Chem120a Final Exam Review

Transcription

1 Chem120a Fial Exam Review David W.H. Sweso December 11, Overview of the semester Before divig ito the ew material that will be o the fial exam, let me try to give you the big picture of what we ve bee doig i this class. For details o older material, look at the reviews from previous exams. The overall goal of this course has bee to teach you some itroductory quatum mechaics, ad show you some of the most importat ways that we apply quatum mechaics to the study of chemistry. So we started off by teachig you the basics of quatum. We talked about some of the ideas which led to the quatum revolutio (ideas like the photoelectric effect ad the de Broglie wavelegth). We itroduced the Schrödiger equatio, which has bee at the heart of the rest of the class. Oce we had the Schrödiger equatio ad its Copehage (probabilistic) iterpretatio, we looked at some of the mathematical techiques which served us throughout the class, such as bra-ket otatio ad the calculatio of commutators. From there, we explored a few simple systems, like the particle i a box, the harmoic oscillator, the hydroge atom, ad the rigid rotor. We saw that these could provide useful explaatios of some real ad relevat systems. However, we saw that we did t have a exact solutio to multi-electro systems. So we developed some approximatio methods: perturbatio theory ad the variatioal method. Oce we had these techiques, we could develop Hartree- Fock ad self-cosistet field (SCF) theories for multi-electro systems. We also saw how atomic states ca be represeted by atomic term symbols. At this poit, we ca start to look at molecules. We saw the valece bod model ad molecular orbital theory. We showed how to approximate molecular orbital theory with Hückel theory. We saw that there were molecular term symbols, similar to atomic term symbols, which ca represet the states. Fially, with all that theory out of the way, we could start to talk about some experimets. The mai experimetal techique of physical chemistry is spectroscopy, ad we foud the base of spectroscopy i quatum mechaics. We saw that spectroscopy allowed us to use light to probe differet degrees of freedom: electroic, vibratioal, rotatioal, ad eve uclear spi (i NMR). We also saw that we could use a dipole approximatio to light, ad from that, we ca calculate selectio rules for each type of spectroscopic trasitio. There are two kids of selectio rules: oe which tells us what kid of molecule ca have a trasitio, ad oe which specifies which trasitios are allowed. Oe of the major recurrig themes i quatum mechaics is certai type of duality that Niels Bohr called complemetarity. The basic idea of complemetarity is that there are two ways of lookig at every problem. The best kow example is the wave-particle duality i 1

2 some situatios, it is best to describe a object as a wave; i others, as a particle. It seems like these two pictures cotradict each other, but i quatum mechaics, both ca be true. That s complemetarity. Oe of the ways we frequetly see complemetarity i quatum mechaics is by cosiderig a view focused o the wavefuctio o oe had, ad o operators o the other. For example, the questio may of you foud most difficult from Prof. Miller s first exam asked you to idetify the two situatios i which a expectatio value is time-idepedet. The first situatio depeded o the wavefuctio (if the wavefuctio is a statioary state), ad the secod situatio depeded o the operator (if the operator commutes with the Hamiltoia). 1 We saw a similar duality whe we discussed approximatio methods. I the variatioal method, we treat the Hamiltoia operator as fixed ad we chage the wavefuctio to better suit it, whereas i perturbatio theory we treat the wavefuctio as fixed, ad act as if the Hamiltoia operator had varied from some exact Hamiltoia. Aother recurrig theme we ve see this semester is the idea of separatio of variables. We first saw this whe we assumed that time ad space were separable i order to do our derivatio of the Schrödiger equatio. We saw it agai whe we dealt with spherically symmetric potetials (ad the particular case of the hydroge atom.) I that case, we separated the agular equatio from the radial equatio, ad the further separated the φ ad θ depedeces of the agular fuctio. 2 I the cotext of the hydroge atom, we also saw a separatio i which we separated the motio of the atom as a whole (i.e., of its ceter of mass) from the iteral (i this case, electroic) motio. 3 Sice the we have implicitly used this separatio so that we ca igore the traslatioal motio of the molecules we re studyig. Aother way of sayig that is that we are choosig the referece frame of the molecule s ceter of mass (we re travellig with the molecule). We talked about the Bor-Oppeheimer approximatio, which makes the assumptio that electroic ad uclear motios are separable. Whe our separatio is approximate, the key to the approximatio begi good is that the timescales of the two motios have to be very differet. I the case of the Bor-Oppeheimer approximatio, we said that sice the masses were very differet, the typical velocities would also be very differet, ad so the Bor-Oppeheimer approximatio is usually a good oe. Later, we talked about makig the rigid rotor-harmoic oscillator (RRHO) approximatio, which says that the vibratios behave exactly like a harmoic oscillator ad that the rotatios behave exactly like a rigid rotor. This implies that the vibratioal ad rotatioal motios are separable. Agai, we made this approximatio based o the very differet timescales rotatios happe o the order of aosecods, whereas vibratios happe o the order of picosecods. That s my overview of the class. The test will, of course, require more detail tha the above. But I thik that if you uderstad this sectio of the review, the you ll take away the big ideas we really wat you to get out of this class. 1 If you take more advaced quatum mechaics classes, you will lear that time depedece i geeral ca be treated either as part of the wavefuctio or as part of the operator. I this class, we ve put the timedepedece i the wavefuctio, which is called the Schrödiger picture. However, you ca also have a situatio with time-depedet operators ad time-idepedet wavefuctios. This is called the Heiseberg picture. 2 This separatio is exact I metio that because may of our separatios are oly approximatios. 3 This is aother rigorous separatio. All the rest are approximatios. 2

