DOING PHYSICS WITH MATLAB QUANTUM MECHANICS SCHRODINGER EQUATION TIME INDEPENDENT BOUND STATES

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1 DOING PHYSICS WITH MATLAB QUANTUM MECHANICS SCHRODINGER EQUATION TIME INDEPENDENT BOUND STATES Ia Cooper School of Physics, Uiversity of Sydey MATLAB SCRIPTS Goto the directory cotaiig the m-scripts for Quatum Mechaics The Matlab scripts are used to solve the Schrodiger Equatio for a variety of potetial eergy fuctios. se_fdm.m m-script se to solve [1D] time idepedet Schrodiger Equatio usig a fiite differece approach where E is etered maually to fid acceptable solutios. simpso1d.m Fuctio to evaluate the area uder a curve usig Simpso s 1/3 rule. Colorcode.m Fuctio to retur the appropriate colour for a wavelegth i the visible rage from 38 m to 78 m. se_wells.m First m-script to be ru whe solvig the Schrodiger equatio usig the Matrix Method. Most of the costats ad all the well parameters are declared i this file. You ca select the type of potetial well from the Commad Widow whe the m-script is ru. You alter the m-script code to chage the parameters that characterize the wells ad you ca Doig Physics with Matlab Quatum Mechaics Boud States 1

2 add to the m-script to defie your ow potetial well. Whe this m-script is ru it clears all variables ad closes all ope Figure Widows. se_solve.m This m-script solves the Schrodiger equatio usig the Matrix Method after you have ru the m-script se_wells.m. The eigevalues ad correspodig eigevectors are foud for the boud states of the selected potetial well. se_psi.m To be ru after se_wells.m ad se_solve.m. A graphical output displays the total eergy, the potetial eergy fuctio, kietic eergy fuctio, eigevector ad probability distributio for a give quatum state. se_measuremets.m To be ru after se_wells.m ad se_solve.m. Evaluates: the expectatio values for a set of dyamic quatities; the iheret quatum-mechaical ucertaities i measuremets ad gives a test of the Heiseberg ucertaity priciple. se_orthoormal.m To be ru after se_wells.m ad se_solve.m. You ca ivestigate the orthoormal characteristic of the eigevectors (statioary state wavefuctios). se_ifwell.m Used to test the accuracy of the Matrix Method. Compares the aalytical ad umerical results for a ifiite square well potetial of width.1 m. gausssiai.m Produces a graphical display of a Gaussia shaped potetial well ad the correspodig force graph. se_statioary.m To be ru after se_wells.m ad se_solve.m. You ca ivestigate ad view the time evolutio of a eergy eigestate ad save the plot as a aimated gif. se_super.m To be ru after se_wells.m ad se_solve.m. You ca ivestigate ad view the time evolutio of a compoud state ad save the plot as a aimated gif. Doig Physics with Matlab Quatum Mechaics Boud States

3 SCHRODINGER EQUATION O a atomic scale, all particles exhibit a wavelike behavior. Particles ca be represeted by wavefuctios which obey a differetial equatio, the Schrodiger Wave Equatio which relates spatial coordiates ad time. You ca gai valuable isight ito quatum mechaics by studyig the solutios to the oe-dimesioal time idepedet Schrodiger Equatio. A wave equatio that describes the behavior of a electro was developed by Schrodiger i 195. He itroduced a wavefuctio ( x, y, z, t). This is a purely mathematical fuctio ad does ot represet ay physical etity. A iterpretatio of the wave fuctio was give by Bor i 196 who suggested that the quatity ( x, y, z, t) represets the probability desity of fidig a electro. For the oe dimesioal case, the probability of fidig the electro at time t somewhere betwee x 1 ad x is give by (1) x * Prob( ) (, ) (, ) x1 t x t x t dx where * is the complex cojugate of the wavefuctio. The value of Prob(t) must lie betwee ad 1 ad so whe we itegrate over all space, the probability of fidig the electro must be 1. * ( x, t) ( x, t) dx 1 I this istace the wavefuctio is said to be ormalized. We ca see how the time-idepedet Schrodiger Equatio i oe dimesio is plausible for a particle of mass m, whose motio is govered by a potetial eergy fuctio U(x) by startig with the classical oe dimesioal wave equatio ad usig the de Broglie relatioship Classical wave equatio ( x, t) 1 ( x, t) x v t Doig Physics with Matlab Quatum Mechaics Boud States 3

4 Mometum (de Broglie) Kietic eergy h p mv k K 1 mv Total eergy E K( x) U( x) i t Wavefuctio ( x, t) ( x) e periodic i time for t coordiate Combiig the above relatioships, the time-idepedet Schrodiger Equatio i oe dimesio ca be expressed as () d ( x) U( x) ( x) E( x) m dx Our goal is to fid solutios of this form of the Schrodiger Equatio for a potetial eergy fuctio which traps the particle withi a regio. The egative slope of the potetial eergy fuctio gives the force o the particle. For the particle to be boud the force actig o the particle is attractive. The solutios must also satisfy the boudary coditios for the wavefuctio. The probability of fidig the particle must be 1, therefore, the wavefuctio must approach zero as the positio from the trapped regio icreases. The impositio of the boudary coditios o the wavefuctio results i a discrete set of values for the total eergy E of the particle ad a correspodig wavefuctio for that eergy, just like a vibratig guitar strig which has a set of ormal modes of vibratio i which there is a harmoic sequece for the vibratio frequecies. The Schrodiger Equatio ca be solved aalytically for oly a few forms of the potetial eergy fuctio. So, we will cosider two umerical approaches to solvig the Schrodiger Equatio: (1) fiite differece method i which the secod derivative is approximated as a differece formula ad () a matrix method where the eigevalues of a matrix gives the total eergies of the particle ad the eigefuctios the correspodig wavefuctios. Doig Physics with Matlab Quatum Mechaics Boud States 4

