Now that we have laws or better postulates we should explore what they imply

Transcription

1 I-1 Theorems from Postulates: Now that we have laws or better postulates we should explore what they mply about workng q.m. problems -- Theorems (Levne 7.2, 7.4) Thm 1 -- egen values of Hermtan operators are real (clearly ths fts well wth Post. 2 restrcton we need real observables let A > = a > < A > = < A >* a > = a * > (a -a *)< > = 0 ==> a I = a *==> a real snce < > s postve defnte Thm 2 -- egen functon of a Hermtan operator can be chosen to be orthogonal.e. f gd = 0 for f g j Α = Α j a j = a j j = a j j = a j j ( a a j ) j = 0, a j * a j snce real, < j> flp order on complex conj. ether a = a j (same state or degenerate) or orthogonal: < j> = 0 f >, j> are degenerate can construct 1 = c + c j j, 2 = c j c j <1 2> = 0, orthogonal by constructon and stll are degenerate egen functons.e. Α(c + c j ) = a (c + c j j )

2 I-2 Wednesday--August 29 Applcatons: The wave functon descrbng the state of a quantum mechancal system can be descrbed as a superposton of n terms of egen functons of an operator ˆ --{f }, snce they form a complete set You have experence wth ths from f Taylor seres: f (x a) = ( n) ( a ) n! (x a) n or f (x) = n n 1 n! n f x n 0 x n so f(x) s represented as lnear combnaton of x n -- power seres More general s Fourer seres : f(x)= n [ s n sn(2 nx) + c n cos(2 nx) ] where expand n sne and cosne fcts or exponetals: n these cases: x n, sn(2πnx), e -2nπx form complete sets 2 xn b n e (Note sn s odd, must add cos f party not odd, elmnate sn f even) So what are the coeffcents? In Taylor : f ( n ) ( x ) n! but n q.m. expand n a set {g } of egen fct of operator α: f(x)= c g (x)= c multply. by g j * and ntegrate : j f = c j = c j = c j Drac delta functon: < j> = j =1 = j = 0 j so c = < f> or f = Σ >< f>,, Where Σ >< s projecton operator pcks out (projects) part of f> that les along > [analogous to dot (scalar) product from vector algebra]

3 I-3 Thm 3 -- f {g } s set of egen fcts of α and f s also an egen fct so that ˆ f = af then f f = c g the only non-zero c are for g whch have egen value of a (degen. wth f). (Alternatevely, f must be a lnear combnaton of degenerate g wth same egenvalue) ˆ f = ˆ c g = c a g = af = a c g c = f g s only non-zero for f = g or f-degenerate wth g, (otherwse f g = 0, orthog) alternatvely: f g are ndep fct, only f a = a wll c be non-zero: ( a a ) c g = 0, Commutaton: A commutator of operators ˆ, ˆ s ˆ, ˆ [ ] = ˆ ˆ ˆ ˆ Note ths s famlar: xp x p x x = h from Post 2 Smple multplcatve or scalar operators commute -- ths s your experence dervatve and matrx operators may or may not commute. [ ] f ˆ, ˆ = 0 we say t commutes Thm 4 -- f ˆ, ˆ are two operators that share a complete set of egen fcts, [ ˆ, ˆ ] = 0???error smple proof not general: let {f} be egen fct ˆ, ˆ ˆ ˆ f = ˆ b f = b ˆ f = b a f = a ( b f ) = a ˆ f = ˆ ( a f ) = ˆ ˆ f [ ˆ ˆ ˆ ˆ ] f = 0 book proof a lttle more general -- g = c f uses commutaton of constants, a, b [ ˆ ˆ ˆ ] g = c [ ]g = c ˆ [ b a ]g = c [ b a a b ]g = 0

