Appendix A. The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System
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1 Appendix A The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System Real quantum mechanical systems have the tendency to become mathematically quite complicated and may discourage a novice in the field from pursuing the detailed steps to understand how the mathematical principles apply to physical systems. Thus, a simple scenario is presented here to illustrate the principles of Quantum Mechanics introduced in Section 1.4. The model to be presented is the so-called particle-in-a-box (henceforth referred to as the PiB ) that is an artificial system, yet with wide-ranging analogies to real systems. This model is very instructive, because it shows in detail how the quantum mechanical formalism works in a situation that is sufficiently simple to carry out the calculations step by step. Furthermore, the symmetry (parity) of the PiB wavefunctions is very similar to that of vibrational wavefunctions discussed in Section 1.4. Finally, the concept of transition from one stationary state to another can be demonstrated using the principles of the transition moment introduced in Section 1.5. A.1 Definition of the Model System The PiB model assumes a particle, such as an electron, to be placed into a potential energy well, or confinement shown in Figure A.1. This confinement (the box ) has zero potential energy for 0 x, where is the length of the box. Outside the box, that is, for x < 0 and for x >, the potential energy is assumed to be infinite. Thus, once the electron is placed inside the box, it has no chance to escape, and one knows for certain that the electron is in the box. Next, the kinetic and potential energy expression will be defined, which subsequently allows writing the Hamiltonian, or the total energy operator of the system. For any quantum mechanical system, the total energy is written as the sum of the kinetic and potential energies, T and V, respectively: E = T + V (A.1) Modern Vibrational Spectroscopy and Micro-Spectroscopy: Theory, Instrumentation and Biomedical Applications, First Edition. Max Diem. 015 John Wiley & Sons, td. Published 015 by John Wiley & Sons, td.
2 366 Modern Vibrational Spectroscopy and Micro-Spectroscopy x 0 x (a) (b) Figure A.1 ( ) (a) Wavefunctions ψ n (x) = nπ xforn = 1,, 3, 4, and 5 drawn at their appropriate energy levels. Energy given in units of h 8m. (b) Plot of the square of the wavefunctions shown in (a) As before, the kinetic energy of the particle is given T = 1 mν or T = p (A.) m where m is the mass of the electron, and p is its linear momentum. Substituting, as before, the quantum mechanical momentum operator, p = ħ d (A.3) i dx into Eq. A., the kinetic energy operator can be written as T = ħ m As pointed out in Section 1.4, all the information one seeks about a quantum mechanical system is contained in a wavefunction ψ. In the case here, this wavefunction is a function of x only and is written as ψ(x). Thus, Eq. A.4 is an instruction that prescribes: to obtain the kinetic energy of a quantum mechanical system, take the second derivative of the wavefunction, and multiply the result by m. The potential energy inside the box is zero; thus, the total energy of the particle inside the box is T = Ê = ħ m Since the potential energy outside the box is infinitely high, the particle cannot be there, because that case would correspond to infinite energy, and the discussion henceforth will deal with the inside of the box. d dx d dx (A.4) (A.5)
3 The Particle in a Box 367 Thus, one may write the total Hamiltonian of the system as Ĥ = T + 0 = ħ m In the notation of linear algebra, an operator/eigenvector/eigenvalue equation is written as: Ĥψ = Eψ d dx (A.6) (A.7) Equation A.7 instructs to define an operator, such as the Hamiltonian shown in Eq. A.5, and operate with it on a set of yet unknown eigenfunctions to obtain the desired energy eigenvalues. The eigenfunctions typically form an n-dimensional vector space in which the eigenvalues are appearing on the diagonal. Thus, Eq. A.7 implies: ψ 1 E ψ 1 ψ 0 E 0 ψ Ĥ ψ 3 = 0 0 E 3 ψ 3 (A.8) ψ ψ 4 Each eigenvalue E i is associated with one eigenfunction. A. Solution of the Particle-in-a-Box Differential Equation Next, one rewrites Eq. A.7 to explicitly include the operator { } ħ d ψ(x) =E ψ(x) m dx which is a simple differential equation: d me ψ(x)+ ψ(x) =0 (A.10) dx Next, the differential equation described by Eq. A.10 needs to be solved. The functions fulfilling this equation have to be of the form that their second derivative equals to the original function, multiplied by a constant. For example, the function y = A cos bx (A.11) could be solution of the differential Eq. A.10, since d y dx = Ab cos bx Here, the term b would correspond to me, and A is a yet undefined amplitude factor. Similarly, (A.9) (A.1) y = A sin cx (A.13) or the sum of Eqs. A.11 and A.13 could be acceptable solutions. For reasons that will become obvious shortly, Eq. A.13 will be used as a trial function to fulfill Eq. A.10: ψ(x) =A sin x (A.14)
