Time Independent Schrodinger Wave Equation

Time independent schmondiger's wave equation. Ercwin Schreonligere in 1940 made use of De Brewine in uncertainity Primeiple Heisenberg's and developed a diftercential equation equation called Schröndigeres Wave equation. The wave nature of material parcticle is expressed by schrondigeni wave eq. Classical Wave eg describing the wave motion of a parctiete along the x-arts can be applied According to classical wave equation. This equation is known as Schrondigers wave eg?. Significance of schrean digeres wave eg?= Eigen Values and Eigen function= schrconaligerii wave equation is and orders Bitbeschreongligen Sequation... So it has many son for 4. many of them arce imaginary & have no significance. The significant value of of will satishby some condition. → The wave feen (4) must be finite & Contineous. → The soi must be valued single val e al a given point there can be nevere be morce than one. value of .. dy → & de must be contineous brun of di dy xy&z respectively. > The probability of bindinge overe all the space 6 to must be equal to unity. Jeg day da, dy.dz = 1 when schrrondigers wave eg? is solved bor energy calculation & Varciable are derived Limitation of schrondigen's Wave This equation. Can be solved exactly hydrogen or hydrogen like atom. for Significance of y It is a wave fum which is coi to the I schriondigens wave ed. It represents amplitude of wave. Value igers wave A The significant of values af ligen function. have diftercer is of 4 These trenctions give significant value of of e. total energy of e. These value arre called ligen values. Significance af → It represents intensity af light. For wave, it represents intensity of & at any Point je It is Represented the probability of binding af dibbercent region. > It leads the idea of achital. Radial Angulare Wave funefion. for the atomwo H. According to the schreendigere eg for the sm (+2)=0- the he above equation can be solved by prubefituting place of Cartesian Co-ordinates (x,yaz). Let take a peint. A having Co-ordinates 1,442 Z in space. Xet OA & the o&& arce the angles. 2=ON = CCOSD & 6 OL= AN = Csino A(z) Y = OM-LK=OL sind=Csinosind 7. OK OL Cosd = resino cost Using this realatiin the schreondigeres equation Can be written in tenems Polar Co-ordinates. Projection of radial time Co-Relation bef? (π,θια) (αψ,2). Where = radial distance O= The angle ben radial distance & Q = The angle the projection of reaclical distance that makes with X-axis. → In order to evalkedte & (7,0,0), the W-branetion Can be written as Ψ€π, θ, Φ) = R(rc) Θ(0) $14) ①R(C) is a brunetion depends upon the distance re trom the nucleus. It depends upon the quantum numbers na 1 but independent of polar Co-ordinater 04. (2) (0) 15 's a tunetion of quantum numbere L& m Which clepends uporn but independent of polare co-ordinates rcq8. ③ $(9) is akenetion of $, which depends onmy on quantiem numbers m biet fordependent af Polar Co-ordmate re4o. Thus 4(π,θ,中)=ROL() Dim (0) Om (9) The separation of es varciable i-e eq Can split the equation into 3 intepenelent eq. Each contains only one varciable.ie dre RCC) + 820 (E+) Ruoko 2 This equation is a benction of re only d When solved giver of. R(r). sino do (sino do(o)) + P()+(+)=0 Sino de STO 20 This. eg is a bunction of a only and when solved gives (o). ③ d²(4) + m²(4)=0 This eg is a bronction of & only & when Solved gives ($(0). 2021 R(n) * The readial to Wave beenection gives and size of orbital. fires the energy the angular wave brunetion is the product of Θ(0) & (P) gives the shape of arcbital. Radial wave brunetim When we solve equcation (3) we get 2 quanteem numbers ng 1 arde obtained Thus readial wave been? Ro, (c) can be written R1.0 or R15 = 2(20) 2. 90 e-9/2 R2,0 cr Ras = 2/2 (20) 3/2 (2-7) 2 ao R2,100 Rap = 22.6/2 (20) 3h Jea₂ Ge Ao Bobre's Radices - 42-atomic no. The Radial dependence of an orbital Can be expressed Captained by plotting a graph bet Dave bus Rn, & distance trom nucleus for belonging e to different Orbitals. And this graph helps to know the probability distance around a point at a distance in from the nucleus. [ Graph bef Ro, & in for the belangiog to differcent orbitals Circaphs indicates that Rn, I can not be related With preobability density, then the maximum proba- -offideg of around any point of a dritance ritrom the nucleus because of the 2 reasons. 1. Rn, is maximum at n=0. 1 of Ron, represents the probability density then the maximum probability of bindinge is mazicem.. but this is not thue. (The probability of binding & at nucleus) 2. In the probability Curves bor as & as orbital, Roje has both tre ve values. But the Probability can nevere have -ve values. Due to the above objection (2) a poot is 011-354 DAYS Radial Probability distribution Curve - In order to understand the concept of rcadial Probability. Let us the space arcound the nucleus is divided into thicked in to a large no. of concentrebe shelle of Radial sheu Ket dir-Thickness of the Space in bef. 2 Concentteic shell.. & dv- Volume af chell. in ben 1&2 concentric Celle. Toln Thickners of space (Concentrove shells) The Radial probability is the total probability teruity of finding the e in a rantical shell ber spheres off readil it & (n+dre) Mathematically. Radiall probability Vol. af spherical shell X probability density je 47re² dr x² it-distance of electronic charge from the nucleus. 2021 Thus, the readial probability districtbution of electron can be obtained by drawing a graph ber the bum 477² R²n, & such graph are called Radial distribution. Cureves ont radial probability distribution Cereve. The radial probability bren? 471m² R² is the produst of 2 bactor The bactore 4inc² increases as reincreeases. & The bactor R² decreases as in increases. 2s Nocle a At r=0 although the bactere R² is maximun 2 get the bactore me² is zerco Cerco Thue the readial probability is zero -at rezo ie nucleus. When it is vercy lange although the beeter 41² is y large yet the probability vercy Thus, the raditel probability. is exore edingly small when it is verry large. The Cureve Can be drawn by remembering. (1) The number of radial nodes in the. Curves is equal to n-1-1. The Cereve staret broom Orrigin. The con erea under the envelope of a Particular Cereve goes on increasing from left to right. such that it becomes maximum under the last envelope. The peak of the Last Grenve gives the distance trom the nucleus wheree the probability of binding eis maximum The distance is called the reactives of. maximum probability. 2021 Angulare Wave bunction When equation 5&6 are solveol When equation 546 ame solved we got centerin Values and that valves depends upon the direction. given by angle && $. know helps For this reason it helps shape of orbital. It also depends upon the value of quantum numbers 1& m In Case of Op, Hatoon the wave been? Or,m. & $ Im are given below.