3 2 Iteral motio of molecules As metioed i the overview, we ca rigorously separate the motio of a molecule as a whole (that is, the traslatioal motio of its ceter of mass) from its iteral degrees of freedom (rotatioal, vibratioal, ad electroic). I this sectio, we re goig to talk a little bit about how we classify these degrees of freedom ad how that affects the molecule. 2.1 The umber of each kid of mode Overall, we have 3N degrees of freedom, where N is the umber of atoms i the molecule. The three comes about because each of those atoms ca move i three dimesios (sice our world is, well, three dimesioal). The ceter of mass of the molecule ca move i three dimesios, so separatig the ceter of mass traslatio from the iteral degrees of freedom will use up 3 of our 3N degrees of freedom. We have 3N 3 left. I geeral, a the rotatio i 3D of a molecule ca be described as a combiatio of 3 idividual rotatio axes. So i the case of a geeral (o-liear) molecule, We subtract off aother 3 degrees of freedom, leavig us with 3N 6. For liear molecules, the rotatio is fully described by 2 rotatios, because the molecule has cylcidrical symmetry about its iteruclear axis. No matter how you rotate about that axis (be it 1 degree or 10 degrees or 180 degrees), the molecule does t chage. So that does t cout as a degree of freedom. Liear molecules have 2 degrees of rotatioal freedom, so that leaves us with 3N 5 degrees of freedom i the liear case. All the remaiig degrees of freedom are vibratioal degrees of freedom. For a liear molecule, there are 3N 5 vibratioal degrees of freedom. For a o-liear molecule, there are 3N 6 vibratioal degrees of freedom. 2.2 Rotatioal tops As metioed above, i geeral a molecule has 3 axes of rotatio. That meas that it also has 3 momets of iertia, ad i geeral, these momets of itertia will ot be idetical. Based o the relatios of those momets of iertia, there are 4 classes of compouds, which tell you what kid of top your molecule is: spherical top, oblate symmetric top, prolate symmetric top, ad asymmetric top. Top Rotatioal Costats Examples Spherical I A = I B = I C CH 4, SF 6 Oblate Symmetric I A = I B < I C bezee Prolate Symmetric I A < I B = I C CH 3 Cl Asymmetric I A < I B < I C H 2 O Table 1: Types of tops. 3

4 2.3 Vibratioal modes Molecular vibratios ca be broke dow ito what we call ormal modes. Normal modes are a special way of combiig the atomic motios such that the ceter of mass does t move ad such that o two ormal modes are coupled to each other. Aother way of sayig that is how Atkis describes them: a ormal mode is a idepedet, sychroous motio of atoms or groups of atoms that may be excited without leadig to excitatio of ay other ormal mode. 4 Idetifyig these ormal modes is a problem of liear algebra, ad we have t discussed the details of it i this class. However, we did show you the ormal modes for a few kids of molecules, ad you should be familiar with those. For CO 2, ad similar liear triatomic molecules, there will be four vibratioal ormal modes. However, two of them are equivalet bedig modes, so oly oe bedig mode has bee illustrated i figure 1. Figure 1: Modes for CO 2 : from left to right symmetric stretch, asymmetric stretch, bedig mode. There are two equivalet bedig modes, although oly oe is pictured here For H 2 O, ad similar oliear triatomic molecules, there will be three vibratioal ormal modes: a symmetric stretch, a asymmetric stretch, ad a bedig mode. These are illustrated i figure 2. Figure 2: Modes for H 2 O: from left to right symmetric stretch, asymmetric stretch, bedig mode. 4 Peter Atkis, Physical Chemistry, 6th Ed. 4

