Wave Mechanics in One Dimension

Wave Mechanics in One Dimension Wave-Particle Duality The wave-like nature of light had been experimentally demonstrated by Thomas Young in 1820, by observing interference through both thin slit diffraction and thin film interference. The early evidence for the wave like nature of light came from interference situations such as the experiment illustrated below: Α Source A Observer Β θ θ θ θ B B travels an extra path length of 2 d sin θ relative to A d d d sin θ d sin θ In this experiment two beams of light from a common distant source reflect from two parallel surfaces spaced a distance d apart. The waves are intially described by by harmonic traveling waves which we assume are initially in phase because of the common source. However as seen in the above figure, Ray B must travel an extra distance compared to Ray A to arrive at the second set of dashed perpendiculars. This is double the base of the one of the indicated right triangles. This extra distance s =2dsin θ will result in a phase shift between the two rays. As one varies the angle θ from 0 o to 90 o the extra path difference varies from 0to2d. Assuming that λ < d/2, it will be possible to find θ values such that the waves emerge in phase or completely out of phase depending on whether the path difference is an integral number of wavelengths (in phase) or an odd number of half wavelengths (out of phase). A modern application of interference is x-ray crystallography. Here we consider interference from a large number of parallel planes. An extreme example would be a huge number of parallel planes in a crystal such as graphite. 1

x- rays crystal planes d d = 0.123 nm for graphite d + thousands of more planes! Because each plane is separated by the same distance d, ifonechoosesan angle θ to insure constructive interference from two adjacent planes, all planes will constructively interfere giving a very tight intensity pattern resembling the below: I 1st order 2nd order 3rd order 1 λ 2 d 2 λ 2 d 3 λ 2 d sin θ The crystal acts as a sort of diffraction grating, and one effectively only observes strong reflections certain angles classified according to the order number n. This phenomena is called Bragg scattering. The angle between diffractive peaks can be used to measure the crystal plane separations at various angles thus allowing one to model the crystalline structure think DNA! Later was realized that light also had a particle nature. Maxwell predicted light pressure and concluded that a beam of light with energy E tranfered a momentum p = E/c when absorbed in a surface. Einstein explained the photoelectric effect by surmising that light did not transfer energy to a photoelectric surface in a smooth continuous manner but rather in the form of photons 2

with discrete quanta of energy. The energy of the photon was E γ = hω where ω = frequency/(2π). Combining the Einstein condition with the Maxwell momentum formula we have: p = E c = hω c = 2π h λ = h λ where h = h 2π Since their discovery by J.J. Thomson, electrons had always been considered as small, charged particles. However in 1927 Davisson & Germer observed the Bragg scattering of electrons and thereby measured the electron wavelength. They found electrons had the exact relationship between momentum and wavelength as photons namely λ = h/p. This relation was proposed earlier by Louis de Broglies to explain the presence of discrete energy levels in the hydrogen atom spectrum as an alternative to the Correspondence Principle argument proposed by Bohr. De Broglie explained the presence of discrete energy levels by imagining the electrons formed circular standing waves with circumferences given by integral multiples of the electron wavelength. This electron wavelength is called the de Broglie wavelength. The wavelength formulas for electrons and photons are identical when expressed in momentum. The only difference is E = cp for photons while E = p 2 /(2m) for non-relativistic electrons. Constructing a wave equation for electrons. To account for the Bragg scattering experiments of Davisson & Germer and G.P. Thomas we should think of electrons as traveling waves in close analogy with x-ray photons. We are therefore possibly looking for waves of the form: Ψ(x, t) =A cos (kx ωt) (1) Thewavenumberk =2π/λ must be chosen according to the de Broglie condition: λ = h p k = 2πp h = p h (2) 3

It is also desirable, to envoke the Einstein condition relating frequency to energy: E = hf= h ω 2π = hω (3) Of course this leaves several questions. What quantity does Ψ(x, t) represent (whats waving anyway)? What is the wave equation for Ψ(x, t)? Lets begin with the wave equation. Try to use the classical wave equation The classical wave equation for sound, mechanical, and electromagnetic waves would read: 1 2 x 2 Ψ(x, t) = 1 2 c 2 Ψ(x, t) (4) t2 Lets try this out for an electron described by Ψ(x, t) =A cos (kx ωt) 2 x 2 A cos (kx ωt) = A k2 cos (kx ωt) or 1 2 ω2 c 2 A cos (kx ωt) = A cos (kx ωt) t2 c2 A k 2 cos (kx ωt) = A ω2 cos (kx ωt) c2 We have a consistent solution as long as: k 2 = ω2 c 2 k = ω c p = E c (5) wherewehaveused hk = p and hω = E from Eq. (2) and (3). 1 c is the speed of wave propagation. For an E&M wave Ψ = E. 4

