Non-Degenerate and Degenerate Time Independent and Time Dependent Perturbation Theory | ECE 6451, Study notes of Electrical and Electronics Engineering

Download Non-Degenerate and Degenerate Time Independent and Time Dependent Perturbation Theory | ECE 6451 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! ECE 6451 - Dr. Alan DoolittleGeorgia Tech Lecture 9 Non-degenerate & Degenerate Time Independent and Time Dependent Perturbation Theory: Reading: Notes and Brennan Chapter 4.1 & 4.2 ECE 6451 - Dr. Alan DoolittleGeorgia Tech Non-degenerate Time Independent Perturbation Theory If the solution to an unperturbed system is known, including Eigenstates, Ψn(0) and Eigen energies, En(0), ... ...then we seek to find the approximate solution for the same system under a slight perturbation (most commonly manifest as a change in the potential of the system). To do this , we expand the Hamiltonian into modified form, Where g is a dimensionless parameter meant to keep track of the degree of “smallness” (we will eventually set g=1, but for now, we keep it) where H’ is the perturbation term in the Hamiltonian. As g→0, then H→Ho, Ψn→Ψn(0) and En→En(0) nnn EH Ψ=Ψ pgHHH += 0 ( ) nnnp EgHH Ψ=Ψ+0 )0()0()0( 0 nnn EH Ψ=Ψ ECE 6451 - Dr. Alan DoolittleGeorgia Tech 1st Order Perturbation Theory Given the term that is 1st order in g, and using the Fundamental Expansion Postulate for Ψn(1), using the basis vectors, Ψj(0) ‘s ( ) 0)0()1()1()0()0()1(0 =Ψ−Ψ−Ψ+Ψ nnnnnpn EEHHg )0()1()0()0()0()0(0 )0( 0 )0( 0 )0()1()0()0()0()0( 0 )0()1()1()0()0()1( 0 )0()1( then, since,But nn j jjnnp j jjj nn nn j jjnnp j jj nnnnnpn j jjn EaEHEa EH EaEHaH EEHH a Ψ+      Ψ=Ψ+      Ψ Ψ=Ψ Ψ+      Ψ=Ψ+      Ψ Ψ+Ψ=Ψ+Ψ Ψ=Ψ ∑∑ ∑∑ ∑ ECE 6451 - Dr. Alan DoolittleGeorgia Tech 1st Order Perturbation Theory Cont’d, Consider the case when m=n, )0()0()1()0()0()0(0 )0(*)0()1()0()0(*)0(0 )0()0( )0(*)0()1()0(*)0()0()0(*)0()0(*)0(0 *)0( )0()1()0()0()0()0(0 or j,m unless toorthogonal is fact that theUsing space allover integrate and by multiply will we,many times done have weAs nmnmnnpmmm nmnmnnpmmm jm nmn j jmjnnpm j jmjj m nn j jjnnp j jjj EaEHEa dvEaEdvHEa dvEdvaEdvHdvEa EaEHEa ΨΨ+=ΨΨ+ ΨΨ+=ΨΨ+ =ΨΨ ΨΨ+      ΨΨ=ΨΨ+      ΨΨ Ψ Ψ+      Ψ=Ψ+      Ψ ∫∫ ∫∫ ∑∫∫ ∑ ∑∑ )0()0( )0()0( )1( )0()0()1()0()0()0(0 )0()0()1()0()0()0(0 nn npn n nnnmmnpnmm nmnmnnpmmm H E EaEHEa EaEHEa ΨΨ ΨΨ = ΨΨ+=ΨΨ+ ⇓ ΨΨ+=ΨΨ+ ECE 6451 - Dr. Alan DoolittleGeorgia Tech 1st Order Perturbation Theory To find the coefficient, am’s consider the case where m≠n, ( ) ( ) nmexcept n m, allfor validis which 0 )0()0( )0()0( )0()0()0()0( )0()0()1()0()0()0(0 = − ΨΨ = +−=ΨΨ ΨΨ+=ΨΨ+ mn npm m mmnnpm nmnmnnpmmm EE H a aEEH EaEHEa ECE 6451 - Dr. Alan DoolittleGeorgia Tech 1st Order Perturbation Theory Things to consider: 1. To calculate the perturbed nth state wavefunction, all other unperturbed wavefunctions must be known. 