Chapter 4: Schroedinger Equation

Chapter 4 Schroedinger Equation Einstein’s relation between particle energy and frequency Eq.(3.83) and de Broglie’s relation between particle momentum and wave number of a corre­ sponding matter wave Eq.(3.84) suggest a wave equation for matter waves. This search for an equation describing matter waves was carried out by Erwin Schroedinger. He was successful in the year 1926. The energy of a classical, nonrelativistic particle with momentum p that is subject to a conservative force derived from a potential V (r) is E= p2 + V (r) . 2m (4.1) For simplicity lets begin first with a constant potential V (r) = V0 = const. This is the force free case. According to Einstein and de Broglie, the dis­ persion relation between ω and k for waves describing the particle motion should be ~2 k2 ~ω = (4.2) + V0 . 2m Note, so far we had a dispersion relation for waves in one dimension, where the wavenumber k(ω), was a function of frequency. For waves in three dimen­ sions the frequency of the wave is rather a function of the three components of the wave vector. Each wave with a given wave vector k has the following time dependence ~k2 V0 ej(k·r−ωt) , with ω = + (4.3) 2m ~ Note, this is a wave with phase fronts traveling to the right. In contrast to our notation used in chapter 2 for rightward traveling electromagnetic waves, we 199 200 CHAPTER 4. SCHROEDINGER EQUATION switched the sign in the exponent. This notation conforms with the physics oriented literature. A superposition of such waves in k−space enables us to construct wave packets in real space Z ³ ´ Ψ (r, t) = φω k, ω ej(k·r−ωt) d3 k dω (4.4) The inverse transform of the above expression is Z ³ ´ 1 Ψ (r, t) e−j(k·r−ωt) d3 r dt, φω k, ω = (2π)4 with à ! ³ ´ ~k2 V0 φω k, ω = φ (k) δ ω − − . 2m ~ (4.5) (4.6) Or we can rewrite the wave function in Eq.(4.4) by carrying out the trivial frequency integration over ω à " à ! #! Z ~k2 V0 Ψ (r, t) = φ (k) exp j k·r − + t d3 k. (4.7) 2m ~ Due to the Fourier relationship between the wave function in space and time coordinates and the wave function in wave vector and frequency coordinates ³ ´ φω k, ω ↔ Ψ (r, t) (4.8) we have ω φω (k, ω) ↔ j ∂Ψ (r, t) , ∂t (4.9) k φω (k, ω) ↔ − j∇ Ψ (r, t) , (4.10) k2 φω (k, ω) ↔ −∆ Ψ (r, t) . (4.11) where ∇ = ex ∂ ∂ ∂ + ey + ez , ∂x ∂y ∂z ∆ = ∇ · ∇ ≡ ∇2 = ∂2 ∂2 ∂2 + + . ∂x2 ∂y 2 ∂z 2 (4.12) (4.13) 201 From the dispersion relation follows by multiplication with the wave function in the wave vector and frequency domain ~ ω φω (k, ω) = ~2 k2 φ (k, ω) + V0 φω (k, ω) . 2m ω (4.14) With the inverse transformation the corresponding equation in the space and tieme domain is j~ ~2 ∂Ψ (r, t) =− ∆ Ψ (r, t) + V0 Ψ (r, t) . ∂t 2m (4.15) Generalization of the above equation for a constant potential to the instance of an arbitrary potential in space leads finally to the Schroedinger equation j~ ∂Ψ (r, t) ~2 =− ∆ Ψ (r, t) + V (r) Ψ (r, t) . ∂t 2m (4.16) Note, the last few pages ar not a derivation of the Schroedinger Equation but rather a motivation for it based on the findings of Einstein and deBroglie. The Schroedinger Equation can not be derived from classical mechanics. But classical mechanics can be rederived from the Schroedinger Equation in some limit. It is the success of this equation in describing the experimentally ob­ served quantum mechanical phenomena correctly, that justifies this equation. The wave function Ψ (r, t) is complex. Note, we will no longer underline complex quantities. Which quantities are complex will be determined from the context. Initially the magnitude square of the wave function |Ψ (r, t)|2 was inter­ preted as the particle density. However, Eq.(4.15) in one spatial dimension is mathematical equivalent to the dispersive wave motion Eq.(2.72), where the space and time variables have been exchanged. The dispersion leads to spreading of the wave function. This would mean that any initially compact particle, which has a well localized particle density, would decay, which does not agree with observations. In the framwork of the "Kopenhagen Interpre­ tation" of Quantum Mechanics, whose meaning we will define later in detail, |Ψ (r, t)|2 dV is the probability to find a particle in the volume dV at position r , if an optimum measurement of the particle position is carried out at time t. The particle is assumed to be point like. Ψ (r, t) itself is then considered to be the probability amplitude to find the particle at position r at time t. Note, that the measurement of physical observables like the position of a particle plays a central role in quantum theory. In contrast to classical 202 CHAPTER 4. SCHROEDINGER EQUATION mechanics where the state of a particle is precisely described by its position and momentum in quantum theory the full information about a particle is represented by its wave function Ψ (r, t). Ψ (r, t) enables to compute the outcome of a measurement of any possible observable related to the particle, like its position or momentum. Before, we discuss this issue in more detail lets look at a few examples to get familiar with the mathematics of quantum mechanics. 4.1 Free Motion Eq.(4.15) describes the motion of a free particle. For simplicity, we consider only a one-dimensional motion along the x-axis. Initially, we might only know the position of the particle with finite precision and therefore we use a Gaussian wave packet with finite width as the initial wave function µ ¶ x2 Ψ (x, t = 0) = A exp − 2 + jk0 x . (4.17) 4σ 0 The probability density to find the particle at position x is a Gaussian dis­ tribution ¶ µ x2 2 2 |Ψ (x, t = 0)| = |A | exp − 2 , (4.18) 2σ 0 σ 20 is the variance of the initial particle position. Since the probability to find the particle somewhere must be one, we can determine the amplitude of the wave function by requireing Z∞ −∞ 1 |Ψ (x, t = 0)|2 dx = 1 → A = √ √ 4 2π σ 0 (4.19) The meaning of the wave number k0 in the wave function (4.17) becomes obvious by expressing the solution to the wave equation by its Fourier trans­ form Z+∞ Ψ (x, t) = φ (k) exp j (kx − ω (k) t) dk −∞ (4.20) 4.1. FREE MOTION 203 or specifically for t = 0 Z+∞ Ψ (x,0) = φ (k) e jkx dk , (4.21) −∞ or 1 φ (k) = 2π Z+∞ Ψ(x,0) e− jkx dx . (4.22) −∞ For the initial