5 FINITE DIFFERENCE METHOD Oe ca use the fiite differece method to solve the Schrodiger Equatio to fid physically acceptable solutios. Oe ca also use the Matlab ode fuctios to solve the Schrodiger Equatio but this is more complex to write the m-script ad ot as versatile as usig the fiite differece method. The fiite differece method allows you to easily ivestigate the wavefuctio depedece upo the total eergy. The heart of the fiite differece method is the approximatio of the secod derivative by the differece formula (3) d x x x x x x ( ) ( ) ( ) ( ) dx x ad the Schrodiger Equatio is expressed as d ( x) m (4) k( x) ( x) k( x) E U( x) dx We will cosider a electro trapped i a potetial well of width L ad depth U as show i Figure 1. I regios where E > U(x), k(x) is real ad ( x) has a siusoidal shape ad this correspods to the classical allowed regio (kietic eergy K > ). I regios where E < U(x), k(x) is imagiary ad ( x) has a expoetial icreasig or decreasig ature ad this correspods to the classical forbidde (kietic eergy K < ). The fuctio d is the curvature of the wavefuctio ad its egative is a measure of the kietic dx eergy of the particle (Figure 1). Doig Physics with Matlab Quatum Mechaics Boud States 5

6 eergy U = E classical allowed regio E > U K > U(x) positive curvature E < U K < E = K + U E < U K < ( x) egative curvature U(x) positive curvature U = -U classical forbidde regio positio x classical forbidde regio curvature d K. E. dx Fig. 1. Potetial well defied by the potetial eergy fuctio U(x). The boud particle has total eergy E ad its wavefuctio is ( x). You ca use a shootig method to fid E that satisfies both the Schrodiger Equatio ad the boudary coditios. We start with ( x ) ad a give value for E ad solve the mi Schrodiger Equatio. The value of E is icreased or decreased util the other boudary coditio, ( x ) is satisfied. If the ed boudary coditio at x = x max is ot max satisfied the the wavefuctio will expoetially diverge ( x xmax ( x) ). Whe a solutio is foud, the wavefuctio decreases expoetially towards the zero as show i Figure. Doig Physics with Matlab Quatum Mechaics Boud States 6

7 Fig.. Physically acceptable solutios are foud oly whe the wavefuctio ( x) coverges to zero at the extreme values of x. The depth of the well is -4 ev ad the width.1 m. [se_fdm.m]. The process of fidig physically acceptable values for E ad the correspodig wavefuctios ( x) ca be automated by coutig the umber of zero crossig of the wavefuctio ad adjustig the value of E util the coditio ( x ) is satisfied. For the lowest eergy state (groud state) E 1, the wavefuctio is zero oly at the extreme values for x ad therefore, the umber of crossig is zero. The 1 st excited state E will have oly 1 crossig ad the th excited state E +1 will have crossigs as show i Figure 3. max I the Fiite Differece Method, we start with 1,,, x x x x N 1,,, k k k k N where N is the maximum umber of x coordiates, x(1) = x mi ad x(n) = x max mi x ad x x x ( x ) (1) () (1) 1 the as x is icremeted, the other values of ( x) are calculated from the equatio ( x k x x x for c = to N-1 c1 c c c 1 Whe a physically accepted solutio is foud for the th state the wavefuctio is ormalized by umerically itegratig the wavefuctio usig Simpso s 1/3 rule Doig Physics with Matlab Quatum Mechaics Boud States 7

8 x( N) x(1) ( x) ( x) dx A ( x) A The m-script se_fdm.m was used to fid the total eergies ad its correspodig wavefuctios for a potetial well of depth -4 ev ad width.1 m. The rage for the x-coordiates was from -.1 m to +.1 m. The value of E was maually adjusted to fid the physically acceptable solutios as show i Figure 3. For this potetial well, there are four boud states. The total eergy for the groud state, = 1 is E 1 = ev. Thus, the bidig eergy of the electro or its ioizatio eergy (eergy eed to free the electro from its boud state) is E B = ev. groud state = 1, E 1 = ev 1 st excited state =, E = ev d excited state = 3, E 3 = ev 3 d excited state = 4, E 4 = -1 ev Fig. 3. The four states for a electro cofied by a potetial well of depth -4 ev ad width.1 m with x mi = -.1 m, x max = +.1 m. [se_fdm.m] The smaller bidig eergy, the less accurate are the results because the rage of x values is ot large eough to accurately defie the expoetial covergece to zero of the wavefuctios as x approaches x max. For the case = 4 as show i Figure 3, E 4 = -1 ev Doig Physics with Matlab Quatum Mechaics Boud States 8

9 whe x max =.1 m ad oe ca see that the expoetial tail is ot very well defied. Whe x max is icreased to. m, the expoetial tail is better defied ad the total eergy is E 4 = ev. Solutios of the Schrodiger Equatio deped upo the width of the well ad the rage of x values. Oe has to make a judgmet based upo the variatio of (x) as x approaches x max i determiig the most suitable rage for the x values. Figure 4 shows the groud state, for the potetial well of depth -4 ev ad width.1 m whe x max is icreased from.1 m to. m. It is ow more difficult to fid the total eergy for this state sice slight variatios i E result i expoetial divergig tails. E = ev E = ev E = ev Fig. 4. The potetial well has a depth -4 ev ad width.1 m. The x values rage from -. m to +. m. The wavefuctio (x) is more sesitive to the total eergy E as the rage of x values is icreased. [se_fd.m]. Doig Physics with Matlab Quatum Mechaics Boud States 9

10 MATRIX METHOD The oe-dimesioal time idepedet Schrodiger equatio ca be expressed as (5) d ( x) U( x) ( x) E( x) m dx d U( x) ( x) E( x) m dx H( x) E( x) where the H is the Hamiltoia operator, which is the operator that correspods to the total eergy of the system. This is a eigevalue equatio. The actio of the operator H o the fuctio returs the origial fuctio multiplied by a costat which could be complex. This eigevalue equatio is geerally satisfied by a particular set of fuctios E, E, 1( x), ( x),, ad a correspodig set of costats 1. These are the eigefuctios ad the correspodig eigevalues of the operator H. The time idepedet Schrodiger equatio of a system is the eergy eigevalue equatio of the system. A eigefuctio ( x ) describes a state of defie eergy E. Whe the eergy of the system i this state is measured, the result will always be E. For the eigefuctio to represet physical sesible solutios, we require ( x ) as x so that the wavefuctio ca be ormalized. For atomic systems it is more coveiet to measure legths i m (aometers) ad eergies i ev (electro volts). We ca use the scalig factors legth: L se = to covert m ito m eergy: E se = to covert J ito ev ad so we ca write Equatio (5) as (6) 1 d m Lse Ese dx U ( x) ( x) E( x) d 1 Cse U ( x) ( x) E( x) where Cse dx m Lse Ese H( x) E( x) Doig Physics with Matlab Quatum Mechaics Boud States 1

