Question

08 The wavefunction can be expanded into the complete set of basis of eigenstates of the Hamiltonian: (7.4) 1°) = cilm) What is the general expression of the off diagonal (i #j) and diagonal (i = j) matrix elements for the Hamiltonian in the basis set of its eigenstates? Hij = (di|h\o ;) (7.5) (hint: Apply the Hamiltonian Ĥ on this arbitrary wavefuction |Y) and its bra version (¥| and use the general expression for time-independent Schrodinger equation (without specifying Ĥ)).

Answer 1

- Schrodinger wave eque . Page no-01 The development of Schrodinger wave equ un Om was a great break fenough for OM, it gives relation de p. . positional Coordinates and the pic of the particle. . Schrodinger in 1926, invowed the concept of De-Broglic that particle of mass (m) moving with velocity. (v) is associated with wave of wavelength (d) given by Idh te The equ for wave associated with moving body in stationary state is given day; 14=40 Sinanot 2 & 4o = amplitude ( displacement from mean position at at any particular paint which us independent of time it, however dependent on coordinates (c,y,z - Position) & to classical Mechanics, the differential equ of wave an be given by 84/0122 = 12/04 + Jeu tou DO

Page no-02) Differentiating equ @ wirst time (t). 34 - 46 68 207 (210) one -46 Sin 210 € (2163) (200) N 4n + Số 2 mot, 6 freem eq" © ¢© 18 = at hay 4 = 40 Sineroth From equ ② 5 Wory --Ummy 4 + une met 4 0 6 The total energy of the system: E Bukiet Pie E = fm² + v 2(E-v) = my 2 multiply bym a ) 12 m (6-v) = m2, 2 multiply by me substituting equ ® in aqr. we get V4 + 8rim (E-V) 4 = 0) or 1174 + 2 (E-v) 4 =0) = 2 72

(page no-03 Hamil tonians Schedinger wave in terms of 74 + 8 nm (£) q = 0 ² y ²4 & E4 - v4 =0 - 6 14 +14 = £4 873m 1-² 7² tv y - E4 SH²m HY = £4] for nth state: - [h4n = Ento

flecosy pagenooy time dependence Io Time Dependent Pen. I time independent Thary, per. Thewery This was proposed lau Tuts was proposed by Schrodinger Boren which deals. 4 the perturbation Hanultonian is with time dependent Static" (rachl & M) be, it has no 7 pechon a time Pudependent Haniltonian (Hol 2 Clynamis [in Physics- it is used Consider a function f(x) which is not an exact isso have function like 40, 41, 42.... Un out satisfies the same boundary condition of 40 141 142 Yo von we f12) > not exact - Ep-Helium L40,4,142 ) Exact soll known ex Hydrogen but both satistics same condition] This function f(x) can be expressed as linear combination of all the functionsie. 4o 14, 142- 4. 4 (x) off f(x) = ? an In (x) - 2 The main problem is to find out the value of an (oefficients) which will help us in

trage no-on) Page No. in determenung arbitary function ( f(x) To find out the value of an, multiply equ♡ by Yom (x) (which is a complex conjugate of any of the wave fruction 40,41,42,43 --- 46) & lutequate it as this satifies the same boundavey condition. (x) f(x) did - | Tost (x) & an 4 (x) dx Now, there are 3 possibilities:- Case-③ = when n=0 then July (2). 4; (x) dx = 0 Case - when n=1 then. 7 @_ en June (x) 4 (2) dx = 0 - ② Case - (1 - when nam then- 4* (x) (W)dx=1 to Ba) & 3D are of no use as nitm but condition of 30 Can be applied to equation to get the value of ancie, & an = { / 4 (2) f(x) dt 4

Time-independent Portwebation !! Consider a AM system for which Schrodinger wave ogu. Cannot be foluent exactly. solved (H4 = Ey) - 6 Page no-og Now the H can be defined as A + Eri Hof He] For Hero perturbed of motibat pel de perturbation paramélies fb A = Ist perturbation terem. Sinularly the general expression for H4 & F for. any complex system can de expressed aes Fellor Tayloots series of expansion or sitricion o no H = 4 Hdh + 8 hod 3 g ... (7) Dis 4= 66 st 4 + 4 + 4z .... o E = E + E + 2 + det 6 x Perturbed part exact Or Un peu twbed part of H, , راوند اور First order Perturbation theory - In order to Ĥ - Hot in derive gst order 4- 4ot it, I-O courection to wave E- tot de la fn 4 energy term, the H, 4 , E can The SWE for exact 848tem be w itten as: is given by; | Howo - to Yo

-01 Ce sowie for the system concern is given by; pagero HY = EU - (ho + dhi) (40 +241) = (fotdE) (46 +d4). ño to + Flody, tdH, 40+ di 4 = to to t fod 4 - + df, 4o + d², 4, Ho 40 + (How+H,40) + 1² (H141) = Eo yot (E04, +6,40) + CE, 4, Ato (12) - H4 = E4 Hoto + d (Ho4, + H140) to (H,4,J = Hototd (Eo 4, + 6,90) tid? (6,4) A is a small quantity hence de terres can be neglected in trick Ho 4, ďt Hi You = 804 18+ £,400 Ho-E04, (E, -Ñ 432 This is the I order perturbation thuany equation. This equ can be used to calculate I order wave function & chergy. first order enerely coverection - Rewriting egu (13), we get - (Horto) 4 = (E-H1) 4. The first order correction for wave for can be given by 4. Ean (14) Value of w, putted in eau ③ -

I Pageno-as (Hoto) Eanto = (f. - H)4 (15) multiplying equ @ 4* & integrating it we get; 140* (Ro - Eo) E an n d E = (4 * (€, - .) 4 dt {an ) &* (+0 - foo) 4, dt = 14+ (E, -A) & de Taking huis of 29" @ I considering following A His = Ean Iyot (40 - 60) 4 d T o two Conditions:
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