Gaussian wavepacket of ¶ µ x2 Ψ (x, 0) = A exp − 2 + jk0 x 4σ 0 (4.23) we obtain £ ¤ Aσ 0 (4.24) φ (k) = √ exp −σ 20 (k − k0 )2 . π This is a Gaussian distribution for the wave number, and therefore momen­ tum, of the particle with its center at k0 . With the dispersion relation ~ k2 (4.25) , 2m with the constant potential V0 set to zero, the wave function at any later time is ∙ ¸ Z+∞ ~k2 Aσ 0 2 2 exp −σ 0 (k − k0 ) − j t + jkx dk. (4.26) Ψ (x, t) = √ 2m π ω= −∞ This is exactly the same Gaussian integral we were studying for dispersive pulse propagation or the diffraction of a Gaussian beam in chapter 2 which results in ⎤ ⎡ ~2σ 20 k02 2 2 x − 4 jσ 0 k0 x + j m t A ⎦ . ³ ´ Ψ (x, t) = q exp ⎣− (4.27) ~t ~ 2 1 + j m2σ2 4σ 0 1 + j m2σ2 t 0 0 As expected the wave packet stays Gaussian. The probability density is ⎡ ⎤ ¡ ¢ 2 ⎢ ⎥ x − ~mk0 t |A|2 2 ⎢ ∙ ¸⎥ . (4.28) exp ⎣ − |Ψ(x, t)| = q ³ ´ 2 ⎦ ~t ~t 2 1 + ( 2mσ 2 )) 2σ 0 1 + 2mσ2 0 0 204 CHAPTER 4. SCHROEDINGER EQUATION With the value for the amplitude A according to Eq.(4.19), the wave packet remains normalized Z+∞ |Ψ(x, t)|2 dx = 1. (4.29) −∞ Using the probability distribution for the particle position, we obtain for its expected value Z+∞ hxi = x |Ψ(x, t)|2 dx (4.30) or −∞ ~ k0 t . (4.31) m Thus the center of the wave packet moves with the velocity of the classical particle ~ k0 υ0 = , (4.32) m which is the group velocity derived from the dispersion relation (4.2) ¯ ¯ ∂ω(k) ¯¯ 1 ∂E(k) ¯¯ = . (4.33) υ0 = ∂k ¯k=k0 ~ ∂k ¯k=k0 hxi = As we will see later, the expected value for the center of mass of the par­ ticle follows Newton’s law, which is called Ehrenfest’s Theorem. For the uncertainty in the particle position q (4.34) ∆x = hx2 i − hxi2 follows for the freely moving particle s ¶2 µ ~t ∆x = σ 0 1 + 2mσ 20 . (4.35) The probability density for the particle position disperses over time. Asymp­ totically one finds ~t . ~t ∆x = for À1 . (4.36) 2 2mσ 0 2mσ 20 Figure 4.1 (a) is a sketch of the complex wave packet and (b) indicates the temporal evolution of the average and variance of the particle center of mass 4.1. FREE MOTION 205 motion described by the complex wave packet. The wave packet disperses faster, if it is initially stronger localised. Figure 4.1: Gaussian wave packet: (a) Real and Imaginary part of the com­ plex wave packet. (b) width and center of mass of the wave packet. Example: Using this one dimensional model, we can estimate how rapidly an elec­ tron moves in a hydrogen atom. If we localize an electron in a box with a size similar to that of a hydrogen atom, i.e. σ 0 = a0 = 0.5 · 10−10 m, without the presence of the proton that holds the electron back from escaping, it will only 2 take t = 2mσ 20 /~ = 2 · 9.81 · 10−31 kg· (0.5 · 10−10 ) m2 /6.626 · 10−34 Js = 46.5as (attoseconds=10−18 sec) until its wave function disperses significantly. This result indicates that electronic motion in atoms occurs on a attosecond time scale. Note, these time scales quickly become very long if macroscopic ob­ jects are described quantum mechanically. For example, for a particle with 206 CHAPTER 4. SCHROEDINGER EQUATION a mass of 1μg localized in a box with dimensions 1μm, the equivalent time for significant dispersion of the wave function is t = 2 · 1019 s = 2Million years. This result gives us a first indication why we are far, far away from encountering quantum mechanical effects in our everyday life and why the mechanics of the micro cosmos, on an atomic or molecular level, is so dif­ ferent from our macroscopic experience. The reason is the smallness of the quantum of action h. The reason for this behaviour is that a well localized particle has a wider momentum or wave number distribution. This is in one to one analogy that an otpical pulse disperses faster in a medium with a given dispersion if it is shorter because of larger spectral width. The wave number spread is (∆k)2 = Z∞ −∞ (k − k0 )2 |φ (k)|2 dk Z∞ . (4.37) |φ (k)|2 dk −∞ Here, we have ∆k = 1 , 2σ 0 (4.38) ∆p = ~ . 2σ 0 (4.39) or for the momentum spread The position-momentum uncertainty product is then s ¶2 µ ~t ~ 1+ . ∆p ∆x = 2 2m20 (4.40) The position-momentum uncertainty product is a minimum at t = 0 and steadily increases from this initial value. As we will show later it is in gen­ erally true that the position-momentum uncertainty product satisfies the condition ~ ∆xi ∆pi > . (4.41) 2 Note, that the index i indicates the coordinate. This is Heisenberg’s uncer­ tainy relation between particle position and moment, which holds for each 4.2. PROBABILITY CONSERVATION AND PROPABILITY CURRENTS207 component individually. Later, we will find other pairs of physical obser­ vavles, which are called conjugate observables and which satiesfy similar uncertainty relations. The product of such quantities is always an action. This is for example also true for the product of energy and frequency and the resulting energy-time uncertainty relation is ∆E ∆t > ~ . 2 (4.42) Note, whereas the position-momentum uncertainy is related to the choice of the particle state described by the wave function, the energy-time uncertainty relation is related to the dynamics of a quantum process. What it means is that a quantum system can only change its state significantly within a time span ∆t, if the state, the quantum system is in, has an energy uncertainty larger than δE > 2~δt . Position and momentum variables that do not belong to the same degree of freedom, such as y, and px are not subject to an uncertainty relation. 