11 Cosider a electro i a potetial well (see Figure 1). For eergy values below the top of the well, the physically acceptable solutio of the time idepedet Schrodiger equatio give a discrete set of eergies which are the eergy eigevalues ad correspodig to each eigevalue there is the eergy eigefuctio. Quatizatio of the eergy levels of boud particles arises aturally from the time idepedet Schrodiger equatio ad the boudary coditios imposed for physically acceptable solutios. The spectral lies observed i atomic systems are the result of trasitios betwee such eergy levels. These eigestates ( x ) represets statioary states ad the total wavefuctio ca be expressed as (7) ( x, t) ( x) e i E t/ This is a state of defiite eergy, if the eergy is measured the the value obtaied will be E. It is called a statioary state, because the probability of locatig a particle i a iterval dx is time idepedet. May problems i physics reduce to solvig a eigevalue equatio, for example, the vibratios of a violi strig. The eigevalues ad eigefuctios ca be easily foud usig the Matlab commad eig. The m-script se_solve.m is used to solve the Schrodiger equatio usig the Matrix Methos. To solve the Equatio (6), we first represet the cotiuous fuctios of x by sets of N discrete quatities expressed as vectors ad matrices. The discrete set of x values is represeted by the vector x the correspodig wavefuctios by the vector. The potetial eergy is give by a (N-)(N-) diagoal matrix [U] with diagoal elemet U. A sample code [se_solve.m] for assigig the diagoal elemets for [U] is U_matrix = zeros(n-,n-); for c = 1 : N- U_matrix (c,c) = U(c); ed Doig Physics with Matlab Quatum Mechaics Boud States 11

12 Next, we have to represet the operator d m dx as a matrix of size (N-)(N-). From the defiitios of the first ad secod derivatives of the fuctio y(x), we ca approximate them by the equatios dy y y dx x x1/ 1 d y 1 dy dy y y y dx x dx dx x 1 1 x x 1/ x 1/ Hece, the secod derivative matrix for N = 6 ca be writte as a 44 [ SD ] x 1 The SD matrix size is (N-)(N-) ad ot NN because the secod derivative of the fuctio ca t be evaluated at the ed poits, = 1 ad = N. The kietic eergy matrix [K] is the defied as [ K] Cse SD We ca ow defie the Hamiltoia matrix as [ H]KU A sample code [se_solve.m] for the geeratig the Hamiltoia matrix is % Make Secod Derivative Matrix off = oes(um-3,1); SD_matrix = (-*eye(um-) + diag(off,1) + diag(off,-1))/dx; % Make KE Matrix K_matrix = Cse * SD_matrix; % Make Hamiltoia Matrix H_matrix = K_matrix + U_matrix; Doig Physics with Matlab Quatum Mechaics Boud States 1

13 Therefore, the Schrodiger Equatio i matrix form is [ H ] E This is a eigevalue equatio i matrix form where the actio of the Hamilto matrix results i each value to be the vector beig multiplied by a multiplied by the set of umbers E. The set of umbers E are called the eigevalues ad set of vectors are the eigevectors. This is a sigle Matlab fuctio that fids both the eigevalues ad eigevectors. The sytax of the commad is [e_fuct, e_values] = eig(h) where e_fuct is a (N-)(N-) matrix with the th colum correspodig to the th eigefuctio ad e_values is a colum vector for the N eigevalues i icreasig order. Oly the egative values of e_values are sigificat. To obtai the complete eigevector we eed to iclude the ed poits where ( xmi ) ( xmax ). A sample Malab code [se_solve.m] to obtai the discrete set of eigevalues ad ormalized eigefuctio is % All Eigevalues 1,,... where E_N < flag = ; = 1; while flag == E() = e_values(,); if E() >, flag = 1; ed; % if = + 1; ed % while E(-1) = []; = -; % Correspodig Eigefuctios 1,,...,: Normalizig the wavefuctio for c = 1 : psi(:,c) = [; e_fuct(:,c); ]; area = simpso1d((psi(:,c).* psi(:,c))',xmi,xmax); psi(:,c) = psi(:,c)/sqrt(area); prob(:,c) = psi(:,c).* psi(:,c); ed % for A potetial well as show i Figure (5) has a miimum at x = ad teds to zero away from the origi. A classical particle would be trapped by this potetial well ad oscillate to ad fro about x = because the force o the particle is always directed towards origi for positio sice F = -du/dx. Doig Physics with Matlab Quatum Mechaics Boud States 13

14 U U = x force o boud electro F du F dx F = x x = Fig. 5. A potetial well which traps a particle because the force actig o the particle is always directed towards the origi. [guassia_p.m aotatio of figure doe i MS Powerpoit]. We ca solve the Schrodiger equatio ( Equatio 5) for a variety of differet potetial eergy fuctios. The m-script SE_wells.m defies most of the costats ad potetial well parameters. Withi the m-script you ca chage the code to modify the potetial wells ad add ew potetial wells. The first step i solvig the Schrodiger Equatio usig the Matrix Method for a boud electro is to ru the file SE_wells.m ad select the type of potetial well. The default potetials iclude: 1 Square well Stepped well 3 Double well 4 Slopig well 5 Trucated Parabolic well 6 Morse Potetial well 7 Parabolic fit to Morse Potetial 8 Lattice Doig Physics with Matlab Quatum Mechaics Boud States 14