4 I-4 Ths theorem very powerful, lets us substtute egen functons to determne propertes. [ ] = 0 Also ths s uncertanty prncple : f ˆ, ˆ then we can determne observables correspondng to ˆ, ˆ wth arbtrary accuracy f not uncertanty relaton tells measurement lmtatons [ ] = ( x h x h x x) = h recall: operator must act on somethng: x, p x (d/dx)xf(x) = f(x) + x(df(x)/dx) Chan rule dfferentaton leaves a noncancellng term Reverse form partcularly mportant Thm 5 -- f ˆ, ˆ [ ] = 0, there exsts a common set {f } of egen functons for both ˆ and ˆ let ˆ f = a f set of egen functons operate β on α egen eqn: ˆ ˆ f = ˆ a f = a ( f ) = ˆ ˆ f = ˆ ˆ f ( ), snce commute snce equal, (βf) must be an egen functon ˆ snce (βf) and (f) have same egen value a, they must be degenerate or relate by const (non-degenerate).e. βf = bf or f must be an egen fct of β = Several general propertes that you can prove: (Atkns 5.4) [ Α ˆ Β ˆ ] = ˆ [ Β, Α ˆ ] [ Α ˆ Β ] b [ Α ˆ, Β ˆ ] Α ˆ C ˆ [ Α, Β ˆ ]ˆ C + Β ˆ Α ˆ, C ˆ [ ] = ˆ [ ] Α ˆ [, Β ˆ + C ˆ ] = [ Α ˆ, Β ˆ ] + [ Β ˆ, C ˆ ] Α ˆ, Β ˆ [ [, C ˆ ] =ΑΒC ΒCΑ ΑCΒ+CΒΑ

5 I-5 Commutaton has form of uncertanty (recall xp x p x x = h <==> x p h 2 ) use expectaton values ˆ = ˆ and mean devaton: ˆ = ˆ snce s a constant: Α ˆ, Β ˆ [ ] = [ Α ˆ, Β ˆ ] = C ˆ let C represent result of commutator can be zero, a constant or operator then ( Α ˆ ) 2 ˆ ( ) C from: ˆ ( Α ) 2 = Α ˆ 2 Α 2 defne root mean square devaton: [ ] Α Β 1 2 C ˆ Α ˆ, Β ˆ ˆ Α 2 Α 2 = Α from : [x,p x ] = h/2π, δxδp x h 2 (precse form) but [x,y] = [x,p y] = 0 no uncertanty Note: E t h/4π not a true uncertanty no true operator, actually from [x,p x ] but a consequence of tme dependent Schroednger Equaton Party --(Levne 7.5)-- functons can be even or odd or nether. even: odd: f(x) = f(-x), f(x,y,z) = f(-x, -y, -z) f(x) = -f(-x) f system has party t s even or odd and that can be represented by party operator ˆ π ˆ π g = cg g - odd - c = -1 g egen fct of ˆ π = g(-x, -y, -z) c - even - c = 1 all possble even/odd fct f [ π ˆ, Η ˆ ] = 0 these egen fct of H ˆ must be even/odd [ π,τ] = 0 2 f ( x) = 2 ( x ) 2 x 2 f (x) but [ π ˆ, V ˆ ] depends on form of potental

6 I-6 e.g. V = 1 2 kx2 (Hooks law sprng) even -- square of coordnate V = e2 r (electrostatc attracton) even --depends only on dstance, not drecton V = ee(x) (change n electrc feld s drectonal - odd) Thm 7 -- If potental V s an even fct, can choose statonary states ψ to be even or odd [ π ˆ, Η ˆ ] = π ˆ, Τ ˆ + V ˆ [ ] = [ π ˆ, Τ ˆ ] + [ π ˆ, V ˆ ] = V = even ex. 1 let V = +1 > x > 1, <x< 1 V = -1 1 > x > -1 ths s even, ψ has defnte party, s egen fct of party operator These concepts are central to applcatons of group theory and symmetry to molecular q.m. problems and spectroscopy. Test: [ π, V]f=π(Vf) V(πf)=V( x)f( x) V(x)f( x) = 0 f V even 0 f V odd or mxed party Probablty ampltudes and superposton of states (Levne 7.6) recall make a measurement on some normalzed state 4 α = = ˆ = c c j j = c * c j a j j = c j p a so measurement s some weghted coverage over several egen values of α p s c 2 or probablty of measurng each a j 2 a j

7 I-7 Thm 8 -- Measurement of property correspondng to ˆ α n ψ has a probablty of a equal to c 2 where c s expanson coeff: ψ = c g and α ˆ g = a g [If a degenerate, probablty s sum of c 2 for degenerate a ] recall: c = ψ --called the probablty ampltude (Levne) Thm 9 Probalty of observng a (for α > = a >) f a s non-degenerate s < ψ> 2 for state ψ -- thus more smlar g and ψ, the more smlar wll be <α> and a Frday--September 1 Tme evoluton of expectaton value: (Atkns, Ch 5.5)