4.2 Probability Conservation and Propabil ity Currents Max Born was the first to introduce the propabilistic interpretation of the wave function found by Schroedinger, that is the propability to find the center of mass of a particle at position r in a volume element dV is given by the magnitude square of the wave function multiplied by dV p (r, t) = |Ψ (r, t)|2 d V . (4.43) If this interpretation makes sense, then the total propability that the parti­ cle can be found somewhere should by 1 and this normalization should not change during the dynamics. We found that this is true for the Gaussian wave packet undergoing free motion. Here, we want to show that this is true under the most general circumstances. We look at the rate of change of the 208 CHAPTER 4. SCHROEDINGER EQUATION probability to find the particle in an arbitrary but fixed volume V = V ol Z d p (r, t) d3 r = (4.44) dt V ol Z ∂ |Ψ (r, t)|2 d3 r = dt ¶ ¶¸ µ ZV ol ∙µ ∂ ∗ ∂ ∗ Ψ (r, t) Ψ (r, t) + Ψ (r, t) d3 r Ψ (r, t) = ∂t ∂t V ol Using the Schroedinger Equation (4.16) for the temporal change of the wave function we obtain Z d p (r, t) d3 r = dt V ol ¶ ¸ Z ∙µ ~ j ∗ ∗ ∗ ∇ · ∇ Ψ (r, t) − V (r) Ψ (r, t) Ψ (r, t) d3 r (4.45) = j2 m ~ V ol ¶¸ µ Z ∙ j ~ ∗ + Ψ (r, t) − ∇ · ∇ Ψ (r, t) + V (r) Ψ (r, t) d3 r j2 m ~ V ol Since the potential V (r) is real the terms related to it cancel. The other two terms can be written of the divergence of a current density Z Z ∂ 3 p (r, t) d r = − ∇ · J (r, t) d3 r, (4.46) ∂t V ol V ol with J (r, t) = ~ (Ψ∗ (r, t) (∇Ψ (r, t)) − Ψ (r, t) (∇Ψ∗ (r, t))) . j2m Eq.(4.46) is true for any volume, i.e. ¸ Z ∙ ∂ p (r, t) + ∇ · J (r, t) d3 r = 0, ∂t V ol (4.47) (4.48) which is only possible if the integrand vanishes ∂ p (r, t) = − ∇ · J (r, t) . ∂t (4.49) Clearly, J (r, t) has the physical meaning of a probability current. The prob­ ability in a volume element changes because of probablity flowing out of 4.2. PROBABILITY CONSERVATION AND PROPABILITY CURRENTS209 this volume element. Note, this is the same local law that we have for the conservation of charge. In fact, if the particle is a charged particle, like an electron is, multiplication of J with −e0 would result in the electrical current associated with the wave function Ψ (r, t) . Gauss’s theorem states Z Z 3 ∇ · J (r, t) d r = J (r, t) dS, (4.50) V ol S where V ol is the volume over which the integration is carried out and S is the surface that encloses the volume with dS an outward pointing surface normal vector. With Gauss’s theorem the local conservation of probability can be transfered to a global result, since Z Z Z d 3 3 (4.51) p (r, t) d r = − ∇ · J (r, t) d r = − J (r, t) dS. dt V ol V ol S ∂ If we choose as the volume the whole space and if Ψ (r, t) and ∂t Ψ (r, t) vanish rapidly enough for r → ∞ such that the probability current vanishes at infinity, the total probability is conserved. These findings proove that the probabilty interpretation of the wave function is a valid interpretation not contradicting basic laws of probability. If the wave function properly normalized at the beginning it will stay normalized. Example The Gaussian wave packet satisfies the condition that the prob­ ability current decays rapidly enough at the surface of a large enough chosen volume so that the normalization is preserved. A monochromatic plane wave does not satisfy this condition. However, the probability current density gives a physical meaning to it. The wave function corresponding to a plan wave Ψ (r, t) = ej(k·r−ωt) , with ω = ~k 2 V0 + 2m ~ (4.52) which is not normalizable, results in a homogenous probability current J (r, t) ~ (Ψ∗ (r, t) (∇Ψ (r, t)) − Ψ (r, t) (∇Ψ∗ (r, t))) (4.53) j2m p ~k = |Ψ (r, t)|2 = = v, m m = 210 CHAPTER 4. SCHROEDINGER EQUATION that is identical to the classical velocity of the particle. Thus a plane wave describes a particle with a precise velocity or momentum but completely unknown position, therefore the related probability current density is com­ pletely homogenous but directed into the direction of v. Such waves describe the initial state in a scattering experiment, where we shoot particles with a 2 ~p2 ~k2 precisely defined velocity v or momentum p or energy E = mv = 2m = 2m 2 onto another object described by a scattering potential, see problem set. The position of these particles is completely unspecified, i.e. |Ψ (r, t)|2 =const. 4.3 Measureability of Physical Quantities (Ob servables) The reason for the more intricate description necessary for microscopic pro­ cesses in comparison with macroscopic processes is simply the fact that these systems are so small that the interaction of the system with an eventual mea­ surement apparatus can no longer be neglected. It turns out this fact is not to overcome by choosing more and more sophisticated measurement apparati but rather is a principle limitation. If this is so, then it eventually doesn’t make sense or it becomes even inconsistent to attribute to a system more precisely defined physical quantities than actually can be retrieved by mea­ surements. This is the physical reason behind the introduction of the wave function in stead of the precisely defined position and momentum of the par­ ticle that we used to deterministically predict the trajectory of a particle in an external field. It is impossible to assign to a microscopic particle a precise position and momentum at the same time. To demonstrate this, we consider the following (Heisenberg) microscope to measure the exact position of a particle. We use light with wavelength λ and focus it strongly with a lense of some focal distance d, see Figure 4.2. From our construction of the Gaussian beam in section 2.4.2, we found that if we generate a focused beam with a waist wo having a Rayleigh range 2 zR = πwλo , the beam is composed of plane waves which have a Gaussian distri­ bution in its transverse k-vector, which has a variance kT2 /2, see Eq.(2.220). The Rayleigh range of the beam is related to the transverse wave number spread of the beam by zR = k0 /kT2 , with k0 = 2π/λ, see (2.221) and there­ after. Note, the intensity profile of the beam has a variance wo2 /4. If a particle 4.3. MEASUREABILITY OF PHYSICAL QUANTITIES (OBSERVABLES)211 crosses the focus of the beam and scatters a single photon, which we detect with the surrounding photo detector arrangement, then it is reasonable to assume that we know the position of the particle in the x-direction, with an uncertainty equal to the uncertainty in the transverse photon or intensity distribution of the beam, i.e. ∆x = wo /2. Photodector p Weak particle beam with precise momentum p Figure 4.2: Determination of particle position with an optical microscope. A weak particle beam with precisely defined moment p of the particles is directed towards the focus of the Gaussian beam. In the focus the particle scatters at least one photon. Detection of the scattered photon with the surrounding photodetector signals, that the position of the particle in xdirection has been determined within the beam waist of the Gaussian beam. However, due to the scattering of the photon a momentum uncertainty has been introduced to the particle state. During the measurement, the √ photon recoil induces a momentum kick with an uncertainty ∆px = ~kT / 2. So even if the momentum of the particle was perfectly know before the measurement, after the additional determina­ tion of its position with a precision ∆x it has at least aquired an uncertainty in its momentum of magnitude ∆px . The product of the uncertainties in postion and momentum after the measurement is ³ √ ´ ~ ∆px · ∆x = ~kT wo / 2 2 = . 