Chapter 3 The Schrödinger Equation and a Particle in a Box

Transcription

1 Chpter 3 The Schrödinger Eqution nd Prticle in Bo Bckground: We re finlly ble to introduce the Schrödinger eqution nd the first quntum mechnicl model prticle in bo. This eqution is the bsis of quntum mechnics nd s we will lern in chpter 4 is postulte rther thn derivble epression. The Schrödinger eqution is the eqution for the wvefunction of prticle - we begin with the clssicl D wve eqution u u with ut (, ) ( )cos( t) & ( ) is the sptil mplitude v t u u ( )cos( t) ( )cos( t) v t v t cos( t) ( ) ( ) cos( t) v t ( ) ( ) cos( t) v cos( t) d ( ) ( ) 0 with v & v v d v d v d 4 ( ) ( ) ( ) ( ) 0 d v d - If we use de Broglie s reltion h nd E = K.E. + P.E. p p h E V( ) p mev( ) m mev( ) - Using h nd substituting bck into our eqution: d 4 d ( ) ( ) ( ) m EV( ) ( ) d d h d mev( ) ( ) ( ) 0 d d mev( ) ( ) ( ) 0 d d ( ) EV( ) ( ) 0 md d ( ) E( ) V( ) ( ) 0 md d ( ) V( ) ( ) E( ) md The lst eqution is the time-independent Schrödinger eqution.

2 Clssicl-Mechnicl quntities re represented by liner opertors in quntum mechnics - In generl, liner opertor  A ht is defined s A[ ˆ cf( ) c ˆ ˆ f( )] ca f( ) ca f( ) where c nd c re constnts - Emples: liner : d d d d  =, cf( ) cf( ) c f( ) c f( ) d d d d non liner : cf cf c f cc f f c f etr term  =, ( ) ( ) ( ) ( ) ( ) ( ) The Schrödinger eqution cn be formulted s n eigenvlue problem - definition of n eigenvlue eqution: A ˆ where = constnt - Emples: -- liner momentum opertor pˆ i ik pˆ ( ) where ( ) e ik ik ik i e iike i ke ik ke k ( ) constnt -- direction opertor ˆ ik ˆ ( ) e constnt ( ) ( ) is not eigenfunction of opertor ˆ - Bck to Schrödinger d ˆ V( ) ( ) ( ) or ( ) ( ) E H E md where Hˆ is clled the Hmiltonin opertor E is the energy of our wvefunction ( ) pˆ d d d KE.. i i nd P. E. = V( ) m m d d m d Wve functions hve probbilistic interprettion - s we will lter point out wvefunctions re often thought of s vectors - the intensity of the wve is the squre of the mplitude or mgnitude of the Wvefunction, I ( ) ( ) ( ) - tody, ( ) ( d ) is the probbility of finding prticle locted btwn nd + d

3 The Energy of prticle in bo is quntized - this will become the model tht UV-vis spectroscopy is bsed on - D cse: prticle is confined to move long one coordinte, -- boundry conditions: ψ(0) = 0 nd ψ() = physiclly we re sying tht the potentil t the ends of our bo is 0 V( ) the prticle is thereby restricted to be locted btwn 0 & or 0 -- How does this effect the Schrödinger eqution? Given our conditions for the potentil, V() d ( ) V( ) ( ) E ( ) md becomes d ( ) E ( ) md d me Or, ( ) ( ) 0 d -- Now, let s look t ψ() --- from Chpter tht possible eigen solution to our Schrödinger eqution is d ( ) e ( ) e ( ) d --- plugging this bck into the eqution: d me me ( ) ( ) 0 ( ) ( ) 0 d me me me or ( ) 0 so i me ik ik -- if we let k then ( ) ce ce Or using Euler s equtions: ( ) Acos( k) Bsin( k) -- Now let s pply the boundry conditions, ψ(0) = 0 nd ψ() = 0 (0) 0 A 0 ( ) 0 B sin( k) 0 so sin( k) 0 So, k n where n,,3, Plugging k bck in: k n gives me me me 8 me k n n n n h h nh n En nd ( ) sin where,,3, n B n 8m -- Gret so wht does ll this men? --- first, energy is quntized given our integer n dependence