4 368 Modern Vibrational Spectroscopy and Micro-Spectroscopy and d me ψ(x) = A sin x = A me ψ(x) (A.15) dx At this point, it should be pointed out that the solutions of any differential equation depend to a large extent on the boundary conditions: the general solution of the differential equation may or may not describe the physical reality of the system, and it is the boundary conditions that force the solutions to be physically meaningful. In the case of the PiB, the boundary conditions are determined by one postulate of quantum mechanics that requires that wavefunctions are continuous. Thus, if the wavefunction outside the box is zero (since the potential energy outside the box is infinitely high and, therefore, the probability of finding the particle outside the box is zero), the wavefunction also must be zero at the inside boundaries of the box. Thus, one may write the boundary conditions for the PiB differential equation as ψ(x) =0 at x = 0 and at x =. (A.16) Because of these conditions, the cosine function proposed as possible solutions (Eq. A.11) of Eq. A.10 was rejected, since the cosine function is nonzero at x = 0. Because of the required continuity at x =, the function ψ(x) =A sin x must be zero at x = as well. This can happen if the amplitude A is zero (this case is of no further interest, since a zero amplitude of the wavefunction implied that the particle is not inside the box), or if sin = 0 The sine function is zero at multiples of π radians; that is, (A.17) = nπ, n = 1,, 3... (A.18) Solving Eq. A.18 for E yields E = n π m = n h (A.19) 8 m Equation A.19 reveals that the energy levels of the particle in a box are quantized, that is, the energy can no longer assume any arbitrary value, but only values of h 8 m, 4 h 8 m, 9 h 8 m, and so on. This quantization is a direct consequence of the boundary conditions, which required wavefunctions to be zero at the edge of the box. Since the energy depends on this quantum number n, one usually writes Eq. A.19 as E(n) = n h (A.0) 8 m Substituting these energy eigenvalues back into Eq. A.14 one obtains ψ(x) =A sin x ( ) mn ψ n (x) =A sin h 1 nπ x = A sin 8 m x (A.14) (A.1)
5 The Particle in a Box 369 A is a still undefined amplitude factor at this point. To determine A, one argues as follows: since the square of the wavefunction is defined as the probability of finding the particle, the square of the wavefunction written in Eq. A.1, integrated over the length of the box, must be unity, since the particle is known to be in the box. This leads to the normalization condition ψ ( ) n (x)dx = 1 = A sin nπ 0 0 x dx (A.) Using the integral relationship sin ax dx = x 1 sin ax 4a (A.3) the amplitude A is obtained as follows: ( ) nπx dx [ = A x A sin [ 0 A ( nπ 4nπ sin A = ( nπx 4nπ sin ) 0 + ] 4nπ sin 0 )] x= x=0 = A [ = 1 ] = 1 (A.4) Thus, the stationary state wavefunctions for the particle in a box can be written in a final form as ( ) nπ ψ n (x) = sin x (A.5) The resulting stationary state (time-independent) wavefunctions and energies are depicted in Figure A.1(a). Although one refers to these wavefunctions as time-independent, they may be considered as standing waves in which the amplitudes oscillate between the extremes shown in Figure A. and resemble the motion of a plugged string. Time independency then refers to the fact that the system will stay in one of these standing wave patterns forever, or until perturbed by electromagnetic radiation. The probability of finding the particle at any given position x is shown in Figure A.1(b). These traces are the squares of the wavefunctions, and depict that for higher levels of n, the probability of finding the particle Figure A. Representation of the particle-in-a-box wavefunctions shown in Figure A.1 as standing waves
6 370 Modern Vibrational Spectroscopy and Micro-Spectroscopy moves away from the center to the periphery of the box. If one draws a vertical line at the center of the box, one finds that the wavefunctions have the same odd/even parity as do the harmonic oscillator wavefunctions in their quadratic potential function (see Figure 1.5). A.3 Orthonormality of the Particle-in-a-Box Wavefunctions The PiB wavefunctions form an orthonormal vector space as defined before for the vibrational wavefunctions (Eq. 1.53), which implies that { = 1 if n = m ψ n (x) ψ m (x)dx = δ mn = (A.6) 0 = 0 if n m The wavefunctions normality was established above by normalizing them (Eq. A.4); in order to demonstrate that they are orthogonal, the integral sin nπx mπx sin 0 dx (A.7) needs to be evaluated. This can be established using the integral relationship sin ax sin bx dx = sin(a + b)x (a + b) + sin(a b)x (a b) (A.8) For any two adjacent wavefunction, say m = 1 and n =, or m = and n = 3, the numerator of the first term in Eq. A.8 will contain the sine function of odd multiples of π, whereas the numerator of the second term will contain the sine function of even multiples of π. Since the sine function of