14 The Postulates of Quantum mechanics

Transcription

1 14 The Postulates of Quantum mechancs Postulate 1: The state of a system s descrbed completely n terms of a state vector Ψ(r, t), whch s quadratcally ntegrable. Postulate 2: To every physcally observable there exst a lnear Hermtan operator. Postulate 3: In any measurement of the observable assocated wth operator Â, the only values that wll ever be observed are the egenvalues a, whch satsfy the egenvalue equaton Âg = a g. Postulate 4: The egenfunctons of operators correspondng to observable forms a complete set. Postulate : If a system s n a state descrbed by a normalzed wave functon Ψ, then the average value of the observable correspondng to  s gven by < A >= Ψ ÂΨdτ. Postulate 6:The wavefuncton or state functon of a system evolves n tme accordng to the tme-dependent Schrdnger equaton ĤΨ(r,t)= Ψ t. 6

2 14.1 Measurement and superposton of states Postulate 4 allows us to expand any state functons n terms of the complete egenfunctons of an operator Ψ(q, t) = c (t)g (q) (1) where Ψ s normalzed and q s the 3n coordnates of the system. The egenfunctons obey the followng egenvalue equatons for the hermtan operator B ˆBg (q) =b g (q) (2) Usng the normalzaton condton we have 1= c g dτ = c c g g dτ (3) c g Assumng that we can nterchange summaton and ntegratons we get c c g g dτ = 1 (4) We now that the egenfunctons are orthogonal snce ˆB s hermtan thus c c δ = c 2 = 1 () and we see that the sum of the square of the coeffcents of a complete set adds to one. Now lets consder the mportance of ths for measurements. We are nterested n determnng the average value of the operator ˆB as < B >=< Ψ ˆB Ψ > (6) =< c g ˆB c g > (7) c c <g ˆB g > (8) c c b <g g > (9) c c b δ (60) = c 2 b (61) 66

3 and the expectaton values s the weghted sum of the egenvalues. The square of the expanson coeffcents are thus the probablty of measurng the egenvalues. We fnd the coeffcents by project the egenfuncton onto the state functons as <g j Ψ >= gj Ψdτ = gj g dτ = c δ j = c j (62) j and we call the ntegral the probablty ampltude. Thus, we see two thngs The frst s that the state functons s gven by a superposton of the egenfunctons of the operator and we fnd the expectaton valued by summng the square of the ampltudes. Ths also means that once we performed a measurement of the system t s collapsed to to an egenfunctons of the system. Ths means that n quantum mechancs the act of performng a measurement perturbs the system. 1 The tme-evoluton of the wave functon Consder a hamltonan that s tme-ndependent. Ths means that the state functon obeys ĤΨ = EΨ (63) and the tme-dependent Schrödnger equatons becomes whch can be ntegrated to gve Ψ t = EΨ (64) Ψ(q, t) = exp( Et/hbar)ψ(q) (6) Therefore, f the hamltonan s tme-ndependent there exst statonary states Ψ n = exp( E n t/ )ψ n (q) (66) where ψ)n s a soluton to the tme-ndependent Schrödnger equaton. We can now expand any functon n terms of these egenfunctons as Ψ = n c n Ψ n = n c n exp( E n t/ )ψ n (q) (67) 67

4 where c n are tme-ndependent. Ths functon s not an egenfuncton of the hamltonan, and thus do not have a defnte energy. To fnd the coeffcents at tme t 0 we evaluate the projecton < ψ j (q) Ψ(q, t 0 ) >= n c n exp( E n t/ ) < ψ j ψ n >= n c n exp( E n t/ )δ jn = c j exp( E j t/ ) (68) Thus, the general soluton to the tme-dependent Schrödnger equaton s gven by Ψ = j < ψ j (q) Ψ(q, t 0 ) > exp( E n (t t 0 )/ )ψ j (q) (69) f the hamltonan s ndependent of tme. 1.1 Example As an example consder a state whch consst of a superposton of the two lowest egenfunctons of the partcle n a 1D box Ψ(x, 0) = ( 1 2/A sn(2πx/a)+ 2 ) sn(πx/z) (70) Our expanson coeffcents are then c 1 =2/ and c 2 =1. and the state of the system at t>0 s gven by Ψ(x, t) = ( 1 2/A exp( ω 2t) sn(2πx/a)+ 2 ) exp( ω 1 t) sn(πx/a) where ω n = E n. The probablty of measurng s then (71) P (E 1 )= c 1 2 =[ 2 exp( ω 1 t)] [frac2 exp( ω 1 t)] = 4/ (72) and s thus constant n tme. What about the expectaton value < E > t>0 = 1 (< exp( ω 2t)φ 2 +2 < exp( ω 1 t)φ 1 ) (73) (Ĥ exp( ω2 t)φ 2 +2Ĥ exp( ω 1t)φ 1 >) (74) 1 ( = < φ 2 Ĥ ψ 2 > +4 < φ 1 Ĥ φ 1 > (7) 68

5 ) + 2 exp( (ω 1 + ω 2 )t)[< φ 1 Ĥ φ 2 > + < φ 2 Ĥ φ 1 >] (76) = 1 (E 2 +4E 1 )=< E > t=0 (77) and also the expectaton value of the energy s constant n tme. 69