2 (4.54) 212 CHAPTER 4. SCHROEDINGER EQUATION Note, this result is exact and is independent of focusing. Tighter focusing will enable us to more precisely determine the position of the particle, but we will introduce more momentum uncertainty due to the photon recoil; the opposite is true for less focusing. Since we can not determine, and therefore, prepare a particle in a state with its position and momentum more precisely determined than this uncertainty product allows, there is nowsuch state and (4.54) is the minimum uncertainty product achievable. The experimental setup can easily be extended to measure the momentum and position of a particle in all three dimensions. For example one can use three focused laser beams at different wavelength, which are orthogonal to each other. Once a particle will fly through the focus and scatters three photons, each of different color. If we knew its momentum initially precisely, we would know afterwards its 3-dimensional position with a position and momentum spread as described by Eq.(4.54). 4.4 Stationary States One of the great mysteries before the advent of quantum mechanics was the orgin of the discrete energy spectra observed in spectroscopic investigations and empirically described by the Bohr-Sommerfeld model of the atom. This mystery is easily explained by the Schroedinger Equation (4.16) j~ ~2 ∂Ψ (r, t) =− ∆ Ψ (r, t) + V (r) Ψ (r, t) . ∂t 2m (4.55) It allows for solutions Ψ (r, t) = ψ (r) ej ωt , (4.56) which have a time independent probability density, i.e. |Ψ (r, t)|2 = |ψ (r)|2 = const., (4.57) which is the reason for calling these states stationary states. Since the right side of the Schroedinger Equation is equal to the total energy of the sys­ tem, these states correspond to energy eigenstates of the system with energy eigenvalues E = ~ω. (4.58) 4.4. STATIONARY STATES 213 These energy eigenstates ψ (r) are eigen solutions to the stationary or time independent Schroedinger Equation ~2 ∆ ψ (r) + V (r) ψ (r) = E ψ (r) . − 2m (4.59) We get familiar with this equation by considering a few one-dimensional examples, before we apply it to the Hydrogen atom. 4.4.1 The One-dimensional Infinite Box Potential A simple example for a quantum mechanical system is an electron that can freely move in one dimension x but only over a finite distance a. Such a situation closely describes an electron that is strongly bound to a molecule with a cigar like shape with length a. The potential describing this situation is the one-dimensional box potential ½ 0, for |x| < a/2 V (x) = , (4.60) ∞, for |x| ≥ a/2 see Figure 4.3. Image removed for copyright purposes. Figure 4.3: One dimensional box potential with infinite barriers. In the interval [−a/2, a/2] the stationary Schroedinger equation is − ~2 d2 ψ (x) = E ψ (x) . 2m dx2 (4.61) 214 CHAPTER 4. SCHROEDINGER EQUATION For |x| ≥ a/2 the wave function must vanish, otherwise the energy eigenvalue can not be finite, i.e. ψ (x = ±a/2) = 0. This is analogous to the electric field solutions for the TE-modes for a planar mirror waveguide and we find r 2 nπx ψn (x) = cos for n = 1, 3, 5 . . . , (4.62) a a r 2 nπx ψn (x) = sin for n = 2, 4, 6 . . . . (4.63) a a The corresponding energy eigenvalues are En = n2 π 2 ~2 . 2ma2 (4.64) We also find that the stationary states constitute an orthogonal system of functions Z+∞ ψm (x)∗ ψn (x) dx = δ mn . (4.65) −∞ In fact this system is complete. Any function in the interval [−a/2, a/2] can be expanded in a superposition of the basis functions ψn (x), which is a Fourier series ∞ X cn ψn (x) (4.66) f (x) = n=0 with cm = Z a/2 ψm (x)∗ f (x) dx, (4.67) −a/2 which is a consequence of the orthogonality relation (4.65). Example: If we approximate the binding potential of a hydrogen atom by a one-dimensional box potential with a width equal to twice the Bohr radius a = 2a0 = 10−10 m, the energy eigenvalues are En = n2 · 35eV. Clearly, the spacing of the energy eigenvalues does not conform with what has been observed experimentaly, compare with section 3.4, however the energy scale is within an order of magnitude. The ionization potential of the hydrogen atom is 13.5eV . 4.4. STATIONARY STATES 4.4.2 215 The One-dimensional Harmonic Oscillator The most important example of a quantum system is the one-dimensional harmonic oscillator. It is the most basic mechanical and electrical system and it describes the dynamics of a mode of the radiation field, see Figure 4.4. Image removed for copyright purposes. Figure 4.4: Elastically bound particle Mechanically, a harmonic oscillation comes about by the elastic force obeying Hook’s law (4.68) F (x) = −Kx, that pulls back a particle with mass m in its equilibrium position. This force is conservative and can be derived from a potential by F (x) = − d V (x) , dx (4.69) with 1 Kx2 . 2 Newton’s law results in the classical equation of motion V (x) = (4.70) mẍ = F (x) , (4.71) ẍ + ω 20 x = 0, (4.72) or with the oscillation frequency ω0 = r K m (4.73) 216 CHAPTER 4. SCHROEDINGER EQUATION The corresponding stationary Schroedinger Equation is µ ¶ d2 ψ (x) 2m 1 2 ψ (x) = 0. + 2 E − Kx dx2 ~ 2 (4.74) This equation is well known in mathematical physics and we want to bring it into standardized form by the scale transformation, i.e. introducing a normalized distance ξ = ax, (4.75) with the scale factor a= µ mK ~2 ¶ 14 = r ω0m = ~ r K . ~ω 0 (4.76) In addition we introduce the energy scale factor γ= 2E . ~ω0 (4.77) Then the stationary Schroedinger Equation for the harmonic oscillator is ¢ d2 ψ (ξ) ¡ + γ − ξ 2 ψ (ξ) = 0. 