2. Since the denominator is the difference in the energy of the unperturbed nth energy and all other unperturbed energies, only those energies close to the unperturbed nth energy significantly contribute to the 1st order correction to the wavefunction. 3. “g” can be set equal to 1 for convenience or rigidly determined by the normalization condition on Ψn. ( ) )0( )0()0( )0()0( )1( )1()0( , m nm mn npm n nnn EE H where g Ψ − ΨΨ =Ψ Ψ+Ψ=Ψ ∑ ≠ )0()0( )0()0( )1( )1()0( , nn npn n nnn H E where gEEE ΨΨ ΨΨ = += ECE 6451 - Dr. Alan DoolittleGeorgia Tech 2nd Order Perturbation Theory Given the term that is 2nd order in g, and using the Fundamental Expansion Postulate for Ψn(2), using the basis vectors, Ψj(0) ‘s, ( ) 0)0()2()1()1()2()0()1()2(02 =Ψ−Ψ−Ψ−Ψ+Ψ+ nnnnnnnpn EEEHHg nmnnmnmjpm j jmm mjjmnn nmn j jmjn j jmjn j jjpm j jjm m nn j jjn j jjn j jjp j jj nnnnnnnpn j jjn EEaEbdvHaEb dvEH dvEdvaEdvbE dvaHdvbH EaEbEaHbH EEEHH b δ δ )2()1()0()0(*)0()0( )0(*)0()0( 0 )0( 0 )0(*)0()2()0(*)0()1()0(*)0()0( )0(*)0()0( 0 *)0( *)0( )0()2()0()1()0()0()0()0( 0 )0()2()1()1()2()0()1()2( 0 )0()2( then, and since,But space, allover gintegratin and by gmultiplyin Again, 0 ++=ΨΨ+ =ΨΨΨ=Ψ ΨΨ+      ΨΨ+      ΨΨ =      ΨΨ+      ΨΨ Ψ Ψ+      Ψ+      Ψ=      Ψ+      Ψ =Ψ−Ψ−Ψ−Ψ+Ψ Ψ=Ψ ∫∑ ∫ ∫∫ ∑∫ ∑ ∫ ∑∫ ∑ ∑∑∑∑ ∑ L L ECE 6451 - Dr. Alan DoolittleGeorgia Tech 2nd Order Perturbation Theory Cont’d, To find the 2nd order energy correction, consider the case of m=n, nmnnmnmjpm j jmm nmnnmnmjpm j jmm EEaEbHaEb EEaEbdvHaEb δ δ )2()1()0()0()0()0( )2()1()0()0(*)0()0( or ++=ΨΨ+ ++=ΨΨ+ ∑ ∫∑ ( ) ( )∑ ∑ ∑ ∑ ∑ ∑ ≠ ≠ ≠ ≠ ≠           − ΨΨ = ΨΨ         − ΨΨ = ΨΨ= ΨΨ−ΨΨ+ΨΨ= −ΨΨ+ΨΨ= = −ΨΨ= nj jn jpn n nj jpm mn npm n nj jpmjn npnnnpnn nj jpmjn n nnnpnn nj jpmjn nnjpm j jn EE H E H EE H E a HaE HaHaHaE E EaHaHaE EaHaE )0()0( 2)0()0( )2( )0()0( )0()0( )0()0( )2( j )0()0()2( )0()0()0()0()0()0()2( )1( )1()0()0()0()0()2( )1()0()0()2( solution,order 1st thefrom for result theInserting solution,order 1st thefrom for result theInserting term,nm thesummation theofout pullingor ECE 6451 - Dr. Alan DoolittleGeorgia Tech 2nd Order Perturbation Theory Cont’d: ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( ) )0( 2)0()0( 2)0()0( )0()0()0()0( )0()0()0()0( )0()0()0()0( )0()0()0()0( 2 )0( )0()0( )0()0( )0( )0(2)0()0( )2(2)1()0( 2)0()0( 2)0()0( )0()0()0()0( )0()0()0()0( )0()0()0()0( )0()0()0()0( 2 1 becomes, equation, original theThus, 2 1 m m nj jn npj mnmn npnnpm nj mnjn jpmnpj m nm mn npm nn m m mm m mnn nnnn nj jn npj mnmn npnnpm nj mnjn jpmnpj m EE H EEEE HH EEEE HH g EE H g bgag gg EE H EEEE HH EEEE HH b Ψ         − ΨΨ −         −− ΨΨΨΨ −         −− ΨΨΨΨ +Ψ − ΨΨ +Ψ=Ψ Ψ+Ψ+Ψ=Ψ Ψ+Ψ+Ψ=Ψ − ΨΨ −         −− ΨΨΨΨ −         −− ΨΨΨΨ = ∑ ∑∑ ∑ ∑∑ ∑∑ ≠≠ ≠ ≠≠ L L ECE 6451 - Dr. Alan DoolittleGeorgia Tech 2nd Order Perturbation Theory In Summary: ( ) ( ) ( )( ) ( )( ) ( )( ) ( ) )0( 2)0()0( 2)0()0( )0()0()0()0( )0()0()0()0( )0()0()0()0( )0()0()0()0( 2 )0( )0()0( )0()0( )0( 2 1 m m nj jn npj mnmn