5 2.4 Symmetry ad motio (ChemBio) Symmetry Poit Groups I wo t go over the details of idetifyig various symmetry elemets, which you ca fid i the first part of chapter 12 of M&S. However, oce you fid the symmetry elemets, you will wat to assig a poit group to the molecule you re examiig. M&S talk about assigig a poit group based o a eumeratio of all the possible symmetry elemets. Aother techique is to use a flow chart such as the oe we saw i class or the oe that ca be foud i Atkis Physical Chemistry, or i iorgaic chemistry textbooks, such as Miessler & Tarr. The aspects we cosider most importat, ad the oes we thik you should be able to recogize quickly, are the groups i the C ad D classes. To idetify these, you first fid the pricipal axis (the axis with the highest degree of rotatioal symmetry). The degree of that axis will be the value of for your poit group. If the molecule has C 2 axes perpedicular to the pricipal axis, the it is a D group. I ot, it is a C group (these are assumig we do t have a highersymmetry group). I C groups, there are two possibilities: horizotal plaes (perpedicular to the pricipal axis), ad vertical plaes (which iclude the pricipal axis). I D groups, there are horizotal plaes, ad the there are dihedral plaes, which bisect the agles made by the C 2 axes of the D groups. Dihedral plaes are essetially a special type of vertical plae. I ay case, if a molecule has a horizotal plae, the it is a C h or a D h group. If ot, it is a C v group if it has a vertical plae, ad a D d group if it has a dihedral plae. Those are the most importat symmetry poit group assigmets to be able to do, although you should also be able to recogize that a tetrahedral molecule, such as methae, gives a T d group Assigig modes from a character table Each symmetry poit group is associated with a character table. The details of how a character table is derived will be left as a exercise for the serious studet, but as a example, let s look at the C 2v character table: C 2v Ê Ĉ 2 ˆσ v ˆσ v A z x 2, y 2, z 2 A R z xy B x, R y xz B y, R x yz Table 2: The character table of the C 2v poit group. As you ca see, there are four major colums of the character table. The first colum gives what is called the represetatio. The secod colum gives the mathematical details of that represetatio. The third colum will tell us some about what kids of motio are ivolved i each represetatio, as well as whether the molecule is IR-active. Fially, the fourth colum will tell us whether the molecule is Rama-active. Now, for out specific molecule (we ll take water so we ca keep playig with the C 2v group) we see what the effects of each of the operatios will be o the axes for each atom. The ext step is to operate each symmetry operatio o the molecule, ad add together the effects o each axis, accordig to the rules: 5

6 Figure 3: Water molecule with axes for each atom. add 0 if the symmetry operatio moved the origi of the axis add 1 if the axis is left uchaged add -1 if the axis is poitig i the directio opposite of its origial directio So for water, (with ˆσ v as the plae cotaiig all three atoms, ad ˆσ v as the other plae of reflexio) that gives us: C 2v H 2 O Ê 0(0) + 9(1) + 0( 1) = 9 Ĉ 2 6(0) + 1(1) + 2( 1) = 1 ˆσ v 6(0) + 2(1) + 1( 1) = 1 ˆσ v 0(0) + 6(1) + 3( 1) = 3 Now we wat to figure out how may degrees of freedom are associated with each represetatio. To do this, we multiple each of the etries from the table above by the correspodig etries from the character table for the represetatio of iterest. The we multiply that by the multiplicity of that symmetry operatio. I the case of C 2v, those multiplicities are all oe. Whe they are ot, they are oted o the character table (e.g., 2Ĉ3 meas the Ĉ3 axis has a twofold multiplicity). After takig those products of the idividual elemets, you add the results up ad divide by the umber of symmetry operatios. This gives you the umber of modes associated with the give represetatio. As a example, let s look at the A 2 represetatio for water. We multiply values we foud above, (9 113) by the values from the character table for the A 2 represetatio, (11 1 1), ad the by the multiplicities of the symmetry operatios, (1111). This gives us ( ). Addig those together dividig by the total umber of symmetry operatios, we obtai ( )/4 = 4/4 = 1. So there is oe mode i water associated with the A 2 represetatio. From the third colum of the character table, we kow that a A 2 represetatio i a C 2v molecule is always associated with a rotatio. Sice we oly have oe A 2 mode, that s what it is. I geeral, we ca use this process to create table 3 for water: 6