2 dξ (4.78) It turns out [4][6], that this equation has only solutions that are bounded, i.e. ψ (ξ → ±∞) = 0, if the normalized energies are (4.79) γ n = 2n + 1. And the corresponding eigensolutions are the Hermite Gaussians, 1 ψn (ξ) = const. Hn (ξ) e− 2 ξ2 , (4.80) which we discovered already as solutions of the paraxial wave equation, see Eqs.(2.298) and (2.299), i.e. dn −ξ2 e dξ n (4.81) H3 (ξ) = 8 ξ 3 − 12 ξ , H4 (ξ) = 16 ξ 4 − 48 ξ 2 + 12 , H5 (ξ) = 32 ξ 5 − 160 ξ 3 + 120 ξ . (4.82) Hn (ξ) = (−1)n eξ H0 (ξ) = 1 , H1 (ξ) = 2 ξ , H2 (ξ) = 4 ξ 2 − 2 , 2 4.4. STATIONARY STATES 217 After denormalization and normalization the stationary wave functions are r a − 12 a2 x2 √ H ψn (x) = (ax) e . (4.83) n 2n π n! Again, we find that the Hermite Gaussians constitute an orthogonal system of functions such that Z+∞ ψm (x)∗ ψn (x) dx = δ mn . (4.84) −∞ Figure 4.5 shows the first six stationary states or energy eigenstates of the harmonic oscillator. Image removed for copyright purposes. Figure 4.5: First six stationary states of the harmonic oscillators. The energy eigenvalues of the stationary states are ¶ µ 1 En = n + ~ω 0 . 2 (4.85) Note, that the energy eigenvalues are equidistant and the difference between two energy eigenstates follows the findings of Planck. An oscillator has dis­ crete energy levels which differ by energy quanta of size ~ω0 , see Figure 218 CHAPTER 4. SCHROEDINGER EQUATION Image removed for copyright purposes. Figure 4.6: Lowest order wavefunctions of the harmonic oscillator and the corresponding energy eigenvalues [3]. .The only difference is, that the whole energy scale is shifted by the energy of half a quantum, which is the lowest energy eigenvalue. Thus the minimum energy, or ground state energy, of a harmonic oscillator is not zero but E0 = 1 ~ω0 .It is obvious, that an oscillator can not have zero energy because its 2 energy is made up of kinetic and potential energy E= 1 p2 + Kx2 . 2m 2 (4.86) Since every state has to fulfill Heisenberg’s uncertainty relation ∆p · ∆p ≥ ~2 , one can show that the state with minimum energy possible has an energy E0 = 12 ~ω 0 , which is true for the ground state ψ 0 (x) according to Eq.(4.83). The stationary states of the harmonic oscillator correspond to states with precisely definied energy but completely undefined phase. If we assume a classical harmonic oscillator with a well defined energy E = 12 Kx20 . Note, that during a harmonic oscillation the energy is periodically converted from potential energy to kinetic energy. Then the oscillator oscillates with a fixed ampltiude x0 x(t) = x0 cos (ω 0 t + ϕ) . (4.87) If the phase is assumed to be random in the interval [-π, π], one finds for the 4.5. THE HYDROGEN ATOM 219 probability density of the position x to be 1 . p (x) = p 2 π x0 − x2 Figure 4.7 shows this probability density corresponding to an energy eigen­ state ψn (x) with quantum large quantum number n = 10. Image removed for copyright purposes. Figure 4.7: Probability density |ψ10 |2 of the harmonic oscillator containing exactly 10 energy quanta. On average, the quantum mechanical probability density agrees with the classical probability density, which is some form of the correspondence principle, which says that for large quantum numbers n the wave functions resume classical properties. 4.5 The Hydrogen Atom The simplest of all atoms is the Hydrogen atom, which is made up of a positively charged proton with rest mass mp = 1.6726231 × 10−27 kg, and a negatively charged electron with rest mass me = 9.1093897 × 10−31 kg. Therefore, the hydrogen atom is the only atom which consists of only two particles. This makes an analytical solution of both the classical as well as the quantum mechanical dynamics of the hydrogen atom possible. All other atomes are composed of a nucleus and more than one electron. According 220 CHAPTER 4. SCHROEDINGER EQUATION Figure 4.8: Bohr Sommerfeld model of the Hydrogen atom. to the Bohr-Somerfeld model of hydrogen, the electron circles the proton on a planetary like orbit, see Figure 4.8.The stationary Schroedinger Equation for the Hydrogen atom is 2m0 (E − V (r)) ψ (r) = 0 (4.88) ~2 The potential is a Coulomb potential between the proton and the electron such that e20 (4.89) V (r) = − 4 π ε0 |r| and the mass is actually the reduced mass ∆ψ (r) + m0 = mp · me mp + me (4.90) that arises when we transform the two body problem between electron and proton into a problem for the center of mass and relative coordinate motion. Due to the large, but finite, mass of the proton, i.e. the proton mass is 1836 times the electron mass, both bodies circle around a common center of mass. The center of mass is very close to the position of the proton and the reduced mass is almost identical to the proton mass. Due to the spherical symmetry of the potential the use of spherical coordinates is advantageous ∙ ¸ µ ¶ ∂ 2 ψ 2 ∂ψ ∂2ψ 1 1 ∂ ∂ψ 1 ∆ψ = (4.91) + + 2 sin ϑ + ∂r2 r ∂r r sin ϑ ∂ϑ ∂ϑ sin2 ϑ ∂ ϕ2 We will derive separate equations for the radial and angular coordinates by assuming trial solutions which are products of functions only depending on 4.5. THE HYDROGEN ATOM 221 one of the coordinates r , ϑ , or ϕ ψ (r, ϑ, ϕ) = R (r) θ (ϑ) φ (ϕ) . (4.92) Substituting this trial solution into the stationary Schroedinger Eq.(4.91) and separating variables leads to radial equation ¶ µ d2 R 2 dR m0 e20 2m0 E α − R=0 , (4.93) + + + r dr dr2 ~2 2πε0 ~2 r r2 the azimuthal equation ¶ ¶ µ µ dθ d 1 m2 θ=0 , sin ϑ + α− sin ϑ dϑ dϑ sin2 ϑ (4.94) and the polar equation d2 φ + m2 φ = 0 , dϕ2 (4.95) where α and m are constants yet to be determined. The polar equation has the complex solutions φ (ϕ) = const. ejmϕ , with m = . . . − 2, −1, 0, 1, 2 . . . (4.96) because of the symmetry of the problem in the polar angle ϕ, i.e. the wavefunction must be periodic in ϕ with period 2π. 4.5.1 Spherical Harmonics The azimuthal equation is transformed by the substitution into ξ = cos ϑ (4.97) ¶ µ ¡ ¢ 2 dθ m2 2 d θ θ=0 . − 2ξ + α − 1−ξ dξ dξ 2 1 − ξ2 (4.98) α = l (l + 1) with, l = 0, 1, 2 . . . (4.99) It turns out, that this equation has only bounded solutions on the interval ξ [−1, 1], if the constant α is a whole number 222 CHAPTER 4. SCHROEDINGER EQUATION and m = −l, −l + 1, . . . − 1, 0, 1 . . . l − 1, l (4.100) For m = 0, Eq.