15 eergy (ev) eergy (ev) eergy (ev) eergy (ev) eergy (ev) eergy (ev) eergy (ev) Potetial Well: SQUARE Potetial Well: STEPPED x (m) x x (m) x Potetial Well: DOUBLE Potetial Well: SLOPING x (m) x (m) Potetial Well: Trucated PARABOLIC 5 Potetial Well: MORSE potetial x (m) x x (m) x 1-1 Potetial Well: Lattice x (m) Fig. 6. Some of the default potetial wells that ca be geerated usig the m-script se_wells.m. Doig Physics with Matlab Quatum Mechaics Boud States 15

16 If you are ot satisfied with the potetial well that is displayed, you ca chage ay of the parameters i the m-script se_wells.m ad ru it agai. The to solve the Schrodiger equatio ru the m-script se_solve.m. Cosider a slopig potetial well with iput parameters set i se_wells.m % Iput parameters um = 81; xmi = -.1; % default value = -.1 m xmax = +.1; % default value = +.1 m U1 = -1; % Depth of LHS well: default = -1 ev; U = -; % Depth of RHS well: default = - ev; x1 =.5; % 1/ width of well: default =.5 m; The solutio of the Schrodiger equatio for this slopig potetial well usig se_solve.m provides the followig iformatio: (1) The eergy eigevalues are displayed i the Commad widow No. boud states foud = 5 Quatum State / Eigevalues E (ev) The eergies are stored i the variable E. E(1) is the first eergy level (groud state), E() is the secod eergy level, ad so. The values ca be displayed at ay time i the Commad Widow by simply typig E >> E E = Doig Physics with Matlab Quatum Mechaics Boud States 16

17 eergy U, E (ev) () A graph of the eergy spectrum as show i Figure (7) for a slopig potetial well Potetial Well: SLOPING positio x (m) Fig. 7. The eergy spectrum for a potetial well produced by the m-script se_solve.m. (3) The eergy eigevectors are give by the array psi with dimesios (um N) where um is the umber of data poits ad N is the umber of eigevalues. For example, to display the eigevector for quatum state =, type psi(:,) ito the Commad Widow. A graphical display of the first 5 eigevectors ad correspodig probability desity distributios are displayed i a Figure Widow as show i Figure (8). Doig Physics with Matlab Quatum Mechaics Boud States 17

18 = 1 = 1 = = = 3 = 3 = 4 = 4 = 5 = 5 Fig. 8. The eergy eigevectors ad probability distributios for slopig potetial well. [se_solve.m]. To view a graphical display of a eigevector ad the probability desity graphs for a give state, you ca ru the m-script se_psi.m from the Commad Widow. The graphical output for a slopig potetial well for quatum state = 3 is show i Figure (9). Variatios with positio of the eigevector ( x ) ; probability desity ( x ) ; eergies U(x), K(x), E are show alog with a probability cloud where each poit displayed shows the positio of the particle after a measuremet is made o the system resultig i the collapse of the wavefuctio. Doig Physics with Matlab Quatum Mechaics Boud States 18

19 prob desity psi? U (ev) wave fuctio psi eergies U (ev) (ev) Potetial Well: SLOPING = 3 E = E (black) U(red) K(gree) positio x (m) positio x (m) positio x (m) Fig. 9. The eigevector, probability desity ad eergy graphs for the quatum state = 3 of a particle trapped i a slopig potetial well. [se_psi.m]. We will agai cosider the slopig well. The slopig potetial well represets a exteral electric field actig o the electro. Classically, the electro is subject to a very large force to the right at the left edge of the well ad a force to the left alog the slopig sectio with the magitude of the force equal to the gradiet of the slope ( F du / dx ). We could also view the potetial as a quatum mechaical versio of a particle slidig alog a frictioless iclied plae with a high wall at the bottom. The classical motio of such a cofied particle is that it would oscillate up ad dow the iclie, havig its largest kietic eergy at the bottom ad decreasig kietic eergy the further up the iclie. The particle would be movig most rapidly at the bottom of the iclie ad most slowly at the top. Therefore, it would sped less time at the bottom tha at the top ad this implies that it is more likely to fid the particle at the top tha at the bottom. Also the particle could be at rest at the bottom. Figure (1) shows the quatum mechaical solutio for the slopig well potetial for the state = 1 (groud state). The results are very differet from the classical predictios. Doig Physics with Matlab Quatum Mechaics Boud States 19

20 prob desity psi? U (ev) wave fuctio psi eergies U (ev) (ev) The miimum eergy of the particle is greater tha zero. There is a o-zero probability of the electro beig i a classically forbidde regio (E K < ), i.e., the electro ca peetrate the walls of the potetial well ad the most likely locatio to fid the electro is towards the bottom ad ot at the top. Potetial Well: SLOPING = 1 E = E (black) U(red) K(gree) positio x (m) positio x (m) positio x (m) Fig. 1. The eigevector, probability desity ad eergy graphs for the quatum state = 1 (groud state) of a particle trapped i a slopig potetial well. [se_psi.m]. There are 5 boud states for the potetial well show i Figures (1, 11). As the quatum umber icreases by 1, a additioal hump of width i the order of half a wavelegth is added to the shape of the wavefuctio. For the state = 1 there is oly 1 hump ad the state = 5, there are 5 humps. The characteristics of the highest boud state ( = 5) is show i Figure (11). This highly excited state has features more i-lie with the classical picture. The particle is more likely to be foud towards the top of the well where its kietic eergy is less. Sice the particle moves at high speed at the bottom ad less at the top, the wavelegth of the wavefuctio must be smallest ear the bottom ad Doig Physics with Matlab Quatum Mechaics Boud States

21 prob desity psi? U (ev) wave fuctio psi eergies U (ev) (ev) icreases towards the top. The groud state defies our classical expectatios but the highly excited states are ofte i broad agreemet with the classical picture. This is ot so surprisig sice classical physics objects have very high quatum umbers. This idea is called the correspodece priciple. Measuremets of the half-wavelegths ca be made usig the giput commad to measure the positios of the peaks ad the the successive distaces betwee the peaks of the probability desity curve, for example, z = giput z(5) z(4) The measured half-wavelegths (m) from Figure (11) as the kietic eergy decreases from left to right are cofirmig that as the kietic eergy decreases the wavelegth must decrease sice p h E p h K m m Potetial Well: SLOPING = 5 E = E (black) U(red) K(gree) positio x (m) positio x (m) positio x (m) Fig. 11. The eigevector, probability desity ad eergy graphs for the excited state, = 5 of a particle trapped i a slopig potetial well. [se_psi.m]. Doig Physics with Matlab Quatum Mechaics Boud States 1

