Toay s Program:. Review of previous lecture. QM free particle a particle i a bo. 3. Priciple of spectral ecompositio. 4. Fourth Postulate Math tools covere toay Lecture #3. Lear how to solve separable ifferetial equatios.. A basis of fuctios, how to epa a fuctio i terms of a set of basis fuctios 3. Ier proucts (ot prouct i vectors) i fuctio space. 4. A basis of fuctios, how to epa a fuctio i terms of a set of basis fuctios 5. The coectio betwee symmetries i the physical system, ivariace of the Hamiltoia a coserve physical quatities (this is oe of the uerlyig themes i this course to be epae o i future lectures) Questios you will by able to aswer by the e of toay s lecture. How to fi statevectors (wavefuctios) for two simple yet importat cases.. Lear how to verify that a give statevector is a eigevector (eigefuctio) of a operator. 3. Fi the spatial a temporal solutios of Schroeiger s equatio for a system which is escribe by a time iepeet Hamiltoia (o eplicit time epeece of the Hamiltoia). 4. Show that the spatial solutio is i fact a eigefuctio of the Hamiltoia. 5. Show how spatial cofiemet of the system leas to eergy iscretizatio. 6. Kow the qualitative iffereces betwee the free particle case a the particle i a bo. 7. What is the coitio that leas to eergy iscretizatio? 8. How to fi the projectio of a fuctio (vector) oto a eigefuctio (elemet of the basis) 9. How to epa a arbitrary wavefuctio i terms of eigefuctios? 0. How to preict the probability of obtaiig a particular measuremet result.
First ample: The Hamiltoia of a free particle: p ˆ P Hclassical = H = m m ˆ P P H = = = ˆ + yˆ + zˆ ˆ + yˆ + zˆ m i i y z y z focus o a D problem for simplicity: To fi the eigefuctios u( t, ) of a particular operator oe ees to solve the followig equatio: = λu( t) Au ˆ t,, The eigevalues of the Hamiltoia operator are calle the eergy let us ow fi a eigefuctio for the operator which we have calle the Hamiltoia. Note: we will use the symbol u for a eigefuctio. = u( t) Hu ˆ t,, m m i i u u m = u ( ) + u ( ) = 0 m u = ae + be So we have ow the eergy eigefuctios a eergy eigevalues for the free particle case. Please ote that these fuctios are oscillatig a sprea over all space a that ca assume ay value positive or equal to zero. The sith postulate allows us to fi the time evolutio of a state usig the Schroiger equatio. ( t, ) ψ ( t, ) ψ = i m t This equatio is a partial ifferetial equatio, which ca be solve usig a separatio of variables metho: substitutig i the above equatio gives, ( t, ) = ( ) ( t) ψ φ ξ t φ ξ i = = mφ ξ t t
The time epeet part becomes: The spatial part becomes, ξ t i ( t ) 0 ( t ) e + ξ = ξ = t i t φ + m φ = = ae + be = u + ik ik 0 φ m k = ach istict correspos to a ifferet vali solutio which has the form: + ik ik (, ) i t ψ t = ae + be e so the complete solutio ca be writte as a superpositio of the iiviual solutios: ψ i i t i= N + iki iki (, ) = ( i + i ) t ae be e
Seco eample: particle i a ifiite potetial well: I] The system: A particle of mass m i a potetial well: II] The classical eergy fuctio of the system: p m (, ) = + V ( ) H p where the potetial eergy is efie as follows V 0 < < = < or > III] Obtaiig the QM Hamiltoia operator: where, Pˆ H p H X P V X V m m (, ) ˆ ( ˆ, ˆ) = + ˆ( ˆ) = + V 0 < < =. < or >
For the momet let us restrict our attetio to the regio of space which has a fiite potetial a cosier the possible eigefuctios a eigevalues of the operator: (, ) Hˆ Xˆ Pˆ = m IV] What are the eergy eigefuctios a eigevalues? (, ) = = H ˆ X ˆ P ˆ u u m u u m m i i u u m = u ( ) + u ( ) = 0 m u = ae + be Note: The uetermie a a b coefficiets imply that there are a iifiite umber of allowe eigefuctios correspoig to every eigevalue (i.e. etermiig oes ot etermie a a b). We will arrow ow this set by usig bouary coitios erive from physical isights ito our problem. Specifically, we will require that our eigefuctios will be equal to 0 at the bouary of the well. At the momet we o ot have a soli justificatio for this other tha the efiitio of the problem which is such that the wavefuctios ee to be equal to zero at the bouaries a therefore we wat to choose a basis set which ca be covietly use to epress the wavefuctios. The problem of fiig the eigefuctios a eigevalues of a liear quatum mechaical operator basically is ietical to solvig a liear ifferetial equatio. As such we ca apply the techiques a theories which have bee evelope to solve ifferetial equatios to our problems. A ifferetial equatio of the type we are cosierig will have a uique solutio provie that the values of the solutio are kow at the bouaries (bouary coitios).