Now the problem is that p = E/c is the correct energy - momentum relation for a photon, but: E = p2 + V (x) (6) 2m is the correct expression for an electron. Clearly we need to modify the our wave equation while still maintaining its desirable features such as simplicity (equate space derivatives to time derivatives) and linearity. A new, improved wave equation for electrons Following our failed attempt to use the classical wave equation, we might begin by casting Eq. (6) in terms of k and ω and multiplying both sides by the wave function Ψ(x, t) : hωψ(x, t) = h2 k 2 Ψ(x, t)+v (x)ψ(x, t) (7) 2m In analogy with the classical wave equation we want to replace k 2 and ω with differential operators. For the case a free particle wave function Ψ(x, t) = A cos (kx ωt) the obvious operator required to create a factor of k is ǩ = / x while to create a factor of ω we require ˇω = / t. 2 Lets try this choice for the case of a free particle (V (x) =0). hωψ(x, t) = h cos (kx ωt) = hω sin (kx ωt) (8) t h 2 k 2 h2 Ψ(x, t) = 2m 2m 2 x 2 cos (kx ωt) = h2 k 2 2m cos (kx ωt) (9) Since we want the condition h 2 k 2 /2m = hω, we would very much like Eq. (8) to equal Eq. (9) aside from possible minus signs. We are actually very close, except for the 90 o phase shift between the sin and cos function. Is there a way of inducing such a phase shift algebraically? 2 We denote operators by placing aˇover quantities such as ˇω and ǩ 5

Euler s representation to the rescue! The solution is to exploit the famous computational trick exploited frequently in physics and engineering of recasting the traveling wave into a complex wave function of the form: exp (ikx iωt) =cos(kx ωt)+i sin (kx ωt) (10) This allows us to simply include a factor of i 1tosaytheˇω operator and thus phase shift the sin (kx ωt) ofeq. (8)backintocos(kx ωt) requiredby Eq. (9). Lets redo Eq. (8) for the real parts (R) with this additional factor of 1: Ψ=exp(ikx iωt) =cos(kx ωt)+i sin (kx ωt) R (i h t ) Ψ = hω cos (kx ωt) (11) ( ) R h2 2 2m x 2 Ψ = h2 k 2 2m cos (kx ωt) (12) This clearly does the trick, since it implies h 2 k 2 /2m = hω which is our desired outcome for a free particle. Direct computation shows that this works for the imaginary part of the wave function as well. For the case of a particle in a potential V (x) wewouldhave: 3 ( hǩ)2 Ψ(x, t)+v (x)ψ(x, t) = hˇωψ(x, t) where 2m ǩ = i x and ˇω = i t Substituting hǩ =ˇp and hω = Ě we have: 3 The exact choice of the i phase for Quantum Measurement. ǩ = i / x will be made clear in the chapter on 6

Schrödinger s Equation ˇp 2 Ψ(x, t)+v (x)ψ(x, t) =ĚΨ(x, t) 2m ˇp = i h x and Ě = i h t (13) Equation (13) is known as Schrödinger s Equation which will form the theoretical foundation for the rest of this course. It has an elegant and fascinating feature of being a statement of the conservation of energy written in terms of operators for p and E. The operators operators must operate on a wave function Ψ(x, t). The presence of the i = 1 in the operator expressions imply that Ψ(x, t) in general will be a complex function with a real and imaginary part. The complex nature of Ψ(x, t) is part of its essence unlike the case in classical physics, we are not treating the complex nature of the wave function as a computational trick with the intention of throwing away the imaginary part as a computational artifact when looking for physical solutions. Rather it is far better to think of the real and imaginary parts of Ψ(x, t) as two complementary fields, in much the same way as having two independent E(x, t) andb(x, t) wave functions describing an electromagnetic wave or having a pressure and displacement waves describe a sound wave. A suggested exercise is to show explicitly that Ψ(x, t) =exp(ikx iωt) does indeed satisfy the Schrödinger Equation. What is waving in the wave function Ψ(x, t)? 7

