The Particle in a Box
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1 Te Particle in a Box Overview ) Te Scrödinger wave equation, H ˆ E, lies at te eart of te quantum mecanical description of atoms. H ) represents an operator (called te Hamiltonian) tat "extracts" te total energy E (te sum of te potential and kinetic energies) from te wave function. Te wave function depends on te x, y, and z coordinates of te electron's position in space. Note tat te Scrödinger equation requires tat wen is operated on by H ), te result is multiplied by a constant, E, wic represents te total energy of te particular state described by a given. As we will see, tere are many possible solutions to te Scrödinger equation for a given system. For example, for te ydrogen atom tere are many functions tat satisfy te Scrödinger equation, eac one corresponding to a particular energy for ydrogen's electron. Eac of tese specific wave functions for te ydrogen atom is called an orbital. In te case of ydrogen we denote tese orbitals as "s", "s", "p", etc. In tis case, solving te Scrödinger equation using te wave function tat describes eac of tese orbitals in ydrogen would yield te relative energies of an electron in suc orbitals. Tis information could ten be used to calculate values requiring tese energies, suc as predicting te spectrum of te atom. Altoug te detailed solution of te Scrödinger equation for te ydrogen atom is not appropriate to te level of tis course, we will illustrate some of te properties of wave mecanics and wave functions by using te wave equation to describe a very simple, ypotetical system commonly called "te particle in a box," a situation were a particle is trapped in a one-dimensional box tat as infinitely ig "sides." It is important to recognize tat tis situation is not an accurate pysical model for te ydrogen atom. Tat is, te ydrogen atom is really not muc like tis particle in a box. Te reasons for treating te particle in a box are tat () it illustrates te matematics of wave mecanics, () it gives an indication of te caracteristics of wave functions, and (3) it sows ow energy quantization arises. Tus tis treatment of a particle in a box illustrates te "flavor" of te wave mecanical description of te ydrogen atom, but it sould not be taken to be an accurate representation of te ydrogen atom itself.
2 Finally, please note tat wile tis discussion utilizes calculus, te logic and conclusions of te derivations presented erein may be understood witout knowledge of te metods used to solve calculus problems. Students will be responsible for te concepts and conclusions of tis material, and te application of te derived equations (see te sample problem at te end), but will not be eld responsible for teir derivation. Tese are presented only as an illustration of te practice of solving problems in quantum mecanics. Figure A scematic representation of a particle in a one-dimensional box wit infinitely ig potential walls. Te Particle in a Box as a Model Consider a panicle wit mass m tat is free to move back and fort along one dimension (we arbitrarily coose x) between te values x 0 and x L (tat is, we are considering a onedimensiona "box" of size L meters). We will assume tat te potential energy V(x) of te particle is zero at all points along its pat, except at te end points x 0 and x L, were V(x) is infinitely large. In effect, we ave a repulsive barrier of infinite strengt at eac end of te box. Tus te particle is trapped in a one-dimensional box wit impenetrable walls (see Fig. ). As we mentioned before, te Scrödinger equation contains te energy operator H ). In tis case, since te potential energy is zero inside te box, te only energy possible is te kinetic energy of te particle as it moves back and fort along te x axis. Te operator for tis kinetic energy is d m were is Planck's constant divided by π, m is te mass of te particle, and d / is te second derivative wit respect to x. Te form of tis operator comes from te description of
3 waves in classical pysics. Inserting tis operator into te Scrödinger equation gives H ˆ ) E d m E were is a function of x ( (x)). We can rearrange tis equation to give d me Our goal is to find specific functions (x) tat satisfy tis equation. Notice tat te solutions to tis equation are functions suc tat d / (constant). Tat is, eac solution must be a function wose second derivative as te same form as te original function. One function tat beaves tis way is te sine function. For example, consider te function A sin(, were A and k are constants. We will now take te second derivative of tis function wit respect to x: d d d sin kx d ( Asin A A ( k cos d cos kx Ak Ak( k sin Ak sin kx Ak sin kx Tus we ave sown tat d ( Asin k ( Asin Tis is just te type of function tat will satisfy te Scrödinger equation for te particle in a box. In fact, wen we compare te general form of te Scrödinger equation d me wit d ( Asin k ( Asin we see tat me k wic can be rearranged to give an expression for energy 3