npnnpm nj mnjn jpmnpj m nm mn npm nn EE H EEEE HH EEEE HH g EE H g Ψ         − ΨΨ −         −− ΨΨΨΨ −         −− ΨΨΨΨ +Ψ − ΨΨ +Ψ=Ψ ∑ ∑∑ ∑ ≠≠ ≠ L L ( )∑≠          − ΨΨ +ΨΨ+= nj jn jpn npnnn EE H gHgEE )0()0( 2)0()0( 2)0()0()0( ECE 6451 - Dr. Alan DoolittleGeorgia Tech Perturbation Theory Consider an important and illustrative example: Small electric field applied to an infinite potential barrier quantum well. What effect does this have on the ground state energy? Ψ(0)n=1 E(0)n=1 Ψn=1 En=1 E(0)n=1 When the electric field is applied, the energy bands bend, resulting a redistribution of the electron in the well toward the right side. Since the energy on this side of the well is lower, the new energy of the ground state is expected to be smaller than the unperturbed ground state. We previously solved this as an asymmetric solution (0<x<L) and had states that only depended on sin functions. For reasons that will become obvious, we redefine our limits as symmetric (- ½ a <x< + ½ a) which will require wave function solutions of the form: -½a +½a0 -½a +½a0Unperturbed Well Perturbed Well ( ) 2 222 (0) 1m )0()0( 2 1E odd mfor cos2 and even mfor sin2 ma m a xm aa xm a mm hπ ππ = =      =Ψ     =Ψ = ECE 6451 - Dr. Alan DoolittleGeorgia Tech Perturbation Theory Consider the Odd indexes in the summation: ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 0cos21cos2 odd, jfor Consider 2 2 * )0()0( 1 )0( 2 2 *)0( 1 )0()0( 1 )0()0( 1 2)0()0( 1)2( )0()0( 1 2)0()0( 1)2( =           −            ==Ψ−Ψ Ψ−Ψ=Ψ−Ψ           − Ψ−Ψ =           − ΨΨ = ∫ ∫ ∑ ∑ + − = + − == ≠ = = ≠ = = dx a xj a xq a xn a xq dxxqxq EE xq E EE H E o a a jon jo a a njon nj jn jon n nj jn jpn n πεπε εε ε ECE 6451 - Dr. Alan DoolittleGeorgia Tech Perturbation Theory Consider the Even indexes in the summation: ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               +       + +      −       − − =Ψ−Ψ =           −            ==Ψ−Ψ Ψ−Ψ=Ψ−Ψ           − Ψ−Ψ =           − ΨΨ = = + − = + − == ≠ = = ≠ = = ∫ ∫ ∑ ∑ 2 1sin 12 1sin 1 2 0sin21cos2 even, jfor Consider 22 )0()0( 1 2 2 * )0()0( 1 )0( 2 2 *)0( 1 )0()0( 1 )0()0( 1 2)0()0( 1)2( )0()0( 1 2)0()0( 1)2( π π π π εε πεπε εε ε j j aj j a a qxq dx a xj a xq a xn a xq dxxqxq EE xq E EE H E o jon o a a jon jo a a njon nj jn jon n nj jn jpn n So the Even indexes in the summation contribute non-zero values! ECE 6451 - Dr. Alan DoolittleGeorgia Tech Perturbation Theory Thus, the correction term for Even indexes is: ( ) ( ) ( ) ( ) ( ) ( )∑ ∑                       − =               +       + +      −       − − =           − ΨΨ = = = odd is j 2 222 2 222 222 )2( odd is j )0()0( 1 2)0()0( 1)2( 22 1 2 1sin 12 1sin 1 2 ma j ma n j j aj j a a q E EE H E o n jn jpn n hh ππ π π π π ε Technically, this is an infinite summation. However, since the denominator increases proportionally to j2, the relative weight of higher order j terms rapidly decreases and the solution converges