22 EXPECTATION VALUES The quatum behavior of boud particles is becomig more importat i electroic devices made usig semicoductor materials such as silico ad gallium. The heart of such devices is a tiy structure called a quatum dot that cosists of a speck of oe semicoductor material (width ~ 1 m, ~1 s atoms) completely eclosed by a much larger semicoductor material. Some of the electros i the tiy speck become detached from their paret atoms ad behave as particles trapped i potetial wells of 1 (quatum wafer), (quatum wire), or 3 (quatum dot) dimesios. These particles are ofte referred to as particles i a box. The boud electros have a discrete eergy spectrum just like electros boud i atoms. Quatum dots are ofte referred to as artificial atoms. The light emittig properties of quatum dots are used i solid state lasers such as DVD players, lightig systems, solar cells, fluorescet markers i biomedical applicatios. Our focus of fidig the physically acceptable solutios of the Schrodiger equatio for boud particles is importat to uderstadig real systems of icreasig techological importace. The wavefuctio that describes a quatum state provides the most complete descriptio that is possible of that state it cotais all the iformatio we ca possibly kow about this quatum state. But the wavefuctio is ot somethig that ca be measured ad therefore has o physical meaig. The iformatio we ca get from the wavefuctio is related oly to the probabilities of the possible outcomes of measuremets made o idetical systems. Hece, there is a itrisic ad uavoidable idetermiism i the quatum world. Whe a measuremet is made o a quatum system, the wavefuctio collapses, the state of the system after the measuremet is ot the same as the state before the measuremet. The Schrodiger equatio does ot describe this collapse of the wavefuctio. So, for experimets desiged to test the predictios of quatum mechaics, you ca t do a sequece of measuremets o a sigle system, rather, you start with a large umber of idetical systems all i the same state ad the perform the same measuremet of each system. Although each system is iitially idetical, the outcomes of Doig Physics with Matlab Quatum Mechaics Boud States

23 the measuremets typically give a spread of results, for example, the measuremet of the positio of the electro i passig through a equivalet of a double slit. We will cosider systems of a sigle electros cofied by a oe dimesioal potetial wells. We already kow that it is a simple matter to use the wavefuctio to fid the probability distributio for the positio of the electro. However, we ca exted this method to fid the probability distributios of other measureable quatities such as positio, mometum ad eergy. This iformatio is cotaied i the shape of the wavefuctio ad we will show how this iformatio ca be extracted. For the electro trapped i the slopig potetial wells (Figures 7, 9, 1, 11) the solutio of the Schrodiger equatio gave a set of eergy eigevalues ad their correspodig eigevectors (statioary states). The eergy eigevalues are the oly allowed eergies of the system. No matter what state the system is i, if we measure its eergy, we always get oe of the eergy eigevalues. These statioary states are special, because they are states of defiite eergy. So i this case, there is zero ucertaity i the measuremets. If the system is described by a statioary state wavefuctio ( xt, ) ad we measure its eergy, we will certaity get the result E for the eergy eigevalue. The eigevectors are said to be orthoormal: (1) the eigevector is ormalized so that the probability is 1 of fidig the electro, ad () ay two eergy eigevectors with differet eergy eigevalues are orthogoal to each other. Mathematically this is stated as: (8) * 1 j ( x) i( x) dx ji Kroecker delta ji j i j i Oe meaig of the wavefuctio is that its magitude measures the probability amplitude of the particle beig foud at some poit. If the wavefuctio correspods to a eigefuctio ( xt, ), the the probability distributio prob (x) is idepedet of time ()ad is give by Doig Physics with Matlab Quatum Mechaics Boud States 3

24 iet iet * * ( ) ( ) ( ) ( ) ( ) prob x x e x e x x ad the probability P of fidig the particle betwee x 1 ad x is x P * ( x) ( x) dx x 1 We ca also calculate probability distributios for other observable quatities i a similar maer to above. The probability distributio is characterized by two measures its expectatio value (estimate of the mea value of the distributio) ad its ucertaity (spread of values aroud the mea). Suppose that a observable quatity A has a discrete set of possible values a 1, a,. The, we ca prepare N idetical systems, all i the same state ad measure the value of A i each system. I geeral, we will get a spread of results. The best estimate of the quatity A is its mea value <A> ad the stadard deviatio Aca be used as a measure of the spread or ucertaity i the measuremets. The expectatio value of a observable A is the quatum-mechaical predictio for the mea value. I a statioary state described by the wavefuctio ( x ), the expectatio value of a observable A is give by (9) A ( x) A( x) dx * ˆ where  is the quatum-mechaical operator correspodig to A. The operator is sadwiched betwee the two wavefuctio * ( x) ad ( x ), ad for this reaso we call the right had side of Equatio (9) a sadwich itegral. The first part of the itegrad is called the bra ( * ( x) ) ad the secod part the ket ( A ˆ ( x) ). Some of the dyamical properties of a system that ca be calculated are show i Table 1. Doig Physics with Matlab Quatum Mechaics Boud States 4

25 Total probability Aˆ 1 * 1 ( x) ( x) dx Mometum * ˆ d A i dx * d p ( x) i ( x) dx dx Potetial Eergy Aˆ U( x) E * P ( x) U( x) ( x) dx Positio  x x * ( x) x( x) dx Kietic Eergy ˆ d A m dx * d EK ( x) ( ) x dx m dx Total eergy ˆ d A H U( x) m dx where H is the Hamiltoia E ( x) H ( x) dx * * d E ( x) U ( x) ( ) x dx m dx Table 1. Operators ad expectatio value itegrals for a rage of dyamical quatities. * Note: For the mometum, the itegrad is imagiary, hece < p> =. The stadard deviatio A is used to describe the spread of the measuremets about the mea value. It is measured by summig the square deviatios from the mea ad the takig the square root. For a system i a give state ( x ), the ucertaity of the observable A is defied by 1/ 1/ (1) A A A A A The fudametals of quatum mechaics were worked out i a remarkably short period of time from 195 to 197. Oe of the fudametal cosequeces of quatum theory is the Heiseberg ucertaity priciple. Oe form of the ucertaity priciple is: I ay system, the ucertaities of positio iequality x ad mometum compoet px obey the (11) xp x Doig Physics with Matlab Quatum Mechaics Boud States 5