7 Rep. Total Rot Tras Vib IR? Rama? A Yes Yes A B B Yes Yes Table 3: Modes of each represetatio of water, ad partitioig of rotatioal, traslatioal, ad vibratioal modes. Rotatioal states are idetified because they have R x, R y, or R z i the third colum of the character table. Traslatioal states have x, y, or z i the third colum of the character table. The umber of vibratioal degrees of freedom is kow because the total umber of states is the sum of the rotatioal, traslatioal, ad vibratioal degrees of freedom. The last thig we do is idetify which vibratioal modes will be active i IR spectroscopy ad which will be active i Rama spectroscopy. For a mode to be active i vibratioal IR, it has to (1) be a vibratioal mode, ad (2) have a x, y, or z i the third colum of the character table. For a mode to be active for vibratioal Rama spectroscopy, it has to (1) be a vibratioal mode, ad (2) have some product of two elemets of the set {x, y, z} i the fourth colum (i.e., x 2, xy, yz are okay; x 3 y, xyz, or othig, are ot). From that iformatio, we re able to costruct table 3 for water. 3 Nuclear magetic resoace (NMR) We have talked about several types of spectroscopy, which are based o differet kids of excitatios. Rotatioal ad vibratioal spectroscopy excite modes of uclear motio. Electroic spectroscopy excite modes related to molecular orbitals. All of these come from the iteractios of various electric dipole momets. NMR is a little differet. It comes about from flippig the spis of uclei. Sice we re dealig with spis, i this case we re lookig at the magetic dipoles istead of the electric dipoles. The basic trasitio we re lookig at is the switch from spi up ( α ) to spi dow ( β ) or vice versa. To calculate NMR spectra, we re goig to use perturbatio theory. We ll take a situatio such as the oe illustrated i figure 4. First we say that each idividual hydrge ucleus has a certai uperturbed Hamiltoia (Ĥ1 for hydroge 1 ad Ĥ2 for hydroge 2). The there is a couplig betwee the magetic dipoles of the two hydroges, which we ll call Ĥ 12. The eergy of iteractio for a sigle ucleus (our uperturbed Hamiltoia) is give by: Ĥ = γ ˆ B Î = γ(b x Î x + B y Î y + B z Î z ) We ca choose our axes such that magetic field is aliged alog the z-axis. That meas that B x = 0 ad B y = 0. 5 Therefore, Ĥ = γb z Î z (1) 5 Note that Îx ad Îy are ot eigefuctios of our wavefuctios (that is, whe they operate o our wavefuctios, they give a differet wavefuctio back). I fact, they completely flip the wavefuctio, so eve if B x or B y is ot zero, their cotributios will lead to a product of the form k α β with k some costat. Sice the 7

8 Figure 4: Two hydroge atoms we re studyig with NMR. We ca cosider the field felt by the the ucleus, B z as the sum of the exteral field, B 0, ad a deshieldig effect, B elec. We ca represet the dishieldig effect as B elec = σb 0. Puttig all of that together, we get Ĥ = γ(b 0 + B elec )Îz = γ(b 0 σb 0 )Îz = γb 0 (1 σ)îz The iteractio betwee the two magetic dipoles will be proportioal to the dot product of the dipoles. We choose the proportioality costat to be h 2 J 12. The choice of h/ 2 gives J 12 i uits of Hertz. So the perturbatio term i the Hamiltoia will be Ĥ 12 = h 2 J 12Î(1) B As you kow from orgaic chemistry, there are two iterestig features of a NMR peak. The first is its chemical shift, which tells us where o the NMR spectrum it shows up. The secod is the splittig of the peak, which, as you kow, ca tell us importat structural iformatio. I class, we ve leared about the quatum mechaical origis of each of these features. The chemical shift comes from the uperturbed Hamiltoia. We have the uperturbed states α(1)α(2), β(1)α(2), α(1)β(2), ad β(1)β(2). Takig the expectatio value of eergy (of two spi states are orthogoal, that gives zero cotributio to the expectatio values. So really, by desigig our field to lie i the z-directio, we re gettig the maximum efficiecy. Aligig it perpedicular to the z-axis would give o cotributio. Î(2) B 8