after just a few terms. Note: Since the denominator is always negative and the numerator is always positive, the 2nd order correction is always negative resulting in a lower energy than the unperturbed ground state as expected. I.e., ( ) ( ) ( )Number Negative0 2)0( 11 )0()0( 1 2)0()0( 12)0( 1 )0( 1 )0( 11 ggEE EE H gHgEE nn nj jn jpn npnnn ++=           − ΨΨ +ΨΨ+= == ≠ = = ==== ∑ ECE 6451 - Dr. Alan DoolittleGeorgia Tech h ..., 3, 2, 1,m any value be can m e wher)0(, =Ψ mn Degenerate Perturbation Theory Lets assume that “h” various unperturbed states result in the same (degenerate) energies. Any specific one of these states, say the mth state can be written as: n still represents the state of the electron while m represents which of the h degenerate states are being considered. Since each of the above unperturbed states is an Eigenfunction of Ho, with h degenerate Eigen energies, En(0), any linear combination of these degenerate basis states is also an Eigenfunction of Ho (see Brennan Chapter 1). Thus, we can construct a new set of basis sets, labeled α, that are linear combinations of the degenerate Eigenstates. h ..., 3, 2, 1,m where 1 )0( , )0( =Ψ=Ψ ∑ = h m mnmn b αα The task is simply to find the appropriate values of the coefficients, bmα. To do this, we simply find the set of bmα’s that force the numerator, to equal zero when the denominator is also zero, thus removing the singularity. ( ) ( ) ( ) ( ) )0( )0()0( )0(*)0( )1( )0( )0()0( )0()0( )1( m nm mn npm n m nm mn npm n EE dxH EE H Ψ − ΨΨ =Ψ Ψ − ΨΨ =Ψ ∑ ∫ ∑ ≠ ≠ α α ECE 6451 - Dr. Alan DoolittleGeorgia Tech                       Ψ Ψ Ψ Ψ Ψ                       =Ψ += == == == )0( )0( 1 )0( , )0( 2,2 )0( 1,1 1 1 1111 111111 n hn hmhn mn mn nnn hhh nh p HH HH HH HHHH H M M LLLLL MOM MOM M MOM M LLL Degenerate Perturbation Theory The problem reduces to a diagonalization of the sub-matrix of index h by defining new basis sets such that,                       Ψ Ψ Ψ Ψ Ψ                       =Ψ += == == == + + )0( )0( 1 )0( , )0( 2,2 )0( 1,1 1 1)1( 11 1)1(111 00 0 000 00 n hn hmhn mn mn nnn h hh nh p HH H H H HHH H M M LLLLL MOM MO ML MMOM M LL α α α ∑ Ψ=Ψ i iic ECE 6451 - Dr. Alan DoolittleGeorgia Tech Time Dependent Perturbation Theory In all previous cases, nothing observable happens because the states are assumed static – unchanging. Most useful systems require transitions between states. For example, optical absorption and electron-hole pair recombination require a change from one state to another. This inherently requires TIME DEPENDENT Perturbation Theory. Ψ(0)n=1 E(0)n=1 Ψn=1 En=1 E(0)n=1 E(0)n=1 Ψn=1 En=1 -½a +½a0 -½a +½a0-½a +½a0 Given wave functions are sluggish in nature (waves do not change instantly when the perturbation changes – consider a water wave as an analogy), an instantaneous change in the perturbation results is a “more” gradual time change in the wave function and thus the distribution of