26 Hece, it is impossible to fid a state i which a particle has both defiite values for positio ad mometum. The classical picture of a particle followig a well-defied trajectory is o loger valid. A system described by a wavefuctio implies a spread of positio ad mometum values ad that a arrow spread i oe of these quatities is offset by a wide spread i the other. The ucertaity priciple is sometimes iterpreted i terms of the disturbace produced by a measuremet, but, this is ot correct. The ucertaity priciple tells us about the idetermiacy that is iheret i a give state before the measuremet is made. Whe a system s wavefuctio correspods to oe of its eigefuctios, the ucertaity i the total eergy E is zero, E. This is show by the followig calculatio, i which the operator for the total eergy E is the Hamiltoia Ĥ : Hˆ E E Hˆ dx E dx E E E * * E Hˆ Hˆ dx E Hˆ dx E * * 1/ 1/ E E E E E I our models, the rage of x values is restricted to the rage, x Mi to x Max ad the wavefuctio is forced to zero at these extreme values of x. So a expectatio value is calculated as x Max (1) A ( bra) ( ket) dx xmi The itegral is evaluated umerically usig Simpso s 1/3 rule by usig the m-script simpso1d.m. The m-script se_measuremets.m ca be used to evaluate the itegrals i Table (1) ad to evaluate the ucertaities. Sample codes from the m-script se_measuremets.m to evaluate the expectatio value for total eergy ad the ucertaity i positio are: Doig Physics with Matlab Quatum Mechaics Boud States 6

27 % total eergy eed to covert uits i calculatios psi1 = gradiet(psi,x); ket = - (hbar^/(*me*e))*gradiet(psi1,x)(lse^*ese) + (U.*Psi')'; braket = bra.* ket; Eavg = simpso1d(braket,xmi,xmax); % positio x & ucertaity i positio ket = x.* Psi; braket = bra.* ket; xavg = simpso1d(braket,xmi,xmax); ket = (x.^).* Psi; braket = bra.* ket; xavg = simpso1d(braket,xmi,xmax); deltax = sqrt(xavg - xavg^) * Lse; % legth uits m to m The m-script se_measuremets.m outputs umerical values of the itegrals to the Commad Widow ad graphical represetatios i a Figure Widow (Figure (1)). For example, the slopig well potetial: Quatum umber, = 3 Eergy, E = ev Total Probability = 1 <x> = m <x^> =.7857 m^ <ip> = e-3 N.s <ip^> = e-47 N^.s^ <U> = ev <K> = ev <E> = ev <K> + <U> = ev deltax =.663e-11 m deltap = e-4 N.s (dx dp)/hbar = Doig Physics with Matlab Quatum Mechaics Boud States 7

28 From the above result, we ca coclude the followig: 1 E = < E > igorig the slight differece i umerical values due to umerical iaccuracies i our modelig. < E > = < K > + < U > 3 The average positio of the electro is slightly to the left of the origi. xp 4 The ucertaity priciple is satisfied The expectatio for mometum is complicated by the factor i ad so we have to calculate < i p > which is the a real quatity ad from the results above, we ca coclude that <i p> =. No measurable physical quatity ca be imagiary, hece the itegral must be zero. It is remarkable that a boud particle has defiite set of discrete values for its total eergy ad this follows simply ad directly from applyig boudary coditios to a wavefuctio that satisfies the Schrodiger equatio. At first, it seems surprisig that the total eergy has a set of uique values ( E ) but the mometum does ot, sice the ucertaity priciple must be satisfied. But a classical vibratig strig i a ormal mode has a uique frequecy of vibratio while ot possessig a uique wavelegth as the trasverse stadig wave is siusoidal alog the strig but is zero at ad outside the fixed ed poits. However, the shape of the strig iside ad outside the fixed eds ca be thought of a superpositio of waves of differet wavelegths with amplitudes ad phases chose so that the disturbace cacels everywhere except i the regio betwee the ed poits. Therefore, the wavefuctio ca be thought of a superpositio of differet wavelegths ad through the debroglie relatioship p h/,the cofied particle has a rage of mometum. Doig Physics with Matlab Quatum Mechaics Boud States 8

29 wave fuctio prob. desity, (m -1 ) bra ket PE, U (ev) Quatum State, = wave fuctio, positio x (m) 5 <x> = m ket, positio x (m) Fig. 1. The graphical output from the m-script se_measuremets.m for the quatum state = 3 of a particle trapped i a slopig potetial well. The lower graphs shows the bra (wavefuctio), the ket ad the area uder the braket curve represetig the itegral for the evaluatio of the expectatio value for positio <x> The blue bar idicates the expectatio value for positio. You ca modify the m-script se_measuremets.m to produce graphical outputs for operators other tha the positio operator ˆx. Doig Physics with Matlab Quatum Mechaics Boud States 9