4 --- n is quntum number such tht n describes the stte of our system ---- for ny n we will hve n En nd ψn for tht stte ---- lso the smller the n the closer our system is to ground stte (g.s.) --- see figure 3. from below so --- We cn determine how much energy is required to promote n e- from stte to the net e.g. to go from n = to n = 4, 6h 4h h E E4 E 8m 8m 8m Since electronic trnsitions often occur in the visible, then let λm = 500 nm which mens = m 34 h Js 9 E4.890 J 8m kg m Wve functions must be normlized - remember we re doing probbilistic pproch here s such the probbility of finding prticle inside our D bo must be no greter thn unity - A normlized wvefunction, ψ(), will dhere to ( ) ( d ) - In our cse we re tlking bout prticle which is confined to the limits of our bo n So, to normlize our wvefunction, n( ) Bsin, the limits of integrtion Are 0 nd : n n( ) n( ) d B sin d B or B 0 the normlized D prticle in bo wvefunction is: ( ) sin n n n n

5 The verge momentum of prticle in bo is zero - Averge/Men -- We cn determine the verge vlue of ny opertor, system ˆ O ( ) Oˆ( ) d this is postulte from Ch-4 Ô, which is pplied to our -- the verge or men position of prticle in constrined to our bo is: n n ( ) ˆ ( ) d sin sin d for ll n This is the midpoint of the bo. - Vrince -- Like the men, the vrince of ny opertor pplied to the system is defined s -- Since we hve, we just need : n n sin sin d 3 0 n n 3 n 4 n 3 - Stndrd Devition, -- s we know the stndrd devition is the n -- for our model, n 3 - We know wht the verge position, let s see bout momentum n d n i n n p sin i sin d sin cos d 0 d 0 0 Which mens tht the prticle is moving in either directions eqully n d n n n p sin sin d 3 d 0 Revisiting Heisenberg nd his uncertinty n n - p p p 0 -- from this we see tht p / which mens the more we try to loclize (or mke the bo smller) the more devition we hve in p -- free prticle hs no bo limiting it, in this cse we will hve no uncertinty in the momentum or

6 n - n 3 -- from this we see tht which mens the smller the bo the smller our devition in position -- for free prticle the uncertinty in position is infinity - once gin, if we hve certinty in position we hve uncertinty in momentum nd vice-vers mthemticlly: p n since n, 3 n n n n 3 3 > we cn sy tht p Unto rel bo 3D time - boundry conditions: 0 (0, yz, ) ( yz,, ) for ll y & z 0 0 yb V( ) (, y, z) (,0, z) (, b, z) for ll & z 0 z c (, y,0) (, y, c) for ll & y elsewhere Once gin the prticle is restricted in ll three directions - the Schrödinger eqution: (, yz, ) E( yz,, ) m y z Lplcin or -- using seprtion of vribles gin we cn derive the solution for the wvefunction nd the energy description see the tet. Appliction to prticle in bo conjugted polymers - For emple C6H8: -- ech crbon tom hs 3 sp orbitls nd -- it turns out tht trnsitions for these conjugted systems re UV-Vis ctive

7 --- the verge C C bond length is.4å this trnsltes to 5.4 Å = 7 Å for the length of our bo --- the first ecited stte is from E3 E4, so Js 9 E J kg 70 m -- this trnsltes to wvelength of: 34 8 hc J s 30 m s m or 69nm E which is in the UV rnge -- let s tlk bout the sttes 9 J -- HOMO (highest occupied moleculr orbitl)/lumo (lowest unoccupied MO) --- for the ground stte the HOMO is ψ3 nd the LUMO is ψ4 --- for the first ecited stte HOMO is ψ4 nd the LUMO is ψ5 -- the gp between the nd bnd is clled the bnd gp --- the smller the bnd gp the more conducting the system is --- -bnd is lso referred to s the vlence bnd --- bnd is lso referred to s the conduction bnd Dirc Nottion br nd ket - this is shorthnd form of sptil integrtion which is commonly used by chemists & physists - d n n n n n n -- n is the br nd is the conjugte of n -- n is the ket n n - Kronecker Delt, nn 0 n n -- this works for normlized wvefunctions

8 -- the n n implies tht ll the sttes re orthogonl to ech other -- these two stipultions combined led to set of wvefunctions which re orthonorml, normlized nd orthogonl wvefunctions