particles. Note, the expectation value of the total potential energy is assumed to change instantly as the perturbation energy changes instantly but the expectation value of the kinetic energy changes “gradually”. Time t<0 Time t=0 Time t>>0 Note: terms like “gradually” and “sluggish” are somewhat misleading in that these changes can often happen in fractions of a nanosecond. However, compared to the instantaneous perturbation these changes are considered “slower”. ECE 6451 - Dr. Alan DoolittleGeorgia Tech Time Dependent Perturbation Theory Important Observations: Since the weighting coefficients are known before the perturbation, knowing the time derivative (rate of change) of the coefficients implies full knowledge of the weighting coefficients after the perturbation occurs. ∑∑ ∫ =∂ ∂ = ∂ ∂ n nm n nm txtxtatadvtxtxtata ),((t)H'),()( i-g)( t or ),((t)H'),()(i-g)( t (0) n (0) m (0) n (0) m φφφφ hh Each individual state is “connected” through the perturbation Hamiltonian. I.e. state “m” is connected to every other state (all n states) via the perturbation Hamiltonian. If the Hamiltonian does not allow a transition from one state to another (i.e. the matrix element is zero) then the weighting coefficient for that state remains unchanged (i.e. static in time). The change in each individual coefficient in time depends on couplings between ALL other available states! ECE 6451 - Dr. Alan DoolittleGeorgia Tech Time Dependent Perturbation Theory Consider an expanded form of the previous equation: MMMM LL h MMMM LL h LL h h = ++++= ∂ ∂ − = ++++= ∂ ∂ − ++++= ∂ ∂ − ⇓ = ∂ ∂ ========== ========== ========== ∑ ),((t)H'),()(),((t)H'),()(),((t)H'),()()( t ),((t)H'),()(),((t)H'),()(),((t)H'),()()( t ),((t)H'),()(),((t)H'),()(),((t)H'),()()( t ),((t)H'),()(i-g)( t (0) jn (0) m (0) 1n (0) m1 (0) 0n (0) jm0 (0) jn (0) 1m (0) 1n (0) 1m1 (0) 0n (0) 1m01 (0) jn (0) 0m (0) 1n (0) 0m1 (0) 0n (0) 0m00 (0) n (0) m txtxtatxtxtatxtxtata gi txtxtatxtxtatxtxtata gi txtxtatxtxtatxtxtata gi txtxtata jjnjnnjm jnnnm jnnnm n nm φφφφφφ φφφφφφ φφφφφφ φφ Considering the vastness of this system of equations a simplification similar to what we did for the unperturbed system is in order. Since the wave function evolution in time is entirely determined by the coefficients, we will expand the coefficients in orders of g. L L +++= +++= ''2')0( 2 2 2)0( jjjj jj jj aggaaa or dg ad g dg da gaa ECE 6451 - Dr. Alan DoolittleGeorgia Tech Time Dependent Perturbation Theory Using this expansion, the previous system of equations becomes: For our purposes we will only consider 1st order time dependent perturbation theory (THANK GOD!). For this we will consider only the linear terms in g. Again, for any choice of g, the terms on the left hand side must equate to the terms of equal magnitude on the right hand side. This is done on the following page... ( ) ( ) ( ) ( ) LLL LLLL h ),((t)H'),( ),((t)H'),(),((t)H'),( t (0) n (0) jm ''2')0( (0) 1n (0) jm '' 1 2' 1 )0( 1 (0) 0n (0) jm '' 0 2' 0 )0( 0 ''2')0( txtxaggaag txtxaggaagtxtxaggaagaggaa i jjjj jjj == ==== ++++ ++++++++=+++ ∂ ∂ − φφ φφφφ ( ) ( ) ( ) ( ) LLL LLLL h ),((t)H'),( ),((t)H'),(),((t)H'),( t (0) n (0) 0m ''2')0( (0) 1n (0) 0m '' 1 2' 1 )0( 1 (0) 0n (0) 0m '' 0 2' 0 )0( 0 '' 0 2' 0 )0( 0 txtxaggaag txtxaggaagtxtxaggaagaggaa i jjjj == ==== ++++ ++++++++=+++ ∂ ∂ − φφ φφφφ ( ) ( ) ( ) ( ) LLL LLLL h ),((t)H'),( ),((t)H'),(),((t)H'),( t (0) n (0) 1m ''2')0( (0) 1n (0) 1m '' 1 2' 1 )0( 1 (0) 0n (0) 1m '' 0 2' 0 )0( 0 '' 1 2' 1 )0( 1 txtxaggaag txtxaggaagtxtxaggaagaggaa i jjjj == ==== ++++ ++++++++=+++ ∂ ∂ − φφ φφφφ MM MM ECE 6451 - Dr. Alan DoolittleGeorgia Tech Time Dependent Perturbation Theory ( ) jmbut 2, 1, 0,mfor )(H')(1 t (0) n (0) m ' ≠=ΨΨ      −= ∂ ∂ = L h ti jm mjexxia ω t=0 t H’ H’(t)=H’u(t) Thus, this general equation, can be directly integrated to result in: 0 general, inbut ion wavefunct totalthe of ionnormalizat throughfound be can 0,for t and, 0 thenaa 0for t Since ' ' '(0) jj ≈ > ==≤ j j j a a a ( ) jmbut 2, 1, 0,mfor u(t) K )(H')(11 (0)n(0)m' ≠=         +ΨΨ        − = = L h ti j mj m mjexxta ω ω The integration constant can be evaluated by restricting a’m(t)=0 at t=0 resulting in, ( ) ( ) jmbut 2, 1, 0,mfor u(t)1 - )(H')(1 (0)n(0)m' ≠=         ΨΨ−= = L h mj ti jm mjexxta ω ω Finally, since the perturbation is defined as small, the jth coefficient is simply, ECE 6451 - Dr. Alan DoolittleGeorgia Tech Time Dependent Perturbation Theory Important Observations: 1) The matrix element in the above expression ( <...> term) is known as the transition matrix element and describes which transitions are allowed and disallowed and how strongly the mth and jth states are coupled. 2) If the transition matrix element integrand is odd (a very common occurrence) the integral is 0 meaning that a transition between the mth and jth state is forbidden. 3) In general, this transition matrix element is responsible for a variety of “selection rules” in atomic, nuclear and semiconductor bulk/quantum well optical spectra (emission or absorption resulting from electron/hole transitions between states). 4) Even though a specific transition is forbidden from say the jth state to the mth state, the mth state may still eventually become populated by indirect (and thus slower) transitions of the form j k m, etc... ( ) ( ) jmbut 2, 1, 0,mfor u(t)1 - )(H')(1 (0)n(0)m' ≠=         ΨΨ−= = L h mj ti jm mjexxta ω ω ECE 6451 - Dr. Alan DoolittleGeorgia Tech Time Dependent Perturbation Theory ( ) ( ) ( ) ( ) ( ) jmbut 2, 1, 0,mfor ee)(A(x))( t jmbut 2, 1, 0,mfor )(u(t)eeA(x))( t jmbut 2, 1, 0,mfor )((t)H')(1 t tiωtiω(0) n (0) m ' (0) n tiωtiω(0) m ' (0) n (0) m ' oo oo ≠=+ΨΨ      −= ∂ ∂ ≠=Ψ+Ψ      −= ∂ ∂ ≠=ΨΨ      −= ∂ ∂ − = = − = L h L h L h ti jm ti jm ti jm mj mj mj exxia exxia exxia ω ω ω t=0 t Example 2: Turning on a “Harmonic Perturbation” at time