4 E k m Wat does tis equation mean? We ave simply specified tat A and k are constants. Wat values can tese constants ave? Note tat if tey could assume any values, tis equation would lead to an infinite number of possible energies tat is, a continuous distribution of energy levels. However, tis is not correct. For reasons we will discuss presently, we find tat only certain energies are allowed. Tat is, tis system is quantized. In fact, te ability of wave mecanics to account for te observed (but initially unexpected) quantization of energy in nature is one of te most important factors in convincing us tat it may be a correct description of te properties of matter. Quantization enters te wave mecanical description of te particle in a box via te boundary conditions. Boundary conditions arise from te pysical requirements of natural systems. Tat is, we must insist tat our descriptions of natural systems make pysical sense. For example, assume tat in describing an aqueous solution containing an acid, we arrive at te expression [H + ] 4.0 x 0-8 M. Te solutions to tis expression are [H + ].0 x 0-4 M and [H + ] -.0 x 0-4 M In doing suc a problem, we automatically reject te second possibility because tere is no pysical meaning for a negative concentration. Wat we ave done ere is apply a type of boundary condition to tis situation. Te boundary conditions for te particle in a box enforce te following facts:. Te particle cannot be outside te box it is bound inside te box.. In a given state te total probability of finding te particle in te box must be (or 00%). 3. Te wave function must be continuous. We ave seen tat te function A sin( satisfies te Scrödinger equation ) H ˆ E. We will now define te constants k and A so tat tis function also satisfies te boundary conditions based on te tree constraints listed above. Because te particle must stay inside te box and because te wave function must be continuous, te value of (x) must be zero at eac wall. Tat is, (0) 0 and (L) 0 4
5 Recall tat te sine function is zero at angles of 0º, 80 (π radians), 360 (π radians), and so on. Tus te function A sin kx is automatically zero wen x 0. Te requirement tat te wave function must also be zero at te oter wall, wic can be stated as (L) A sin (kl) 0, means tat k is limited to te values of nπ/l, were n is an integer (,, 3,... ). Tat is, nπ ( x) Asin L nπ ten ( L) Asin L Asin( nπ ) 0 L To assign te value of te constant A, we need to introduce a new idea. In te application of wave mecanics to te description of matter, scientists ave learned to associate te square of te wave function ( ) wit probability. As we will discuss in more detail below, tis means tat te square of te wave function evaluated at a given point gives te relative probability of finding a particle near tat point. In fact, te tree dimensional pictures of orbitals (suc as s, s, p, etc.) tat we are familiar wit for atoms comes from plotting te regions in space were te values of are greater tan 0.9 (or 90%). Tis concept is relevant to te boundary conditions for te particle in a box because te total probability in a given state must be (or 00%). To be more precise, te probability of finding te particle on a segment of te x axis of lengt surrounding point x is (x). Because tere is one particle in te box te sum of all of tese probabilities along te x axis from x 0 to x L must be. We sum tese probabilities over te lengt of te box (from x 0 to x L) by integration from x 0 to x L: Total probability of finding te particle in te box L 0 ( x) Substituting (x) A sin [(nπ/l)x], we ave L nπ L ( x) A sin 0 L L nπ or sin L A 0 0 Te value of te integral is L/, wic means tat 5
6 L and A L A Now tat we know te allowed values of k and A, we can specify te wave function for te particle in a one-dimensional box as nπ ( x) sin We can also substitute te value of k into te expression for energy: k E m ( nπ / L) m Substituting / π gives n E were n,, 3, 4,... Note tat tis analysis leads to a series of solutions to te Scrödinger equation, were eac function corresponds to a given energy state: n Function Energy 3 4 π sin E π sin 4 E ml 3π sin 9 E 3 3 4π sin 6 E ml 4 4 M M M Notice someting very important about tese results. Te application of te boundary conditions as led to a series of quantized energy levels. Tat is, only certain energies are allowed for te particle bound in te box. Tis result fits very nicely wit te experimental evidence, suc as te ydrogen emission spectrum, tat nature does not allow continuous energy 6
