Lecture 1. time, say t=0, to find the wavefunction at any subsequent time t. This can be carried out by

Transcription

1 Lectue The Schödinge equation In quantum mechanics, the fundamenta quantity that descibes both the patice-ike and waveike chaacteistics of patices is wavefunction, Ψ(. The pobabiity of finding a patice in the infinitesima voume, dv, about in space is P( dv = Ψ( dv (. whee Ψ ( x, = Ψ( x, Ψ( x, * is measuabe and it is just the pobabiity pe unit voume o pobabiity density. Howeve, (, itsef is not measuabe. The sum of the pobabiity ove the entie space must be o entie space Ψ( dv =. (. This equation is caed nomaization condition. The fundamenta pobem of quantum mechanics is: Given the wavefunction at an initia time, say t=, to find the wavefunction at any subsequent time t. This can be caied out by soving the Schödinge equation, h m Ψ( V ( Ψ + ( = ih, t (.3 whee 34 h =.5 J s is the Panck constant, m is the mass of the patice and V( is the potentia fied in which the patice is moving. The Schödinge equation fo coection of patices ike many eectons and nucei in a moecue is vey simia. In this case, the wavefunction, Ψ = Ψ,,,..., is a function of the ( 3 t coodinates of a the eectons and nucei in the moecue and time, t. If V in the equation is time independent, we can attempt to sepaate the time dependent pat of Ψ = Ψ( fom the space dependent pat by witing

2 Ψ ( = ( φ(. (.4 Substituting.4 into.3, we obtain two equations, fo ( and φ(, espectivey. The equation fo φ( is d t ih φ ( = Eφ( t (.5 dt xe i E t and its soution is Ψ( xt, = ( h. The equation fo time-dependent wavefunction, (, is h ( + V( ( = E(. (.6 m v This is the time independent Schödinge equation. Since Ψ (, = (, the pobabiity is time independent. Fo this eason soutions in sepaabe fom ae caed STATIONARY states - the pobabiity is static and enegy is conseved. If denoting H h = + V ( (.7 m caed Hamitonian opeato o simpy Hamitonian of the patice, then the time-independent Schödinge equation becomes ˆ H ( = E (. (.8 This is an eigen equation: an equation in which an opeato acting on a function poduces a mutipe of the function itsef. The set of functions fo which the eigen equation hods ae eigenfunctions, descibing the diffeent stationay states of the system, and the associated E fo the eigenfunction ae eigenvaues, coesponding to the enegies of the diffeent stationay states. We wish to detemine the wavefunction of a moecua system, but we wi stat with the simpe hydogen atom.

3 The Hydogen atom The hydogen atom pobem epesents one of the few pobems in quantum mechanics that can be soved exacty. It is the pototype system fo the many compex ions and atoms of the heavie eements. Indeed, ou study of hydogen atom utimatey wi enabe us to undestand the peiodic tabe of the eements. The hydogen atom pobem is aso vey usefu in addessing the pobem of doping in semiconductos as we as exciton pobem. Finay the concepts such as obitas and enegy eves wi be cucia fo undestanding behavio of moecues. The Hamitonian of H atom is Hˆ h = m e 4πε (.9 whee the fist tem is the kinetic enegy opeato and the second tem is due to eectostatic attaction between the eecton and the nuceus. m in the above expession is the educed mass of the eecton and nuceus, but it is vey cose to the eecton mass since the nucues (poton mass is about times the eecton mass. Since the potentia enegy tem depends ony on the distance between the eecton and the nucues, it is easie to sove the Schodinge eq. in spheica coodinates, which takes the fom of h ( m ˆ + m e (, = E (, 4πε (. whee ˆ = h { (sinθ + } (. sinθ θ θ sin θ ϕ is associated with the squae of the angua momentum. Note that the ˆ tem contains the angedependent of the wavefunction and satisfies

4 ˆ m m Y ( = ( + h Y (, (. m whee Y ( is spheica hamonics, =,,,., caed angua momentum quantum numbe, m=-, -+,, caed magnetic quantum numbe. Accodingy we can wite the compete soution of Eq.. as poducts of angua and adia eigenfunctions as nm m (, = R ( Y ( (.3 n whee Rn is specia functions, and n=,, 3,, is caed the pincipa quantum numbe, =,,,. n- and m=-, -+,,. The enegy eigenvaues of H-atom is given ae given by = e n a 3.6 = ev n E n 4πε (.4 whee a 4πεh = me = o.53a is the Boh adius. Shes and Subshes: A states with the same pincipa numbe n ae said to fom a SHELL. These shes ae identified by the ettes K, L, M,, which designate the states fo which n=,, 3, Likewise, states having the same vaue of both n and ae said to fom a SUBSHELL. The ettes, s, p, d, f, ae used to designate the states fo which =,,,

5 Hydogen atom gound state: The gound state of a one-eecton atom with atomic numbe Z, fo which n=, = and m=, has enegy, E = 3. 6eV and wavefunction, / a, = R( Y ( = e (.5 3/ a π ( Notice that the wavefunction does not depend on ange, it is spheicay symmetic. In fact any state with = is spheicay symmetic and caed s-state. Hydogen atom Lowest Excited states: The owest excited states ae fou degeneate states,,,, -. A of them have the same pincipa quantum numbe, n=, hence the same enegy 3.6 E = ev = 3.4 ev and the wavefunctions ae / a = R Y = ( e : s state. (.6 (, ( ( 3 / π (a a

6 / a, = R( Y ( = e cosθ : p 5 / state. (.7 π (a ( (, ( (, / a iφ = R Y θ ϕ = e sinθe : p 5 / state. (.8 8 π (a (,, ( (, / a iφ θ ϕ = R Y θ ϕ = e sinθe : p 5 / - state. (.9 8 π (a The wavefunctions given by 3. and 3. ae often combined to fom the so-caed p x and p y states, ~ sinθ cos x (. px ~ ( φ ~ + ~ sinθ sin y. (. py ~ ( φ ~ Hydogen atom Second Lowest Excited states: The second owest excited states ae 9 degeneate states, 3, 3, 3, 3-, 3, 3, 3, 3-, 3-. A of them have the same pincipa quantum numbe, n=3, hence the same enegy 3.6 = ev and the wavefunctions ae given in the Tabe. 3 E

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8 Fig..

9 Homewok.. What is the pobabiity of finding an eecton in a hydogen s obita inside the nuceus? The diamete of a poton is about - cm.. Find the maxima in P n ( fo the s, s and p obitas. Woud you expect p o s to be moe avaiabe fo covaent bond fomation? Whee P n is the adia pobabiity density, defined by the statement that P n (d is the pobabiity of finding the eecton between and +d. It can be witten as, P π π n ( = dφ nm sinθdθdφ = Rn (.