State function: eigenfunctions of hermitian operators> normalization, orthogonality completeness


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1 Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators> normalzaton, orthogonalty completeness egenvalues and expectaton values of operators Tme ndependent Schroednger equaton and statonary states. Probablty current.
2 Schroednger equaton. Schroednger equaton s a wave equaton, whch lnks tme evoluton of the wave functon of the state to the Hamltonan of the state. For most of systems Hamltonan represents total energy of the system T+V= knetc +potental. Hamltonan s defned also classcally, and equatons of motons for classcal systems can be wrtten usng dervatves of the Hamltonan. Classcally there s no need for a concept of the wave functon of the state, as any state can be totally specfed by gvng momentum and postons of all partcles.
3  Frst I remnd you about a flat wave. Ths s the wave functon descrbng a free partcle.  I wll show that the flat wave s a soluton of the free Schroednger equaton.  It useful to test operators and propertes of wave functons on a flat wave to understand what they really mean.
4 A nonrelatvstc partcle has the followng Hamltonan> energy: H=[ p2 2 m V r ] where knetc energy s E k = m v 2 2 = p2 2 m we know that a free partcle ( propagatng n a place wthout potental) can be descrbed by a flat wave or a combnaton of flat waves wave packet m r,t = Ae r,t = 2 h we note that: r,t t = 2 h h p r Et where h =E, =h/ p, ħ= h h p e p r Et d 3 p 3 2 h h p E e p r Et d 3 p
5 Thus r,t t = 2 h 3 2 h p p2 2 m e Now remnd ourselves Laplace operator h p r Et d 3 p 2 = 2 x 2 2 y 2 2 z 2 and note that for example : 2 r,t = 2 ħ x 2 2 r,t = 2 ħ thus h t ħ 2 p p x 2 e ħ 2 p p 2 e =[ h2 2 m 2 ] ħ p r Et d 3 p ħ p r Et d 3 p Ths s Schroednger equaton for free partcle
6 We can also note that dfferentatng over x (for example): h r,t = 2 h x and defne momentum operators h t h t = H =[ p 2 2 m V r ] h t 3 2 h p p x e p r Et d 3 p p x h =[ h2 2 m 2 V r ] x, p y h y, p z h z =[ h2 2 m 2 ] h t = H =[ p 2 2 m ] Natural extenson to the stuaton wth potental ( nonfree partcle)
7 Postulates of Quantum Mechancs A ) For every classcal observable a lnear operator. It can depend on momentum and poston operators. Momentum operator (x) s proportonal to dervatve over x F q 1, q 2,.. q n, p 1, p 2,.. p n : B) A system s fully descrbed by a wavefuncton whch fulflls wave equaton, Schroednger for nonrelatvstc Hamltonan C) expectaton value of an operator F q n =q n, p n = ħ h t = p H =[ 2 2 m V r ] corresponds to the expectaton ( mean value) of the measurement result of the varable F taken over a bg number of ndependent measurements (ths s more dffcult then t sounds) D) the only possble results of sngle measurements of the varable F are the egenvalues of the operator F F = f (example of spn ½, z projectons ) f = constant < F >= * F d q n (after a measurement the system collapses to the state wth well defned varable f)
8 What's new here? You have heard ths before on kvantefyskk og statstk mekankk The dfference s that now we are tryng to wrte our wave functon n a more general way. It can be a functon of space varables, or momentum or perhaps even more general varables ( and tme). In practce we wll start wth space representaton, then we wll dscuss momentum representaton, and then other general Drac representaton of the sate and vector representaton ( matrx mechancs)
9 Ad C. < F >= * F d d =dq 1... d q n In general we ntegrate over the set varables the wave functon of the system s defned on, and normalzed on. In space representaton of quantum mechancs we dscuss rght now ths s space coordnates. For example an expectaton value of a poston of a partcle wll be: < r t >= * r,t r r,t d 3 r < r t >= r r,t 2 d 3 r proper normalzaton 1= r,t 2 d 3 r what about momentum? Ths s probablty densty to fnd the state n locaton r
10 Hermtan operators. The expectaton values of an operator representng any real varable must be real < F >= * F d = F * d =< F > * Defnton of hermtan operator: The operator F s hermtan f : 1* F 2 d = 2 F 1 * d If the operator s hermtan ts expectaton value s REAL that s a requrement for operators assocated wth observables Defnton of adjoned operator to F 1 * F 2 d = 2 F 1 * d Hermtan operators are selfadjoned meanng : F = F example, check p_x operator.
11 Ad hermtan, prove of hermetcty condton < F >= * F d = F * d (alpha, any constant number take real functons PSI1, PSI2 1 e a 2 * F 1 e a 2 d = 1 e a 2 F 1 e a F 2 * d 1 * F 1 d e a 1 * F 2 e a 2 * F 1 2 * F 2 d = = 1 F 1 * d e a 1 F 2 * e a 2 F 1 * 2 F 2 * d 1* F 2 d = 2 F 1 * d 2* F 1 d = 1 F 2 * d
12 Commutator of operators Commutators: are mportant! for example varables who's operators commute can be measured smultaneously for the system. [ A, B ]= A B B A Lets check [ x, p x ] = x p x p x x x p x p x x =x ħ x ħ x x = x ħ x ħ x x ħ [ x, p x ]= ħ =ħ x poston varable, and p_x do not commute. What does that mean for subsequent measurements of px and x?