30 ACCUARY OF THE NUMERICAL MODELLING Numerical results may ot always be accurate. Where possible you eed to check that the results are OK. Firstly, you should always vary the umber of poits used i the modelig. I solvig the Schrodiger equatio by the Matrix Method, the variable um which is set i the m-script se_wells.m determies the size of the vectors ad arrays. Geerally, if this umber is too low the results are ot accurate. You should start with a small umber for um ad icrease it util there is a covergece i the results. If um is too large, the the executio time i ruig the m-scripts could be excessive ad/or too much memory is used. Secodly, if possible you should compare the predictios of a umerical method with the kow results predicted by aalytical methods. The eergy eigevalues for a ifiite square well ca be easily foud aalytically, so we ca compare those results with our umerical model. Cosider a system with a electro of mass m e cofied to a oe dimesioal regio of legth L. The potetial well called a ifiite square well has ifiitely high walls which trap the particle, but the force o the electro is zero betwee the walls. The potetial eergy fuctio ad boudary coditios o the wavefuctios are U( x) L / x L / U( x) x L / ad x L / ( L/ ) ( L/ ) The classical predictios for such a system, is that the total eergy of the trapped particle ca have ay positive value ad sice the potetial eergy iside the well is zero, the kietic eergy of the particle equals the total eergy. The particle ca have zero total eergy (zero kietic eergy) ad if its kietic eergy is greater tha zero the particle will bouce back ad forth betwee the impeetrable walls with costat speed. Icreasig the eergy simply icreases the speed but o matter how great the speed, the particle ca t be outside the walls, the regio outside the walls is said to be classically forbidde. The predictios of quatum mechaics are very differet. I solvig of the Schrodiger equatio it is foud that the shape of the wavefuctio iside the walls is siusoidal ad Doig Physics with Matlab Quatum Mechaics Boud States 3

31 the wavefuctio must be zero at the walls sice the potetial eergy is ifiite at each wall. Therefore, the oly acceptable physical solutios are those wavefuctios where iteger multiple of half-wavelegths of a sie curve fit ito the distace L betwee the walls: L L 1,,3, The total eergy E is related to the mometum p ad the mometum related to a wavelegth (de Broglie relatioship) hece, we ca determie a expressio for the total eergy as a fuctio of (priciple quatum umber) ad the legth L L p h E p m (13) h E 8mL 1,,3, This is a mometous result ad very differet from the predictios of classical physical. The priciple quatum umber is restricted to iteger values, therefore, the boud particle has oly a discrete set of values for its total eergy ad these are the oly possible values for the total eergy of the system. The lowest value is = 1, ad the lowest allowed eergy is E h m L 1 / 8. No particle i a ifiite square well ca have a eergy lower tha this, there is o state of zero total eergy. This eergy is called the groud state eergy. If L, the the spacig betwee eergy E which implies that the particle is behavig more like a free classical particle which ca have ay positive eergy. We ca compare the predictios from the aalytical model ad the Matrix Method for a electro cofied to a ifiite square well potetial of width.1 m. To use the Matrix Method, modify m-script se_wells.m to produce a ifiite square well potetial the ru it: Ifiite square well parameters set i se_wells.m um = 81 Doig Physics with Matlab Quatum Mechaics Boud States 31

32 Optio 1 (case 1) square well xmi = -.5 xmax = +.5 U1 = The ru the m-scipt se_solve.m. The scree output states that o eigevalues are foud. This is because the code is expectig egative values for the total eergy. However, we ca extract the required values for the total eergy maually usig the commad e_values(mm,mm) i the Commad Widow. Do this ad record the first 1 eigevalues for mm = 1,,, 1. These values are stored i the m-script se_ifwells.m. Ru the m- script se_ifwells.m. Note, whe this file is ru, all variables are cleared from the workspace. The file se_ifwells.m calculates the first 1 eergy levels usig Equatio (13), outputs the results to the Commad Widow ad plots the umerical ad aalytical values i a Figure Widow (Figure (13)): E (ev) Emm (ev) Doig Physics with Matlab Quatum Mechaics Boud States 3

33 Total Eergy E (ev) 4 35 aalytical Matrix Method Fig. 13. The eergy eigevalues for a ifiite square well of width.1 m calculated from Equatio (13) ad the Matrix Method usig the m- scripts se_wells.m ad se_solve.m. [se_ifwell.m] The discrepacy betwee the two sets of results is better tha.%. Therefore, we ca have some cofidece i the results from the Matrix Method for other potetial well fuctios Quatum No. The wavefuctios for the ifiite square well ca be plotted after ruig se_solve.m by typig the followig commads i the Commad Widow, for example, to display the eigevector for the state = 8 plot(e_fuct(:,8),'liewidth',3) axis off Figure (14) show the eigevectors for the first 1 quatum states of a ifiite square well. The eigefuctio is zero outside the well ad is of a siusoidal shape iside the well. For the th state, half-wavelegths of a siusoidal curve fit ito the width of the well. The shape of the wavefuctio is the same as the shape of a vibratig guitar strig for ormal modes of vibratio. Doig Physics with Matlab Quatum Mechaics Boud States 33

34 Fig. 14. The eergy eigevectors for a ifiite square well for the first 1 quatum states. [se_wells.m se_solve.m] Whe a potetial well is of fiite depth, the wavefuctio exteds ito the classically forbidde regio ad so there is some probability of fidig the particle outside the well which i classical terms i impossible. For a particular state of a ifiite square well, the wavelegth is costat (idepedet of x). This is ot the case whe the potetial iside the well chages with x, for example, the wavelegth icreases with decreasig potetial i the slopig well (Figure (9,1)) ad so the shape of the wavefuctio is ot simply siusoidal. Doig Physics with Matlab Quatum Mechaics Boud States 34

35 STATIONARY STATES AND SUPERPOISITION PRINCIPLE I wave mechaics, the state of a system is described by a wavefuctio that satisfies the Schrodiger equatio ad the boudary coditios imposed o the system. For a oe dimesioal system, the Schrodiger equatio for a potetial eergy fuctio U(x) is (14) ( x, t) ( x, t) i U( x) ( x, t) t m x We will cosider oly those solutios that are described by a state of defiite eergy by the method of separatio of variables where the wavefuctio is expressed as (15) ( x, t) ( x) T( t) The substitutio of this wavefuctio ito the Schrodiger equatio, gives two ordiary differetial equatios ad for quatum state : (16) m d ( x) U( x) ( x) E ( x) time idepedet Schrodiger equatio dx (17) i T () t ET() t t = 1,, 3, The solutio of the ordiary differetial equatio (17) is i E t (18) T ( t) exp For statioary states which have a defiite total eergy, the wavefuctio is i E t (19) ( x, t) ( x)exp If the eergy of the system is measured, the value E will certaily be obtaied. This is a special type of solutio of the Schrodiger equatio. Applyig Bor s rule, which states that the probability of fidig the particles i a small iterval dx cetered o x for the state is * (, ) (, ) (, ) ( ) probability x t dx x t x t dx x dx Doig Physics with Matlab Quatum Mechaics Boud States 35