9 the uperturbed Hamiltoia operator) for the state α(1)α(2), we fid: H αα = α(1)α(2) γb 0 (1 σ 1 )Î(1) z γb 0 (1 σ 2 )Î(2) z α(1)α(2) ) = γb 0 ( α(1)α(2) (1 σ 1 )Î(1) z α(1)α(2) + α(1)α(2) (1 σ 2 )Î(2) z α(1)α(2) = γb 0 ( α(1)α(2) (1 σ 1) 2 α(1)α(2) + α(1)α(2) (1 σ 2) ) 2 α(1)α(2) = γb 0 2 ((1 σ 1) + (1 σ 2 )) = γb 0 2 (2 σ 1 σ 2 ) Similarly, we obtai the eergies for the other states ad get the results illustrated i figure 5. β(1)β(2) α(1)β(2) 2 γb 0(2 σ 1 σ 2 ) 2 γb 0(σ 1 σ 2 ) β(1)α(2) 2 γb 0(σ 1 σ 2 ) α(1)α(2) 2 γb 0(2 σ 1 σ 2 ) Figure 5: Eergies ad allowed trasitios for NMR calculated from the uperturbed Hamiltoia. This gives us four trasitios, but they come i two sets of two. We ca t tell the differece because i spectroscopy, we measure differeces of eergy betwee states, ot the exact locatios of the states themselves. The resultig spectrum is show i figure 6. γb 0 σ 1 σ 2 E Figure 6: The spectrum of our example, with splittig tured off (i.e., eglectig the perturbatio) 9

10 Oce the chemical shift has bee determied, we tur o the perturbatio of our Hamiltoia from the spi-spi iteractios. We calculate the effect of the perturbatio o each of our eergy states by takig the expectatio value of the perturbatio with each state. For example, the α(1)α(2) state gives us: Ĥ12 = α(1)α(2) h αα 2 J12Î(1) B Î(2) B α(1)α(2) = hj 12 2 α(1)α(2) Î(1) x Î(2) x + Î(1) y Î y (2) + Î(1) z Î z (2) α(1)α(2) = hj ( 12 2 α(1)α(2) Î(1) x Î(2) x α(1)α(2) + α(1)α(2) Î(1) y Î y (2) α(1)α(2) ) + α(1)α(2) Î(1) z Î z (2) α(1)α(2) = hj ( 12 2 α(1)α(2) 2 2 β(1)β(2) + α(1)α(2) i i 2 2 β(1)β(2) ) + α(1)α(2) 2 2 α(1)α(2) = hj ) 12 ( = hj 12 4 After calculatig similar correctios for the other states, we obtai we obtai eergies as illustrated i figure 7, which also shows a compariso with the uperturbed eergy states. β(1)β(2) 2 γb 0(2 σ 1 σ 2 ) + hj12 4 α(1)β(2) 2 γb 0(σ 1 σ 2 ) hj12 4 β(1)α(2) α(1)α(2) 2 γb 0(σ 1 σ 2 ) hj γb 0(2 σ 1 σ 2 ) + hj12 4 Figure 7: Eergy states ad allowed trasitios or NMR calculated with the spi-spi perturbatio term, ad compariso to the uperturbed eergy levels. This splits the trasitio eergies, ad gives us a spectrum as depicted i figure 8. Note that the splittig is idepedet of the exteral field, although the chemical shift does deped o the exteral field. 10