The bouary coitios i this problem are: u =± = 0 + ik ik k k k k u 0 cos si cos si 0 = ae + be = a + i b i + = k k ( a+ b) cos + ( a b) si = 0 a= b k π = =,3,5... or a = b k π = =,4,6... The form of the solutio is, π o ccosk = ccos u( ) u( ) = π eve sik = si k m π = = k (, ) ψ = cos = cos = cos = ψ m m (, ) = Hˆ Xˆ Pˆ a k a k a k Hˆ Xˆ Pˆ u u Because H is a liear operator ay superpositio of solutios is also a solutio. The bouary coitios have lea to a quatizatio of the eergy levels:
k π = =,,3,4,5... = π h = m 8 m V] Fiig Schroiger s equatio a the wave fuctio: (, ) ψ (, ) = ψ (, ) ψ (, ) = ψ (, ) Hˆ Xˆ Pˆ t i t t i t t m t We have alreay solve this equatio a the solutios are: Let us focus o the spatial compoet: ψ + ik ik (, ) i t t = ae + be e ψ k = m + ( ik ik = ae + be ) ote: ψ ( ) is ot the complete wave fuctio! Where c, are some ormalizatio costat. How o we ormalize a vector basis The costat therefore is v cv i? i c = ( v, v ) ( v, v ) ( u( ), u( ) ) i i i i = = (, ) c c u u / = cos π = c= / c
usig: = ( ) cos = + cos ( ) si cos / / / / / / c π cos + = c c π + cos = c c = = Discussio: Compariso betwee the eigevalues a eigevectors of the free particle a particle i a bo. Oe has cotiuous spectrum the other is iscrete. The iscrete character was a result of the bouary coitios the fact the particle was cofie to a particular regio i space. = 0 Free particle Particle i a bo Hamiltoia eigefuctios Hamiltoia eigevalues k ik = u ( ) u e k m = k π o ccosk = ccos = π eve sik = si k π =, k = m
Poits to emphasize:. iscrete vs. cotiuous spectrum. epeece of eergy level separatio o size of well. 3. umber of oes vs. eergy of eigefuctio 4. solutios are either o or eve 5. lowest eergy solutio is eve
Pricipal of spectral ecompositio Cosier a system whose state is characterize at a give time by the wavefuctio ψ rt,. We wat to preict the result of a measuremet at this time of a physical quatity a associate with the observable A. The preictio of a possible outcome will be i terms of probabilities. We will ow give a set of rules which allow us to preict the probability of obtaiig i a measuremet ay eigevalue of A. Let us first assume that the spectrum of A is etirely iscrete. If all the eigevalues a of A are o-egeerate there is associate with each of them a uique eigevector u ( ) : Sice A is Hermitia the set of u = ˆ Au au is a basis i the wavefuctio space. That meas that ay wave fuctio, ψ ( rt, ) cu ( t, ) = Defie a ier prouct betwee two fuctios ϕ ψ = ϕ ψ Remi ourselves of the geometrical iterpretatio (i D for simplicity) Look at two vectors ψ, ϕ (psi a phi), the ier prouct results i a umber (coul be comple scalar) tells us what the projectio of oe o the other is. Now the projectio is iepeet of the basis which you ecie to represet your vector i. ϕψ = c how o we represet a vector ψ i a particular basis: we choose a particular set of vectors which spa the vector space ϕ, ϕ usually we choose basis set to be orthoormal. the we fi the projectio of ψ i the irectio of each basis vector. ϕ ψ = c ϕ ψ = c
Write the origial fuctio as a liear combiatio of the basis vectors. I geeral ψ ψ = cϕ + c ϕ N = ciϕ Fiig the eigevectors a eigevalues of operators, iscuss the geometrical iterpretatio of eigevectors a eigevalues scalig. i= i ample: Particle of mass m i a ifiite potetial well We saw that the eigefuctios are of the form: u k ( ) π o cosk = cos = π eve sik = si m π = = Suppose you ha ow a wave fuctio ψ ( ) = + 8 4 3 3.4. 0.8 0.6 0.4 0. 0 - -0.8-0.6-0.4-0. 0 0. 0.4 0.6 0.8 Coul you epress it i terms of your eigefuctios?
ψ 3 05 3 π π = cos si 7 + = a + b 8 4 Fourth Postulate (iscrete o-egeerate): Whe the physical quatity a is measure o a system i the ormalize state ψ ( t) the probability P( a ) of obtaiig the oegeerate eigevalue a of the correspoig observable is where u = ψ =, ψ P a u u is the ormalize eigevector of A associate with the eigevalue a.