figure 10.1: The first four modes of vibration of a stretched string fixed at both ends. 10.3 Particle in a Box One simplest way of restricting a quantum-mechanical system is to restrict it to a one- dimensional ‘box’. The walls of the box are assumed to be perfectly elastic, and rigid, and the particle moves freely between these walls. V(x) 0 L x Figure 10.2 Potential energy for s particle in a one-dimensional box. The potential energy is infinite at x – 0 and x = L. If the particle has energy E and momentum P 2mE, then the harmonic waves traveling to the right and the left along the x-axis are, respectively, Et x i 2mE e (10.13) and Et x i 2mE e (10.14) Employing the superposition principle (from equations (10.13) and (10.14) we obtain Et x Et x i 2mE i 2mE x,t Ae Be (10.15) where for the sake of generality we have multiplied Eqns. (10.13) and (10.14) by constant A and B respectively. A standing wave will result provided A B. At x =0 and at x = L, ψ must be zero at all times. Thus 0,t 0 at x 0 (10.16) and L,t 0 at x L (10.17) Equations (10.16) and (10.17) are the boundary conditions. These boundary conditions are the same as those for a vibrating string fixed at both ends. From equations (10.16) and (10.17) we obtain from equation (10.15) AeiEt / BeiEt / 0 at x = 0 (10.18) and Et L L i 2mE i 2mE Ae Be 0 at x = L (10.19) From equation (10.18) we obtain B = -A, and therefore equation (10.19) reduces to L L i 2mE i 2mE iEt / Ae e e 0 (10.20) This equation is equal to L 2iA sin 2mE 0 (10.21) from the fact that ei ei sin 2i For the wave function to vanish (become zero) at the boundaries (see Fig. 10.2) we demand that the argument of the sine function be an integer multiple of π, Remember Sin = 0 if = 0, , 2, 3,4 etc., where = 180 L 2mE n (10.22) or 2 2 2 n n = 1,2,3 (10.23) En 2 2mL Equation (10.23) shows that the energies of the particles in a box are quantized. The lowest 2 2 2 energy is obtained by setting n = 1, and is the energy of the ground state E1 /2mL Shown in Figure 10.3 are the energy levels for a particle in a box as predicted by equation (10.23). The quantized energies (E1, E2, E3, etc) of the systems are called energy eigenvalues, and the states that correspond to the definite values of quantized energy are called energy eigenstates.