11 J 12 J 12 γb 0 σ 1 σ 2 E Figure 8: The first-order spectrum for our example (with splittig icluded). As you kow from orgaic chemistry, the more deshielded a ucleus is (that is, the fewer electros aroud it), the further it is to the left o the spectrum. You also kow from orgaic chemistry that i geeral we observe a +1 splittig patter i first-order NMR spectra (where is the umber of eighborig uclei). 4 Spectroscopy i Complex Systems (ChemBio) Much of what we talked about it this class was focused o gas phase spectroscopy of small molecules. That is, of course, the simplest way to look at this stuff. However, may problems of iterest either ivolve lorge molecules or take place i codesed phases (e.g., liquids) or both. Studyig these systems ca ivolve some differet techique, ad there are some of the features of gas-phase spectra are t preset i more complex systems. 4.1 Eergy trasfer i the codesed phase Whe we do spectroscopy i the gas phase, the idividual molecules are t iteractig with each other (at least, for the most part). That s o loger the case i the codesed phase, so we have to cosider the possibility that eergy trasfer betwee molecules could affect the spectra we obtai. Collisios with solvet molecules happe o average about oce a picosecod, which is about the same timescale as vibratioal motio. Because of this dissapatio of eergy, vibroic spectroscopy is very difficult i the codesed phase. You ca put your molecule i a excited vibratioal state of a excited electroic state, but the vibratioal eergy will quickly relax due to collisios with the solvet, ad it will fall to the groud vibratioal state of the excited electroic state before fluorescig back dow to the groud electroic state. A similar effect ca also happe i complex molecules due to what is called itramolecular vibratioal rearragemet, i which the distributio of eergy amog the various modes of motio chages. There is also a iterestig effect i codesed phase spectroscopy which allows you to observe chages is the structure of the solvet. If you have a polar solvet (such as water) the the solvet molecules will have preferred orietatios aroud a polar solute. If, as a result of a spectroscopic trasitio, you chage the directio of the dipole momet of the solute, it will take some time for the solvet to readjust. Chagig the dipole is a electroic chage, ad so it happes much faster tha the uclear motios of the solvet (this ca be see as a maifestatio of the Bor-Oppeheimer approximatio). 11

12 4.2 Spectroscopy of Proteis Quechig i Proteis The cocept of fluorescece quechig is actually a very geeral oe, but we talked about it i class i terms of proteis. The basic idea is that you may have a molecule (or a part of a large molecule, such as a protei) that ormally fluoresces. However, by havig a quechig aget ear the fluorescet group o the molecule, we ca cause it to lose eergy more quickly. As we saw i class, the quechig aget could be aother group o the same molecule, or it could be a separate etity i solutio. Quechig is particularly iterestig because certai biologically active molecules are fluorescet or ca have fluorescet groups attached to them (called taggig ). Fluorescece is also rather easy to observe, which makes it a coveiet techique Circular dichroism From orgaic chemistry, you kow that chiral molecules ca chage the orietatio of polarized light. That fact is the basic idea behid the spectroscopic techique kow as circular dichroism. Circular dichroism is usually used o large (geerally biological) molecules. It provides a way to study the secodary structure (α-helices, β-sheets, etc.) of such molecules. By passig liearly polarized light through a quarter-wave plate, we obtai circularly polarized light. Circularly polarized light ca be either right-polarized or left-polarized, ad each will iteract differetly with a chiral sample. My measurig the differece i absorbace betwee the two, ca obtai a spectrum which hold iformatio about the secodary structure. I order to iterpret this spectrum, we compare it to kow samples (i.e., we would use a kow α-helix, a kow β-sheet, ad a kow radom coil sample, ad see if we could make our sample s spectrum by creatig a liear combiatio of those kow spectra). 5 Time-Depedet Perturbatio Theory (ChemPhys) (writte by Shervi Fatehi) The purpose of this little cameo is to hit the high poits of time-depedet perturbatio theory i geeral the expressios that you should kow ad the relatios that you should be comfortable with idepedetly of whether they have to do with light-matter iteractio or some other time-depedet process. Let s jump right i. Suppose that we have a system described by Ĥ = Ĥ0 + Ĥ1(t). The first term is a 0 th order Hamiltoia describig a isolated atom, molecule, or other system of iterest to us, ad we kow the solutios to the associated Schrödiger equatio, Ĥ 0 φ = E φ. The secod term is a 1 st order Hamiltoia that might describe aythig from a electric field to aother particle that s goig to zoom by. Although it cotais time depedece, we explicitly ote that it will ot cotai time derivatives t this depedece is just a parametric oe, such that chagig t chages positio-related quatities. For example, the 1 st order Hamiltoia might just be a polyomial with time-depedet coefficiets. Give that this is the case, we would like to study the possibility that the presece of this time-depedet Hamiltoia will lead to trasitios from statioary state φ i to statioary 12