13 Egenfunctons and egenvalues: The result of an operator workng on a functon s usually a dfferent functon. If there exst a set of functons that : F n = f n n we call them egenfunctons of the operator F, and fn are egenvalues (constants) The set of egenvalues ( or spectrum of the operator) can be dscrete, contnuos or mxed. Example of contnuos energy operator (Hamltonan) for a free partcle, dscrete= hamltona for harmonc oscllator, mxed hamltonan for hydrogen atom. p x = ħ x ħ x f = f f f =const exp fx ħ That's the form of egenfunctons of momentum operator. They are not quadratcaly ntegrable and have to be normalzed n a dfferent way.
14 Expectaton value vs. egenvalue of an operator. < F >= * F d The expectaton ( mean value) of the measurement of the varable F taken over a bg number of ndependent measurements ( n practce over large number of dentcally prepared states an ensamble ). But the only possble results of measurements of the varable F are the egenvalues of the operator F F = f (example of spn ½, z projectons ) If the system s n the state descrbed by the egenfuncton of the operator,the expectaton value of the measurement s equal to the egenvalue. As the egenvalue s the only possble measurement results, that s what we wll always get! < F >= * F d = n* F n d = f n n * n = f n for the egenstate number n Ths proof s straghtforward for normalzed quadratcntegrable egenfunctons, can be also proved for dfferent type of normalzaton
15 Orthogonalty of egenfunctons For a hermetc operator, egenfunctons correspondng to dfferent egenvalues are orthogonal n* F m d = m F n * d f m n * m d = f n * m n * d f m f n n * m d =0 f f m f n then n * m d =0 Drac notaton: Scalar product n * m d = n m n* F m d = n F m n F m = F n m Hermtan operator n Drac notaton
16 We try to normalze our set of egenfunctons of an operator typcally choosng a constant to multply the functons wth. n * m d = nm 0 for n = m, orthogonalty 1 for n=m, normalzaton Normalzaton of egenfunctons wth contnuous spectrum of egenvalues has to be a bt dfferent functons are not quadratcally ntegrable f * ' f d = f ' f =< f' f> f x =c exp fx ħ * f ' x f x d x= 1 2 ħ C must be c= 1 2 ħ Drac delta Example momentum egenfunctons, here just 1dm e x f f ' ħ dx= f ' f to get the normalzaton rght
17 More about Drac delta functon: Drac delta functon : f x x x ' dx= f x ' f x x dx= f 0 propertes : possble representaton: 1 x dx=1 x =lm 0 x =lm 1 2 x = 1 2 exp xy dy exp xy y dy =lm 0 x 2 2
18 x = x 2 2 =0.01 =0.1 =0.5 x
19 Completeness We assume that every reasonable, quadratcally ntegrable functon can be expressed as a lnear combnaton of egenfunctons of a hermtan operator F g= c g >= For egenfunctons wth contnuous spectrum of egenvalues we can represent a functon g n the followng way : (egenfunctons have to form a complete bass set) g= c f f df g >= df c f f >= df <f g> f > f * ' f d = f ' f =< f' f> then we have : c f = f * g d =< f g> we must also have : f * r ' f r d f = r ' r c >= < g> > for orthonormal set of egenfunctons the coeffcents are a scalar product of the functon n queston and approprate egenfuncton n * g d = c n * d =c n c n = n * g d =<n g>
20 Completeness, contnuaton: Lets now consder that our functons are normalzed n the normal space, so f * ' f d = f * ' r f r d 3 r= f ' x f x f ' y f y f ' z f z etc.. g r = c r g r = g r ' we must n analogy have for contnuous spectrum: c n = n * r ' g r ' d 3 r ' * r ' r d 3 r ' f * r ' f r d f = r ' r * r ' r = r r ' 3dm Drac delta
21 What s the nterpretaton of expanson coeffcents cn? We see that modulus squared of an expanson coeffcent wll correspond to a probablty to measure certan egenvalue: Lets take spectrum of egenfunctons of gven operator F and expand a functon of state (g) nto t : g= c and check what s the expectaton value of the operator F for the state descrbed by g < F >= g * F g d = n < F >= n < F >= n c n * n * f c = n c n * c f,n = c n * n * F c 2 f c c n * c f n * d but we know that from nterpretaton of expectaton value we must have < F >= P f where P_ s the probablty that the value f_ wll be measured.
22 P f df = c f 2 df = f * g d 2 df where F f = f f Thus we have. The probablty that measurng observable assocated wth F on a state descrbed by a wave functon g wll gve a result f_ s the followng: P = c 2 = * g d 2 where F = f For contnuos spectrum of F we can prove n analogy that: F = f c f 2 df The probablty that measurng observable assocated wth F on a state descrbed by a wave functon g wll gve a result beetween f and f+df s
23 Statonary states: If the Hamltonan does not contan tme explctly we can try to separate the solutons n to tme dependent part and coordnates dependent part. We obtan that there s new constant nvolved proportonal to the tme dervatve of the tme dependent functon dvded by the functon tself. We call t ENERGY. We obtan tmendependent Shroednger equaton for the part whch does not depend on tme. t, r r T t dt t /dt ħ =const E T t H r =E r Et/ħ T =C e h T t t r =[ h2 2 m 2 r V r r ]T t What statonary means n practce? Expectaton values of operators do not depend on tme. ( for normal operators whch do not contan tme dervatves ) Prove t!
24 Probablty current : Probablty nterpretaton for the partcle wave functon: modulus square of t s a probablty to fnd a partcle n a gven place (probablty densty). THUS: Integral over space has to gve 1. However locally spacal probablty densty can change wth tme. We can defne the probablty current, useful when dscussng movement of partcles r,t = * = * * = t t t t ħ 2 m * 2 2 * = change of probablty densty wth tme (at a gven place) s related to the outflow of the current. t = ħ 2 m * * = j j= ħ 2 m * * =R * ħ m ħ 2 m * * h t h2 =[ 2 m 2 V r ] h * h2 =[ t 2 m 2 V r ] *
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