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Solving Schrodinger Equation • If V(x,t)=v(x) than can separate variables 2 2m 2 ( x ,t ) x 2 V ( x ) i t assume ( x, t ) ( x ) (t ) 2 2m d 2 dx 2 2 d 2 2 mdx 2 V ( x ) (t ) ( x ) i V 1 id dt G G is separation constant valid any x or t Gives 2 ordinary diff. Eqns. P460 - Sch. wave eqn. 1 d dt Solutions to Schrod Eqn • Gives energy eigenvalues and eigenfunctions (wave functions). These are quantum states. • Linear combinations of eigenfunctions are also solutions. For discrete solutions ( x, t ) c11 c2 2 ......cn n each i i e iEi t / If H Hermitian i orthogonal * i j dx ij normalized 2 c i 1 P460 - Sch. wave eqn. 2 id dt G (t ) e iGt / G=E if 2 energy states, interference/oscillation 2 d 2 V E 2 2mdx ( x, t ) ( x )e iEt / 1D time independent Scrod. Eqn. Solve: know U(x) and boundary conditions want mathematically well-behaved. Do not want: ( x) x 2 x 2 No discontinuities. Usually except if V=0 or =0 in certain regions P460 - Sch. wave eqn. 3 Linear Operators • Operator converts one function into another Of ( x) f ( x) x 2 d f ( x) Of ( x) dx • an operator is linear if (to see, substitute in a function) if O[ f1 ( x) f 2 ( x)] Of 1 ( x) Of 2 ( x) linear ex : O d dx • linear suppositions of eigenfunctions also solution if operator is linear……use “Linear algebra” concepts. Often use linear algebra to solve non-linear functions…. P460 - Sch. wave eqn. 4 Solutions to Schrod Eqn • Depending on conditions, can have either discrete or continuous solutions or a combination ( x, t ) iEn t / C u ( x ) e n n n C ( E )u E ( x )e iEn t / dE • where Cn and C(E) are determined by taking the dot product of an arbitrary function with the eigenfunctions u. Any function in the space can be made from linear combinations P460 - Sch. wave eqn. 5 Solutions to Schrod Eqn • Linear combinations of eigenfunctions are also solutions. Assume two energies ( x, t ) c11 c2 2 c1 1e iE1t / c2 2 e iE2t / assume know wave function at t=0 ( x,0) 1 2 5 7 2 7 • at later times the state can oscillate between the two states probability to be at any x has a time dependence | ( x, t ) | | c1 1 ( x) | 2 c1c2 ( 2 e * 1 i ( E2 E1 ) t / 2 | c2 2 ( x) | 1e P460 - Sch. wave eqn. * 2 2 i ( E1 E2 ) t / 6 ) Example 3-1 • Boundary conditions (including the functions being mathematically well behaved) can cause only certain, discrete eigenfunctions d f ( ) f ( ) d with f ( ) f ( 2 ) i • solve eigenvalue equation i 1 d f ( ) d f ( ) eigenvalue or i d f ( ) d f ( ) int egrate ln f ( ) i cons tan t or f ( ) f (0)e i • impose the periodic condition to find the allowed eigenvalues e i ( 2 ) 1 0,1,2, etc P460 - Sch. wave eqn. 7 Square Well Potential • Start with the simplest potential V ( x ) V0 V ( x) 0 | x | | x | a 2 V0 finite or a 2 (" in" the well ) For value ( x) 0 for | x | a 2 V is finite Boundary condition is that is continuous:give: out ( a2 ) in ( a2 ) 0 if V0 V -a/2 a/2 P460 - Sch. wave eqn. 0 8 Infinite Square Well Potential • Solve S.E. where V=0 2 2 d 2 m dx 2 E A sin kx, B cos kx, Ceikx Boundary condition quanitizes k/E, 2 classes Odd Even =Bcos(knx) =Asin(knx) kn=n/a kn=n/a n=1,3,5... n=2,4,6... (x)=(-x) (x)=-(-x) En p2 2m 2k 2 2m 22n 2 2 ma 2 h 2n 2 8 ma 2 as n 0 E min E1 0 P460 - Sch. wave eqn. 9 Parity • Parity operator P x -x (mirror) P ( x ) ( x ) • determine eigenvalues Pu ( x) u ( x) P 2u ( x) Pu ( x) 2u ( x ) but P[ Pu ( x)] Pu ( x ) u ( x ) 2 1 1 even and odd functions are eigenfunctions of P Odd : Px x Even : Px 2 x 2 P sin x sin x P P cos x cos x P x 2 x 2 x 2 x 2 • any function can be split into even and odd ( x) 12 [ ( x) ( x)] 12 [ ( x) ( x )] ( x) ( x) ( x) 1 2 (1 P ) P ( x ) ( x ) ( x ) 12 (1 P ) P460 - Sch. wave eqn. 10 Parity • If V(x) is an even function then H is also even then H and P commute [ H , P ] HP PH 0 • and parity is a constant. If the initial state is even it stays even, odd stays odd. Semi-prove: • time development of a wavefunction is given by i H ( x, t ) t • do the same for P when [H,P]=0 i ( P ) H [ P ( x , t )] P[ H ( x , t )] t • and so a state of definite parity (+,-) doesn’t change parity over time; parity is conserved (strong and EM forces conserve, weak force does not) P460 - Sch. wave eqn. 11 Infinite Square Well Potential • Need to normalize the wavefunction. Look up in integral tables 2 | ( x ) | dx A a 2 A2 sin 2 nx a dx 1 a 2 2/a What is the minimum energy of an electron confined to a nucleus? Let a = 10-14m = 10 F Emin 2 2 2 ma 2 ( hc ) 2 8 mc 2 a 2 (1240MeVF ) 2 8.51MeV (10 F ) 2 4000 MeV relativist ic Emin m2 p 2 k hc 2a redo m 2 (k ) 2 1240MeV F 210 F P460 - Sch. wave eqn. 60 MeV 12 Infinite Square Well Density of States • The density of states is an important item in determining the probability that an interaction or decay will occur • it is defined as dn (E) • for the infinite well n number of states dE 8ma 2 n E cE h2 dn c 1 c 2ndn cdE dE 2n 2 E 2 • For electron with a = 1mm, what is the number of states within 0.0001 eV about 0.01 eV? 8 511000eV (. 1cm ) 2 c 2.7 1012 eV 1 4 2 (1.24 10 eVcm ) dn 1 n E dE 2 c 1 E E 2 P460 - Sch. wave eqn. 2.7 1012 .0001eV 820 .01eV 13 Example 3-5 • Particle in box with width a and a wavefunction of ( x ) A( x / a ) 0 x a / 2 ( x ) A(1 x / a ) a / 2 x a A 12 / a • Find the probability that a measurement of the energy gives the eigenvalue En Au n n ( x) un 2 a sin nx a n a An ( x )u n a/2 dx 2 0 2 0 24 12 x a a 2 nx sin dx a a 1 ( 1) n 1 2 n • With only n=odd only from the symmetry • The probability to be in state n is then | An |2 96 Pr ob1 .986 4n 4 Pr ob3 P460 - Sch. wave eqn. .986 .012 34 14 Free particle wavefunction • If V=0 everywhere then solutions are A cos k x , A sin k x , e ikx , e ikx E p2 2m 2k 2 2m • but the exponentials are also eigenfunctions of the momentum operator pop i x pop ( eikx ) i ik eikx eigenvalue k p pop ( e ikx ) i ik e ikx k p • can use to describe left and right traveling waves • book describes different normalization factors P460 - Sch. wave eqn. 15