(4.98) is Legendre’s Differential Equation and the solutions are the Legendre-Polynomialsm [5] P3 (ξ) = 52 ξ 3 − 32 ξ , P4 (ξ) = 35 ξ 4 − 15 ξ 2 + 38 , 8 4 63 5 35 3 P5 (ξ) = 8 ξ − 4 ξ + 15 ξ . 8 P0 (ξ) = 1 , P1 (ξ) = ξ , P2 (ξ) = 32 ξ 2 − 21 , (4.101) For m 6= 0, Eq.(4.98) is the associated Legendre’s Differential Equation and the solutions are the associated Legendre-Polynomials, which can be gener­ ated from the Legendre-Polynomials by ¡ ¢m/2 dm P1 (ξ) . (4.102) P1m (ξ) = 1 − ξ 2 dξ m Overall the angular functions can be combined to form the spherical harmon­ ics s (2l + 1) (l − |m|)! m jm ϕ P1 (cos ϑ) e , (4.103) Y1m (ϑ, ϕ) = (−1)m 4π (l + |m|)! which play an important role whenever a partial differential equation that contains the Laplace operator is solved in spherical coordinates. The spheri­ cal harmonics form a system of orthogonal functions on the full volume angle 4π, i.e. ϑ [0, π] and ϕ [−π, π] Zπ Z2π 0 0 Ylm∗ (ϑ, ϕ) Ylm (ϑ, ϕ) sin ϑ dϑ dϕ = δ ll0 , δ mm0 . 0 (4.104) 0 Therefore, a function of the angular variable (ϑ, ϕ) can be expanded in spher­ ical harmonics. The spherical harmonics with negative azimuthal number -m can be expressed in terms of those with positive azimuthal number m. Y1−m (ϑ, ϕ) = (−1)m (Ylm (ϑ, ϕ))∗ . (4.105) The lowest order spherical harmonics are listed in Table 4.1. Figure 4.9 shows a cut through the spherical harmonics Y1m (ϑ, ϕ) along the meridional plane. 4.5. THE HYDROGEN ATOM Y00 (ϑ, ϕ) = √14π , Y10 (ϑ, ϕ) = 223 q 3 4π cos ϑ , q 3 Y11 (ϑ, ϕ) = − 8π sin ϑ ejϕ , q q q 2 5 15 15 (3 cos2 ϑ − 1) , Y12 (ϑ, ϕ) =- 8π sin ϑ cos ϕ ejϕ , Y2 (ϑ, ϕ) = 32π sin2 ϑ e2jϕ , Y02 (ϑ, ϕ) = 16π Y03 Y23 q 7 (ϑ, ϕ) = 16π (5 cos3 ϑ − 3 cos ϑ) , q 105 (ϑ, ϕ) = 32π sin2 ϑ cos ϑ ej2ϕ , Y13 Y33 q 21 (ϑ, ϕ) =- 64π sin ϑ (5 cos2 ϑ − 1) ejϕ , q 35 (ϑ, ϕ) =- 64π sin3 ϑ ej3ϕ . Table 4.1: Lowest order spherical harmonics Figure 4.9: Lowest order spherical harmonics Y1m (ϑ, ϕ) , along the meridional plane, i.e. ϕ = 0. 224 4.5.2 CHAPTER 4. SCHROEDINGER EQUATION Radial Wave Functions Obviously, the spherical harmonics are related to the angular momentum L of the particle, because after choosing the spherical harmonic with indices l, m the radial Equation (4.93) is µ ¶ d2 R 2 dR 2m0 E m0 e20 l (l + 1) − + + + R = 0. (4.106) dr2 r dr ~2 2πε0 ~2 r r2 The radial equation has in addition to the 1/r Coulomb potential the cen­ trifugal potential ~2 l (l + 1) L2 Erot = = , (4.107) 2m0 r2 2m0 r2 ¯ ¯ ¯ ¯ which is the rotation energy of a particle with angular momentum ¯L¯ = p l (l + 1)~ and moment of inertia m0 r2 . Thus quantum mechanically, the particle can no longer access arbitrary values for angular momentum. ¯ ¯the p ¯ ¯ The angular momentum can only have values ¯L¯ = l (l + 1)~ with l = 0, 1, 2, .... For large radii, the radial equation simplifies to d2 R 2m0 E + R = 0, dr2 ~2 (4.108) which indicates that the radial wave function must decay exponentially for large radii. Therefore, we rescale the radius accoring to ρ = Ar with A2 = − 8m0 E , because E < 0, ~2 (4.109) (4.110) and form the trial solution R (ρ) = ρs w (ρ) e−ρ/2 . (4.111) Substitution into Eq.(4.109) leads to the following differential equation for w (ρ) ρ2 d2 w dw + ρ [2 (s + 1) − ρ] + [ρ (λ − s − 1) + s (s + 1) − l (l + 1)] w = 0, 2 dρ dρ (4.112) 4.5. THE HYDROGEN ATOM 225 with √ m0 e2 m0 e2 √ √ λ= = . 2πε0 ~2 A 4 2πε0 ~ −E Evaluation of this differential equation at ρ = 0 leads to (4.113) l = s, and we are left with the much simpler equation ρ d2 w dw + [2 (l + 1) − ρ] + (λ − l − 1) w = 0 . 2 dρ dρ (4.114) One way to solve this equation is by using a polynomial trial solution. w (ρ) = b0 + b1 ρ + b2 ρ2 + . . . bp ρp (4.115) Substitution into Eq.(4.114) leads to the following recursion relation for the coefficients k+l+1−λ bk+1 = (4.116) bk (k + 1) (k + 2l + 2) For λ=p+l+1 (4.117) the recursion breaks off and we obtain a polynomial of finite order. If λ is not an integer the polynomial does not stop and the corresponding series converges against a w (ρ) that has an asymptotic behavior w (ρ) ˜eρ , which leads to a radial function not normalizable. Thus we have the condition λ ≡ n, with n > l + 1 (4.118) and in total w (ρ) = L21+1 n−l+1 (ρ) (4.119) with the Laguerre Polynomials Lrs (x) = s X q=0 (−1)q (s + r)!2 xq . (s−q) ! (r + q)! q! (4.120) The lowest order Laguerre Polynomials are summarized in Table 4.2 The radial wave function is then a Laguerre function −ρ/2 Fn1 (ρ) = ρ1 L21+1 , n−l=1 (ρ) e (4.121) 226 CHAPTER 4. SCHROEDINGER EQUATION L10 (x) = 1 , L11 (x) = 4 − 2x , L13 (x) = 96 − 144x + 48x2 − 4x3 L21 (x) = 18 − 6x , L33 (x) = 6 , L12 (x) = 18 − 18x + 3x2 , , L20 (x) = 2 , L22 (x) = 144 − 96 + 12x2 , L31 (x) = 96 − 24x , L40 (x) = 24 . Table 4.2: Lowest order Laguerre Polynomials and they again form an orthogonal system of functions Z∞ Fnl (ρ) Fn0 l (ρ) ρ2 dρ = 0 2n [(n + l)!]3 δnn0 . (n − l − 1)! (4.122) We now reverse the normalization of the radial coordinate and from Eqs.(4.109,4.110) and (4.113) we find 2r ρ= (4.123) na0 with the Bohr radius 4πε0 ~2 , (4.124) a0 = 2 e0 m0 which we found already in the Bohr-Sommerfeld model, see section 3.4. The radial wave function is then Rn1 (r) = Nnl Fnl (ρ) . (4.125) And the normalization factor is determined by Z∞ Rnl (r) Rn0 l (r) r2 dr = δ n,n0 , (4.126) 0 which gives 2 Nnl = 2 n s (n − l − 1)! −3/2 a0 . [(n + l)!]3 (4.127) 4.5. THE HYDROGEN ATOM 227 The radial wave functions of the hydrogen atom are listed in Table 4.3 and plots of the lowest order radial wave functions are presented in Figure 4.10 R10 (r) = √2 3 e−r/a0 , a0 R20 (r). = R21 (r) = 1 r −r/2a0 e √ √ 2 6 a30 a0 R30 (r) = 1 √ √ 81 3 a30 R31 (r) = 4 √ √ 81 6 a30 1 √ √ 2 2 a30 ³ 2− r a0 ´ e−r/2a0 ³ ´ 2 27 − 18 ar0 + 2 ar 2 e−r/3a0 0 ³ 6− r a0 ´ r −r/3a0 e , a0 R32 (r) = 4 r2 −r/3a0 √ √ 2e 81 30 a30 a0 Table 4.3: Lowest order radial wavefunctions Rn,l (r). Figure 4.10: Radial wavefunctions Rnl (r) of the hydrogen atom. 228 CHAPTER 4. SCHROEDINGER EQUATION 4.5.3 Stationary States of Hydrogen In total we found the stationary states, or the energy eigenfunctions, of the hydrogen atom. Those are ψnlm (r, ϑ, ϕ) = Rnl (r) Ylm (ϑ, ϕ) . (4.128) The lower order wave functions are listed in Table 4.4 and plots of the re­ sulting probability densities of the lowest order energy eigenstates of the hydrogen atom are shown in Figure 4.11 ψ100 (r, ϑ, ϕ) = 1 −r/a0 √ √ 3e π a0 ψ200 (r, ϑ, ϕ) = 1 √ √ 4 2π a30 ψ210 (r, ϑ, ϕ) = 1 r −r/2a0 e √ √ 4 2π a30 a0 ψ21±1 (r, ϑ, ϕ) = ψ300 (r, ϑ, ϕ) = ³ 2− r a0 ´ 1 r −r/2a0 e √ √ 8 π a30 a0 1 √ √ 81 3π a30 e−r/2a0 cos ϑ sin ϑe±jϕ , ³ ´ r r2 27 − 18 a0 + 2 a2 e−r/3a0 0 Table 4.4: Lowest order hydrogen wavefunctions ψn,l,m (r, ϑ, ϕ). 