Again we will argue from analogy with classical waves. The square of a classical wave function is the intensity. For the case of an electromagnetic wave, the square of the electric field is proportional to the energy density. If we think of that energy as carried in the form of photons of fixed energy hω, the square of the electric field becomes proportional to the photon density or the probability density of finding a photon in a particular location. Since probability densities are postive definite, our best bet is to identify the squared compex modulus or Ψ(x, t) 2 =Ψ (x, t)ψ(x, t) with the PDF (probability density function) of finding the electron at a given x location. 4 For the case of one dimension, the probability of finding an electron between x and x + x is given by: Prob(x, x + x) = x+ x x dx Ψ (x, t)ψ(x, t) (14) As implied in Equation (14), this probability will generally be dependent on time. As we will show below, for a wide range of useful situations, the PDF s are time independent which correponds to a static or stationary solution. Normalization We will generally use normalized wave functions which reinforce the connection of Ψ(x, t) Ψ(x, t) with the PDF of finding an electron at a given x location. Assuming that the probability of the electron at any location from <x<x must be 1, the PDF function must obey: + dx PDF(x) = 1 (15) A properly normalized wave function therefore obeys: + 1= dxψ(x, t) Ψ(x, t) (16) 4 This is called the Born interpretation of the Schrödinger wave function. 8

Because the Schrödinger equation is linear and homogeneous one can scale any solution by a constant to obtain a new solution. Hence if Ψ(x, t) isasolution, N Ψ(x, t) is a solution as well. This latter observation can usually be used to obtain a properly normalized wave function from an initially unnormalized wave function by solving for the coefficient N which satisfies Eq. (16). + N 2 dxψ(x, t) Ψ(x, t) =1 N = 1 + dxψ(x, t) Ψ(x, t) (17) An unfortunate exception to the normalization procedure is the traveling wave Ψ(x, t) =exp(ikx iωt) solution, since in this case Ψ(x, t) Ψ(x, t) =1andthe normalizing integral + dx Ψ(x, t) Ψ(x, t) +. Stationary solutions and the time independent S.E. We being by writing the Schrödinger equation as a differential equation rather than an operator equation: 2 h2 2m x 2 Ψ(x, t)+v (x)ψ(x, t) =i h Ψ(x, t) (18) t The classic technique for obtaining a class of solutions of linear partial differential equations of this form is called separation of variables where we write the wave function as a product of space and time functions: Ψ(x, t) = ψ(x)h(t). For the case of the Schrödinger Equation these solutions are called stationary solutions. Substituting Ψ(x, t) = ψ(x)h(t) into Eq.(13) we have: ( ) H(t) h2 2 2m x 2 ψ(x)+vψ(x) = ψ(x) (i h t ) H(t) (19) The trick is to rearrange Eq. (19) by separate the t and x dependence and setting 9

both forms to a constant of separation. 5 ( 1 h2 ψ(x) 2m ) 2 x 2 ψ(x)+vψ(x) = E (20) E = 1 (i h t ) H(t) H(t) (21) Eq. (21) is written as a PDE (partial differential equation). However since there is no dependence on x we can write it as an ordinary DE which can be easily solved by re-arrangement and explicit integration: dh H = E i h dt H(t) =exp ( īh ) Et (22) Comparing the time dependence implied by Eq. (22) with the time dependence of our complex traveling wave solution Ψ(x, t) =exp(ikx iωt) =exp(ikx) exp ( iωt) we see that E = hω and hence our separation variable is associated with the total energy E of electron. Our stationary solution becomes: Ψ(x, t) =ψ(x) exp ( īh ) Et where ψ(x) solves a slightly re-arranged form of Eq. (20) called the time independent Schrödinger Equation. Time independent SE 2 h2 2m x2ψ(x)+v (x)ψ(x) =Eψ(x) (23) 5 We will use E for the constant of separation. We will identify it with the energy shortly. 10