odd and even multiples of π is zero, the total integral described by Eq. A.7 is zero. An identical argument can be presented for any n m case. A.4 Dipole-Allowed Transitions for the Particle in a Box The PiB model system may also be used to study transitions between stationary states, induced by electromagnetic radiation. In complete analogy to the perturbation discussed for the harmonic oscillator, the transition moment ψ m μ ψ n (A.9) needs to be evaluated for the PiB wavefunctions. For the transition from n = 1ton =, Eq. A.9 becomes μ = e sin πx πx x sin 0 dx (A.30) In analogy with the graphical interpretation of the harmonic oscillator transition moment, Figure A.3 demonstrates the plot of ψ 1,ψ and the transition operator. The latter is represented by a straight line, since μ = ex for a one-dimensional system. The shaded area under the curve in Figure A.3(b), when integrated from 0 to, is nonzero. This can also be established from the parity argument introduced earlier: since ψ and the dipole operator μ = ex are odd functions, their product will be even. This product, multiplied by the ground state ψ 1, which has even parity, will result in an overall transition moment with even parity. For the particle in a box, this leads to the following selection rules: transitions with n =±1, ±3, ±5, and so on are allowed, whereas transitions with n =±, ±4, and so on are forbidden because the transition moment
7 The Particle in a Box 371 (b) (a) 0 Figure A.3 (a) Plot of particle-in-a-box wavefunction ψ 1 (light gray),ψ (gray) and transition moment operator μ. (b): product of the three functions shown in (a) integrals are zero. Thus, one encounters here the transitions being allowed or forbidden depending on the symmetry of the wavefunctions. The transition moment is the quantity that needs to be evaluated in order to determine whether or not a transition may occur. A.5 Real-World PiBs Although the PiB was introduced here as a model to demonstrate quantum mechanical principles in a mathematically manageable system, there are physical examples that can be treated adequately using the PiB formalism. One of these is frequently incorporated as an experiment in physical chemistry laboratories, and involves a conjugated dye such as 1,6-diphenyl-1,3,5-hexatriene. In this molecule, the ewis structure suggests three double and four single bonds. The UV-vis absorption spectrum shows one broad absorption peak that is, in this experiment, commonly assigned as a PiB transition of one electron from the n = 3ton = 4 PiB energy level. Here, one assumes that the six conjugated π-electrons occupy the energy levels 1,, and 3. Modeling this system with a conjugated length of about 1 nm, corresponding to seven alternating single and double bonds, reproduces the observed transition frequency satisfactorily. Certain quantum dot structures can also be modeled by a two-dimensional particle in a box. Quantum dots are typically manufactured by creating n-type semiconductors whose electrons are free to move over the entire size of the dot. In these quantum dots, the energy levels of the free electrons follow a -D PiB model; consequently, the color of electronic transitions can be tuned by changing the size of the quantum dot. Finally, the super lattice found in quantum cascade lasers (QCs) crystal gives rise to energy levels that can be explained using a PiB model. In these super-lattice structures, layers of high, and low potential energy alternate, produced by different doping levels in the layers. These potential wells mimic a Pib with finite potential energy barriers (see Figure A.4). Furthermore, these energy wells are superimposed on a sloping potential energy background. The slope of the background has two consequences: first, the wavefunctions are distorted to have higher amplitudes at the left side of each potential well, and second, the electrons in the ground state of each well can tunnel through the barrier between the wells to end up in an excited state of the next lower
8 37 Modern Vibrational Spectroscopy and Micro-Spectroscopy Potential energy e hv e hv e hv attice dimension Figure A.4 The super-lattice structure in a quantum cascade laser modeled by successive PiB energy states energy well. Electrons are injected at a potential energy marked by the -symbol, and undergo a transition as indicated by the left-most down arrow. During this transition, an (infrared) photon is emitted and subsequently, the electron may tunnel through the finite-heights barrier and arrives in the next quantum well, and undergo another transition. The emission and tunneling processes are repeated as many times as there are quantum wells in the superstructure. The term cascade in QC is due to the fact that one electron can undergo many consecutive emission process in the super-lattice structure. By placing the super-lattice crystal into an optic cavity, stimulated emission from the excited states into the ground states of each well can be achieved.
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