t=0. This case is important as many excitations such as electromagnetic radiation, ac electric and magnetic fields all can be periodic in nature. This problem proceeds identical to the previous example except: can be directly integrated to result in: 0for t 0 and 0for t 0 '' >≈≤= jj aa Where the integration constant was again evaluated by restricting a’m(t)=0 at t=0. Finally, since the perturbation is defined as small, the jth coefficient is simply, ( )u(t)eeA(x)(t)H' tiωtiω oo −+= ( ) ( )( ) ( ) ( )( ) ( ) jmbut 2, 1, 0,mfor u(t)ω 1- ω 1-)(A(x))( o tω o tω(0) n (0) m' oo ≠=         − − +        ΨΨ− = −+ = L h mj i mj i j m mjmj eexxta ωω ωω ECE 6451 - Dr. Alan DoolittleGeorgia Tech Lo g{ (a ’ m (t) )* (a ’ m (t) )} o (0) j (0) m ωEE h−≈ t≤0 t>0 t>>0 o (0) j (0) m ωEE h+≈ (0) jE The probability of all states including “far off-resonant” states becoming occupied increases with time. ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) o (0) j (0) m2 omj omj 2 2 2(0) jn (0) m' m *' m o (0) j (0) m2 omj omj 2 2 2(0) jn (0) m' m *' m ωEfor E u(t) ωω tωω 2 1sin(x)ΨA(x)(x)Ψ 4tata ωEfor E u(t) ωω tωω 2 1sin(x)ΨA(x)(x)Ψ 4tata h h h h −≈             +       +           ≈ +≈             −       −           ≈ = = and Time Dependent Perturbation Theory ECE 6451 - Dr. Alan DoolittleGeorgia Tech Instead of discrete state to state transitions, it is often useful to consider a discrete state to “band of states” transition. Examples of this are donor states to conduction band transitions, acceptor states to valence band transitions or simply defect/impurity states to either conduction/valence band transitions. These bands can be described by their density per unit energy (see lecture 7) or since E=ħω, the density (number) per frequency ω centered around the transition frequency = ρ(ωmj). Assuming that this density of states, ρ(ωmj), does not change quickly with ωmj, we can find the new probability density of state m by integrating the previous expression over ωmj. For example, Time Dependent Perturbation Theory ( )( ) ( )( ) ( ) ( ) ( ) o(0)j(0)mmjmj2 omj omj 2 2 2(0) jn (0) m' m *' m ωEfor E ωω 2 1 tωω 2 1sin(x)ΨA(x)(x)Ψ tata h h +≈                 −       −           ≈ ∫ = ωωρ d ECE 6451 - Dr. Alan DoolittleGeorgia Tech Making some assumptions about this function: 1) The transition matrix element is a slowly varying function of ωmj. 2) The density of states, ρ(ωmj), also does not change quickly with ωmj. 3) Both of the above two conditions can be achieved by noting that since the [sin2...] function sharpens in in ωmj with increasing time, we can always wait long enough in time to make this part of the integrand the most rapidly varying portion of the integrand in ωmj. ... Time Dependent Perturbation Theory ( )( ) ( )( ) ( ) ( ) ( ) o(0)j(0)mmjmj2 omj omj 2 2 2(0) jn (0) m' m *' m ωEfor E ωω 2 1 tωω 2 1sin(x)ΨA(x)(x)Ψ tata h h +≈                 −       −           ≈ ∫ = ωωρ d