4.5.4 Energy Spectrum of Hydrogen We haven’t yet discussed the energy eigenspectrum of hydrogen. Eqs.(4.113) and (4.118) we find this to be E=− m0 e4 1 , 8 ε20 h2 n2 From (4.129) which also agrees with the energy spectrum of the Bohr-Sommerfeld model, see section 3.4. The lowest energy eigenstate is E1 = − m0 e4 = −13.7eV. 8 ε20 h2 (4.130) 4.5. THE HYDROGEN ATOM 229 Image removed for copyright purposes. Figure 4.11: Probability densities of the lowest order hydrogen wavefunctions. (The density is presented along the meridial plane). 230 CHAPTER 4. SCHROEDINGER EQUATION ψ310 (r, ϑ, ϕ) = 1 √ √ 81 π a30 ψ31±1 (r, ϑ, ϕ) = ψ320 (r, ϑ, ϕ) = ³ 6− 1 √ √ 81 π a30 r a0 ³ 6− ´ r a0 e−r/3a0 cos ϑ ´ 1 r2 −r/3a0 √ √ 2e 81 6π a30 a0 r −r/3a0 e a0 sin ϑe±jϕ (3 cos2 ϑ − 1) ψ32±1 (r, ϑ, ϕ) = 1 r2 −r/3a0 √ √ 2e 81 π a30 a0 ψ32±2 (r, ϑ, ϕ) = 1 r2 −r/3a0 √ √ 2e 162 3π a30 a0 sin ϑ cos ϑe±jϕ , sin2 e±2jϕ Table 4.5: Lowest order hydrogen wavefunctions ψn,l,m (r, ϑ, ϕ).continued. The energy eigenvalues constitute a sequence that converges for large n → ∞ towards 0, which corresponds to removing the electron from the atom. The energy to do so is E∞1 − E1 = 13.7eV. Figure 4.12 shows the energy levels and the term diagram of the hydrogen atom and how the Lyman, Balmer, Paschen, Brackett and Pfund series arise from it. Each wavefunction is uniquely described by the set of quantum numbers (n,l,m). The first quantum number n specifies the energy eigen value En . As we will show in problem sets, the second quantum number l determines the eigenvalue of the squared angular momentum operator L2 with eigenvalues L2 ψnlm (r, ϑ, ϕ) = l(l + 1)~2 ψnlm (r, ϑ, ϕ) , (4.131) and the third quantum number m detemines the eigenvalue of the operator describing the z-component of the angular momentum operator Lz ψnlm (r, ϑ, ϕ) = m~ ψnlm (r, ϑ, ϕ) . (4.132) 4.5. THE HYDROGEN ATOM 231 Image removed for copyright purposes. Figure 4.12: Energy levels and term diagram for the hydrogen atom [3] In fact, the description of the electron wave functions is not yet complete, because the electron has an internal degree of freedom, that is its spin. The spin is an internal angular momentum of the electron that carries a magnetic moment with it. The Stern-Gerlach experiment shows that this degree of freedom has two eigenstates, i.e. the spin can be oriented parallel or anti­ parallel to the direction of an applied magnetic field. The values of the internal angluar mometum with respect to the quantization axis defined by an external field, that shall be chosen along the z-axis, are s = ±~/2. Thus the energy eigenstates of an electron in hydrogen are uniquely characterized by four quantum numbers, n, l, m, and s. As Figure 4.12 shows, the energy spectrum is degenerate, i.e. for n > 1, there exist to each energy eigenvalue several eigenfunctions, that are only uniquely characterized by the additional quantum numbers for angular momentum and spin. This is called degeneracy because there exist to a given energy eigenvalue several states. 232 4.6 CHAPTER 4. SCHROEDINGER EQUATION Wave Mechanics In this section, we generalize the concepts we have learned in the previous sections. The goal here is to give a broader description of quantum mechanics in terms of wave functions that are solutions to the Schroedinger Equation. In classical mechanics the particle state is determined by its position and momentum and the state evolution is determined by Newton’s law. In quantum mechanics the particle state is completely described by its wave function and the state evolution is determined by the Schroedinger equation. The wave function as a complete description of the particle enables us to compute expected values of physical quantities of the particle when a cor­ responding measurement is performed. The measurement results are real numbers, like the energy, o4 position or momentum the particle has in this state. The physically measureable quantities are called observables. In clas­ sical mechanics these observables or real variables like x for position, p for momentum or functions thereof, like the energy, which is called the Hamil­ 2 tonian H(p, x) = 2pm + V (x) in classical mechanics. For simplicity, we state the results only for one-dimensional systems but it is straight forward to ex­ tend these results to multi-dimensional sytems. In quantum mechanics these observables become operators: position operator ~ ∂ p = : momentum operator j ∂x ~2 ∂ 2 H(p, x) = − + V (x) : Hamiltonian operator 2m ∂x2 x : (4.133) (4.134) (4.135) If we carry out measurements of these observables, the result is a real number in each measurement and after many measurements on identical systems we can make a statistics of these measurements and the statistics is completely described by the moments of the observable. 4.6.1 Position Statistics The statistical interpretation of quantum mechanics enables us to compute the expected value of the position operator or any of its moments according 4.6. WAVE MECHANICS 233 to hxi = hxm i = Z ∞ Z−∞ ∞ Ψ∗ (x, t) x Ψ (x, t) dx (4.136) Ψ∗ (x, t) xm Ψ (x, t) dx (4.137) −∞ The expectation value of functions of operators can always be evaluated by defining the operator by its Taylor expansion Z ∞ hf (x)i = Ψ∗ (x, t) f (x) Ψ (x, t) dx (4.138) −∞ + *∞ X 1 f (n) (0) xn = n! n=0 ¿Z ∞ À ∞ X 1 (n) ∗ n f (0) Ψ (x, t) x Ψ (x, t) dx = n! −∞ n=0 4.6.2 Momentum Statistics The momentum statistics is then Z ∞ ~ ∂ Ψ (x, t) dx Ψ∗ (x, t) hpi = j ∂x −∞ (4.139) which can be written in terms of the wave function in the wave number space, which we define now for symmetry reasons as the Fourier transform of the wave function where the 2π is symmetrically distributed between Fourier and inverse Fourier transform Z ∞ 1 Ψ (x, t) e−jkx dx, (4.140) φ (k, t) = √ 2π −∞ Z ∞ 1 φ (k, t) ejkx dk. (4.141) Ψ (x, t) = √ 2π −∞ Using the differentiation theorem of the Fourier transform and the generalized Parseval relation Z ∞ Z ∞ ∗ φ1 (k) φ2 (k) dk = Ψ∗1 (x) Ψ2 (x) dx (4.142) −∞ −∞ 234 CHAPTER 4. SCHROEDINGER EQUATION we find Z hpi = ∞ Z−∞ ∞ = φ∗ (k, t) ~k φ (k, t) dk −∞ ~k |φ (k, t)|2 dk. (4.143) (4.144) The introduction of the symmetrically defined expectation value of an oper­ ator according Eq.