7 levels for bound systems, as classical pysics ad led us to expect. Note tat te energies are quantized, because te boundary conditions require tat n assume only integer values. Consequently, we call n te quantum number for tis system. We can diagram te solutions to te particle-in-a-box problem conveniently by sowing a plot of te wave function tat corresponds to eac energy level. Te energy level, wave function, and probability distribution are sown in Fig. for te first tree levels. Figure (a) Te first tree energy levels for a particle in a onedimensional box in increments of /( ). (b) Te wave functions for te first tree levels plotted as a function of x. Note tat te maximum value is / L in eac case. (c) Te square of te wave functions for te first tree levels plotted as a function of x. Note tat te maximum value is /L in eac case. Note tat eac wave function goes to zero at te edges of te box, as required by te boundary conditions. Anoter way to say tis is tat te standing waves tat represent te particle must ave wavelengts suc tat an integral number of alf-wavelengts exactly equals te size of te box. Waves wit any oter wavelengts could not exist because tey would destructively interfere over time. Also, note from Figure tat te probability distribution is significantly different for te tree levels. For n {te lowest energy or ground state} te particle is most likely to be found near te center of te box. In contrast, for n te particle as zero probability of being found in te center of te box. Tis zero point is called a node. Notice tat te number of nodes increases wit n. Anoter interesting caracteristic of te particle in a box is tat te particle cannot ave zero energy (tat is, n cannot equal zero). For example, if n were equal to zero, 0 would be zero everywere in te box (sin 0 0). Tis would mean tat 0 would also be zero. In tis 7
8 case tere could be no particle in te box, wic contradicts te boundary conditions. Tis fact tat te particle must ave a nonzero energy in its ground state is a caracteristic of all particles wit quantized energies. In addition, for te particle in a box, a value of zero for te energy would mean tat te particle was sitting still (zero kinetic energy). Tis condition would violate te uncertainty principle, because we would simultaneously know te exact values of te momentum (zero) and te position of te particle. For similar reasons all quantized particles must possess a minimum energy, often called te zero-point energy. Wile te particle in te box is a simple one-dimensional model, te same sort of analysis presented ere is applied to an electron in a sperical potential well around a positively carged nucleus wen solving for te energies and wave functions of te ydrogen atom. Altoug te particle in te box is not a good model for te atom, it does approximate te energy levels found in simple covalent bonds (were a electron sared between two nuclei is very similar to one trapped in a "box") and it elps to explain resonance stability. Bot of tese applications of te particle in te box will be explored later in class. Sample Problem Assume tat an electron is confined to a one-dimensional box.50 nm in lengt. Calculate te lowest tree energy levels for tis electron, and calculate te wavelengt of ligt necessary to promote te electron from te ground state to te first excited state. Solution To solve tis problem, we need to substitute appropriate values into te general expression for energy: n E Te mass of an electron (m) is kg; te dimension of te box (L) is.50 nm, or m; and te value of Planck's constant is J s. For n we get. E 34 ) ( J s) 3 (8)(9. 0 kg)(.50 0 ( m) J Similarly, for n we get E J And for n 3 we get 8
9 E J Note tat since E n n n E ten E ( ) 4E and E 3 9E To calculate te wavelengt of ligt necessary to excite te electron from level to level (te first excited state), we first need to obtain te energy difference between te two levels: E E E ( n (3)( n ) J) Ten we find te wavelengt using Plank's equation c E λ Inserting te appropriate values gives 34 8 c ( J s)( m/s) λ -0 E J m 470 nm J 9
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