We call this the stationary solution since the probability density function is independent of time indicating that the electron is stationary as you can see by direct computation: PDF(x) =Ψ (x, t)ψ (x, t) = ( ψ(x)exp ( īh )) ( Et ψ(x)exp ( īh )) Et = ψ (x)ψ(x) (24) In such stationary solutions the exp ( iet/ h) = exp( iωt) part simply changed the phase of ψ(x) and did not affect its modulus. An important final word is that since the Schrödinger equation is linear you can always make linear combinations of solutions to obtain new valid solutions. In all cases which we will consider in this course we will always be able to build the SE solutions we need from a sum of stationary solutions: Ψ(x, t) = i a i ψ i (x)exp( iω i t) (25) The complex a i constants (called amplitudes) will often be selected in order to match required boundary conditions. An example which we will encounter shortly involves the superposition of two complex traveling waves free particle solutions: Ψ 1 (x, t) =exp(ikx iωt) and Ψ 2 (x, t) =exp( ikx iωt) wherewenotethatbothψ 1 and Ψ 2 have the same time dependence exp ( iωt) dependence required by Eq. (23) and we use a single ω for both solutions since the free particles satisfy E = p 2 /2m or ω = hk 2 /2m = h( k) 2 /2m. Wenote(by setting the exponential argument equal to constant to follow a fixed phase) that Ψ 1 (x, t) represents a traveling wave which travels along the +ˆx axis and Ψ 2 (x, t) 11

represents a traveling wave which travels along the ˆx axis. Schrödinger s equation insures us that The linearity of Ψ 1 +Ψ 2 2 =cos(kx)exp( iωt) (26) Ψ 1 Ψ 2 2i =sin(kx)exp( iωt) (27) are also solutions of the free particle SE. It is easy to verify that standing wave 6 solutions such as cos (kx)exp( iωt) satisfies the time dependent Schrödinger equation (with V (x) = 0) where as our original (Eq. (1)), motivating form cos (kx ωt) doesnot! Itisalsoeasytoverifythatψ(x) =cos(kx) or sin(kx) satisfies the free particle time independent Schrödinger equation. Normalization of non-stationary states With reference to Eq. (18), it would clearly be disasterous if Ψ(x, t) werea valid solution of the time dependent SE and the normalization integral N 2 = + dxψ(x, t) Ψ(x, t) changed as a function of time. If N had a time dependence, the properly normalized wave function N Ψ(x, t) would pick up an additional time dependence and in general would no longer be a solution of the time dependent SE. In the homework, I will ask you to explore the property of SE solutions which prevents this time dependent normalization. Assuming this problem doesn t exist, one frequently normalizes at t = 0 and has a normalizing factor of: 1 N = + dx Ψ(x, 0) Ψ(x, 0) For the case of stationary states this becomes: (28) N = 1 + dx ψ(x) ψ(x) (29) 6 Classical standing waves are the sum or difference two oppositely directed traveling waves 12

Important Points 1. The wave equation for matter waves although analogous to the classical wave equation has important differences which requires the use of complex wave functions. 2. In Schrödinger quantum mechanics physical quantities such momentum and energy are replaced by operators such as ˇp = i h / x and Ě = i h / t. The wave equation follows from conservation of energy written as an operator expression ˇp 2 /2m + V (x) =Ě operating on a complex wave function Ψ(x, t) 3. The squared modulus of the wave function, Ψ(x, t) Ψ(x, t), describes the time dependent probability density function for finding an electron at a given x location. We generally normalize the wave function via 1 = + dx Ψ(x, 0) Ψ(x, 0). 4. The linearity of the SE allows us to frequently build up general solutions of the time dependent SE by linear combinations of the stationary solutions of the (factorized) form Ψ(x, t) = ψ(x) exp( iωt) where ψ(x) satisfies a time independent SE. The normalization condition becomes 1= + dxψ(x) ψ(x). In the de Broglie picture of the Bohr atom, the wave nature of the electron created and a standing wave condition lead to a discrete spectrum of bound state energy levels for the hydrogen atom. Although the mechanism is somewhat different, there is a discrete spectrum of SE stationary states for electrons bound in localizing potentials. We will explore this discrete energy spectrum for several solutions of the time independent Schrödinger equation in the next lecture. Later we will concentrate on the nature of quantum measurements and the deeper role of probabilistic nature of Ψ(x, t) 13