13 state φ f (as we saw was true for the electric field). This is straightforward as soo as we recall that ay state may be described as a liear combiatio of states formig a complete set (e.g., the solutios to the 0 th Hamiltoia). We may therefore write some geeral state as Ψ(t) = c (t)e it E φ where the expoetial term is time evolutio arisig from Ĥ0 ad the coefficiet c (t) cotais the time depedece arisig from Ĥ0 as well as our iitial coditios (more o this i a momet). Let s pick a particular state, i, to be the iitial state of our system, ad let s say that we choose the time for our system to be some time t i typical choices for this are 0 (if, for example, we have a time-depdet Hamiltoia that ca be switched o at some arbitrary time, like a electromagetic field) or (if we have a time-depedet Hamiltoia that s always o but with varyig stregth, as would be true for a scatterig process). This meas that c (0) = δ i. We d like to kow what the probability of beig i some other state f will be at time t f typical choices here are either just some arbitrary t (if we wat to have a geeral expressio) or (if we just wat to look at the process as a whole over a highly exteded period). I other words, we re lookig for the trasitio amplitude φ f Ψ(t), which we may the relate to a trasitio probability. We ca write this out as φ f Ψ(t) = c (t)e it E φ f φ = c (t)e it E δ f = c f (t)e it E f ad therefore the trasitio probability will be the modulus squared of the coefficiet, c f (t) 2, sice the expoetial cacels out. The problem, the, is to determie those coefficiets. We ca do this by substitutig Ψ(t) ito the time-depedet Schrödiger equatio: i Ψ(t) = Ĥ Ψ(t) [ t ] i c (t)e it E φ = [Ĥ0 Ĥ1(t)] [ ] + c (t)e it E φ t i. ċ (t)e it E φ = c (t)e it EĤ 1(t) φ where the dot o the coefficiet idicates a time derivative. 13

14 We ow have a equatio for the coefficiets. To fid the coefficiet for some particular state f, however, we eed to project oto it usig the shervbra φ f : i ċ (t)e it E φ f φ = c (t)e it φ E f Ĥ 1 (t) φ i ċ (t)e it E δ fi = c (t)e it φ E f Ĥ 1 (t) φ i ċ f (t)e it E f = c (t)e it φ E f Ĥ 1 (t) φ ċ f (t) = 1 c (t)e it (E f E ) φ f Ĥ 1 (t) φ i ċ f (t) = 1 c (t)e iω ft φ f Ĥ 1 (t) φ i where we ve defied the trasitio frequecy ω f E f E. Now all that s left to do is itegrate over [t i, t f ]. tf t i ċ f (t)dt = 1 i c f (t f ) c f (t i ) = 1 i tf t i tf t i c (t)e iω ft φ f Ĥ 1 (t) φ dt c (t)e iω ft φ f Ĥ 1 (t) φ dt c f (t f ) = c f (t i ) + 1 i tf t i c (t)e iω ft φ f Ĥ 1 (t) φ dt Here we ivoke the approximatio that makes this approach a perturbatio theory: we are goig to assume that Ĥ1(t) is a weak eough disturbace to the system that the coefficiets i our expasio are t goig to chage very much over time c i (t) will be roughly 1 for all time, ad c f (t) for ay other state will be close to 0. We ca therefore plug i c (t) c (t i ) = δ i o the right to get c f (t f ) = δ fi + 1 tf i t i c f (t f ) = δ fi + 1 tf e iω fit i t i δ i e iω ft φ f Ĥ 1 (t) φ dt φ f Ĥ 1 (t) φ i dt ad this is our fial result. We ow have the coefficiets to first order i time-depedet perturbatio theory, at least for all times. With that said, you should kow these three relatios for the exam: The coefficiet of state f at time t f, give that the system started i state i at time t i, will be c f (t f ) = δ fi + 1 tf e iω fit φ f Ĥ 1 (t) φ i dt i t i 14

15 The trasitio probability for goig from state i at time t i to state f at time t f will be P fi (t f ) = c f (t f ) 2 = δ fi + 1 i tf The rate of trasitios from i to f at time t f will be t i P fi (t f ) = dp fi dt e iω fit φ f Ĥ 1 (t) φ i dt Take together, these will allow you to tackle ay time-depedet perturbatio theory problem that comes your way. 2 15