36 Hece, the probability is idepedet of time t ad this is why it is called a statioary state. The real ad imagiary parts of the wavefuctio chage with time but ot the modulus. All parts oscillate i phase (like a statioary ormal mode of a vibratig guitar strig). The cocept of statioary states is ot oe that is familiar to us i the classical world as othig is movig at all, the probability of detectig a electro i ay give regio ever chages. All the expectatio values listed i Table (1) are also idepedet of time. Each statioary state wavefuctio as described by Equatio (19) describes a complex stadig wave. A stadig wave is oe that oscillates without propagatio through space, with all poits of the stadig wave vibratig i phase. The frequecy ad period of the vibratio are E E h f T h E The stadig wave has odes fixed i space. As icreases, the total eergy E ad umber of odes icreases, while the period of vibratio T decreases. There is zero probability of locatig the particle boud i the well at each ode. We ca o loger thik of the particle movig from place to place i the well. I quatum physics, a particle has o positio util it is measured, so it makes o sese to thik of the particle movig about. There is o problem associated with the particle havig to travel through a odal poit, sice the particle has o positio ad velocity. Whe a measuremet is made, it oly tells us the positio ad velocity at the istat after the measuremet. Before the measuremet, the state is simply described by the wavefuctio which gives us the most complete descriptio that is possible of the state of the system, but it does ot give us iformatio about the positio ad velocity of the particle. Doig Physics with Matlab Quatum Mechaics Boud States 36

37 Figure (15) show the time evolutio for a umber of time steps of the real ad imagiary parts of the wavefuctio ad the probability desity for the statioary state = 3 of the trucated harmoic oscillator. The phase of the real ad imagiary parts chage with time but the probability desity is idepedet of time. A aimatio of the chages with time of the wavefuctio ca be observed by ruig the m-script se_statioary.m. real imagiary prob. desity Fig. 15. The time evolutio of the wavefuctio s real ad imagiary parts for the statioary state = 3 for a trucated parabolic well. The probability desity does ot chage with time. [se_wells.m se_solve.m se_statioary.m] A aimated gif [se_statioary.m] of the time evolutio of the statioary state show i Figure (15) ca be viewed at Doig Physics with Matlab Quatum Mechaics Boud States 37

38 Wavefuctios have a very importat property i that they obey the superpositio priciple: If a set of wavefuctios are solutios of the Schrodiger equatio, the ay liear combiatio of the wavefuctio is also a solutio If the wavefuctios a where a c are arbitrary complex umbers a are ormalized wavefuctio ad is also ormalized the 1 The probability of fidig a particle i a give legth elemet dx is * ( x, t) ( x, t) dx Whe the particle is i a eigestates with a defiite eergy E the * ( x, t) ( x, t) dx *( x) ( x) dx costat ad so the probability distributio does ot chage with time as show i Figure (15). Also, the expectatio value for the positio of the particle x is ot a fuctio of time. However, whe the particle is i a compoud state (more tha oe a is ozero), the probability desity ( xt, ), the expectatio value for positio x ad probability of locatig the particle at each positio all vary with time. The expectatio value for the total eergy is * E ( x, t) i ( x, t) dx t Usig the fact that the eigefuctios are orthoormal * 1 j ( x) i( x) dx ji Kroecker delta ji j i j i ad doig some algebra it is ot difficult to show that the expectatio value for the total eergy is Doig Physics with Matlab Quatum Mechaics Boud States 38

39 () E a E where E c are the eigevalues Ay measuremet of the total eergy of the system will always give oe of the eigevalues E ad ot E. If may idetical system with the compoud wavefuctio were prepared ad the total eergy measured for each system tha the average value those measuremets would be very close to the expectatio value E. Ru se_wells.m ad se_solve.m for the default trucated parabolic well: % parabolic ***************************************************** case 5 xmi = -.; % default = -. m xmax = +.; % default = +. m x1 =.; % width default =. m; U1 = -4; % well depth default = -4 ev; The eergy eigevalues for this potetial well from se_solve.m are: No. boud states foud = 5 Quatum State / Eigevalues E (ev) The m-script se_super.m adds ay two of the eigefuctios to form a compoud state. To ivestigate the additio of the eigestates 1 ad, i the code for se_super.m eter q(1) = 1; q() = ; % states to be summed ac(1) =.5; ac() = sqrt(1-ac(1)^); % coefficiets to form the compoud state 1 (1) (, ).5 1( )exp i E t i E t x t x.866 ( x)exp The expectatio value (ev) for this compoud state is displayed i the Commad Widow Eavg_s = see equatio () Figure (16) shows the time evolutio of the compoud state at times t =, T/4, T/, 3T/4, T where T is the period of the oscillatio of the wavefuctio. The red curves are the real parts ad the blue curves the imagiary part of the compoud wavefuctio. The black Doig Physics with Matlab Quatum Mechaics Boud States 39

40 curves shows the variatio with time of the probability desity. For the compoud state where 1 1 1the probability curve oscillates back ad forth i the potetial well with a period T where T h E E 1 real real t = imagiary imagiary t = T/4 prob. desity prob. desity real real real imagiary t = T/ t = imagiary 3T/4 t = T imagiary prob. desity prob. desity prob. desity Fig. 16. The time evolutio of the wavefuctio s real ad imagiary parts for a compoud state give by Equatio (1) for a trucated parabolic well. The probability desity chages with time as the charge distributio oscillates back ad forward i the potetial well. Red curves real part, blue curves imagiary part ad black curves probability desity. [se_wells.m se_solve.m se_super.m] A aimated gif [se_super.m] of the time evolutio of the statioary state show i Figure (16) ca be viewed at Therefore, the expectatio of the positio of the electro x oscillates with the period T as show i Figure (17). Doig Physics with Matlab Quatum Mechaics Boud States 4