(4.136), where x can stand for any operator can be carried out using the wave function in the position space or the wave number space using the corresponding represenation of the wave function and of the oper­ ator. 4.6.3 Energy Statistics The analysis for the measurement of position or moment carries over to every observable in an analogous way. Thus the expectation value of the energy is Z ∞ Ψ∗ (x, t) H(x, p) Ψ (x, t) dx (4.145) −∞ µ ¶ Z ∞ 1 ∂2 ∗ = Ψ (x, t) − + V (x) Ψ (x, t) dx.(4.146) 2m~2 ∂x2 −∞ hH(x, p)i = If the system is in an energy eigenstate, i.e. Ψ (x, t) = ψn (x) ej ωn t (4.147) H(x, p) ψn (x) = En ψn (x) , (4.148) with we obtain hH(x, p)i = Z ∞ Ψ∗ (x, t) En Ψ (x, t) dx = En . (4.149) −∞ If the system is in a superposition of energy eigenstates Ψ (x, t) = ∞ X cn ψn (x)ej ωn t . (4.150) n=0 we obtain hH(x, p)i = ∞ X n=0 En |cn |2 . (4.151) 4.6. WAVE MECHANICS 4.6.4 235 Arbitrary Observable There may also occur observables that are not simple to translate from the classical to the quantum domain, such as the product pcl · xcl = xcl · pcl (4.152) Classically it does not matter which variable comes first. However, if we tranfer this expression into quantum mechanics, the corresponding operator depends on the odering, for example ~ ∂ (xΨ (x, t)) = j ∂x ~ ~ ∂ = Ψ (x, t) + x Ψ (x, t) , j j ∂x ¶ µ ~ + xqm · pqm Ψ (x, t) . = j pqm · xqm Ψ (x, t) = (4.153) (4.154) (4.155) The decision of which expression represents the correct quantum mechanical operator or eventually even a linear combination of the possible expressions, has to be based on a close examination of the actual measurement apparatus that would measure the corresponding observable. Finally, the expression also has to deliver results that are in agreement with experimental findings. If we have an operator that is a function of x and p and we have decided on a unique expression in terms of a power expansion in x and p g(x, p) → gop (x, ~ ∂ ) j ∂x (4.156) then we can compute its expected value either in the space domain or the wave number domain Z ∞ ~ ∂ hgop i = ) Ψ (x, t) dx (4.157) Ψ∗ (x, t) gop (x, j ∂x −∞ Z ∞ ∂ φ∗ (k, t) gop (j , ~k) φ (k, t) dk (4.158) = ∂k −∞ That is this operator can be represented either in real space or in k-space as ∂ ∂ ) or gop (j ∂k , ~k). gop (x, ~j ∂x 236 4.6.5 CHAPTER 4. SCHROEDINGER EQUATION Eigenfunctions and Eigenvalues of Operators A differential operator has in general eigenfunctions and corresponding eigen­ values gop (x, ~ ∂ ) ψ (x) = gn ψn (x) , j ∂x n (4.159) where gn is the eigenvalue to the eigenfunction ψn (x) . An example for a differential operator is the Hamiltonian operator describing a partical moving in a potential 1 ∂2 + V (x) (4.160) 2m~2 ∂x2 the corresponding eigenvalue equation is the stationary Schroedinger Equa­ tion Hop ψn (x) = En ψn (x) . (4.161) Hop = − Thus the energy levels of a quantum system are the eigenvalues of the corre­ sponding Hamiltonian operator. The operator for which Z Z ∗ ψn (x) (Hop ψm (x)) dx = (Hop ψn (x))∗ ψm (x) dx, (4.162) for arbitrary wave functions ψn and ψm is called a hermitian operation. From this equation we find immediately that the expected values of a hermitian operator are real, which also has the consequence that the eigenvalues of hermitian operators are real. This is important since operators that represent observables must have real expected values and real eigenvalues since these are results of physical measurements, which are real. Thus observables are represented by hermitian operators. This is easy to proove. Let’s assume we have found two eigenfunctions and the corresponding eigen values gop ψm = gm ψm , gop ψn = gn ψn . Then Z ψ∗n gop ψm dx = gm Z ψ∗n ψm . (4.163) (4.164) (4.165) 4.6. WAVE MECHANICS 237 By taking advantage of the fact that the operator is hermitian we can also write Z Z Z ∗ ∗ ∗ (4.166) ψn gop ψm dx = (gop ψn ) ψm dx = gn ψ∗n ψm dx The right sides of Eqs.(4.165) and (4.166) must be equal Z ∗ (gm − gn ) ψ∗n ψm dx = 0 (4.167) If n = m the integral can not vanish and Eq.(4.167) enforces gn = gn∗ , i.e. the corresponding eigenvalues are real. If n 6= m and the corresponding eigenvalues are not degenerate, i.e. different eigenfunctions have different eigenvalues, then Eq.(4.167) enforces that the eigenfunctions are orthogonal to each other Z ψ∗n ψm dx = 0, for n 6= m. (4.168) Thus, if there is no degneracy, the eigenfunctions of a hermitian operator are orthogonal to each other. If there is degeneracy, one can always choose an or­ thogonal set of eigenfunctions. If the eigenfunctions are properly normalized R ∗ ψn ψn dx = 1, then the eigenfunctions build an orthonormal system Z ψ∗n ψm dx = δ nm , (4.169) and are complete, i.e. any arbitrary function f (x) can be expressed as a superposition of the orthonormal basis functions ψn (x) f (x) = ∞ X cn ψn (x) . (4.170) n=0 Thus we can freely change the basis in which we describe a certain physical problem. To account fully for this fact, we no longer wish to use wave me­ chanics, ie. express the wave function as a function in position space or in k-space. Instead we will utilize a vector in an abstract function space, i.e. a Hilbert space. In this way, we can formulate a physical problem, without us­ ing a fixed representation for the state of the system (wave function) and the corresponding operator representations. This description enables us to make full use of the mathematical structure of Hilbert spaces and the algebraic properties of operators. Bibliography [1] Introduction to Quantum Mechanics, Griffiths, David J., Prentice Hall, 1995. [2] Quantum Mechanics I, C. Cohen-Tannoudji, B. Diu, F. Laloe, John Wiley and Sons, Inc., 1978. [3] The Physics of Atoms and Quanta, Haken and Wolf, Springer Verlag 1994. [4] Practical Quantum Mechanics, S. Flügge, Springer Verlag, Berlin, 1999. [5] Handbook of Mathematical Functions, Abramowitz and Stegun, Dover Publications, NY 1970. 239