Lecture 14 and 15: Algebraic approach to the SHO. 1 Algebraic Solution of the Oscillator 1. 2 Operator manipulation and the spectrum 4

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1 Lecture 14 ad 15: Algebraic approach to the SHO B. Zwiebach April 5, 2016 Cotets 1 Algebraic Solutio of the Oscillator 1 2 Operator maipulatio ad the spectrum 4 1 Algebraic Solutio of the Oscillator We have already see how to calculate eergy eigestates for the simple harmoic oscillator by solvig a secod-order differetial equatio, the time-idepedet Schrödiger equatio. Let us ow try to factorize the harmoic oscillator Hamiltoia. By this we mea, roughly, writig the Hamiltoia as the productof a operator times its Hermitia cojugate. As a first step we rewrite the Hamiltoia as Ĥ = 1 pˆ2 2 2 xˆ ω m Motivated by the idetity a 2 +b 2 = a iba+ib, holdig for umbers a ad b, we examie if the expressio i parethesis ca be writte as a product ipˆ ipˆ 2 pˆ2 i xˆ xˆ+ = xˆ + xp m 2 + ˆˆ pˆxˆ, ω pˆ2 = xˆ2 + m 2 ω2 1, where the extra terms arise because xˆ ad pˆ, as opposed to umbers, do ot commute. We ow defie the right-most factor i the above product to be V: Sice xˆ ad pˆ are Hermitia operators, we the have ipˆ V xˆ+, 1.3 ipˆ V = xˆ, 1.4 ad this is the left-most factor i the product! We ca therefore rewrite 1.2 as ad therefore back i the Hamiltoia 1.1 we fid, 2 pˆ2 xˆ + V, 2 2 = V m ω Ĥ = V V ω

2 This is a factorized form of the Hamiltoia: up to a additive costat E 0, Hˆ is the product of a positive costat times the operator product V V. We ote that the commutator of V ad V is simple ] ipˆ ipˆ ] i i 2 V,V = xˆ+,xˆ = xˆ,pˆ]+ pˆ,xˆ] = This implies that V, V ] = This suggests the defiitio of uit-free operator operators â ad â : â V, â V. 2 Due to the scalig we have â,â ] = The operator â is called aihilatio operator ad â is called a creatio operator. The justificatio for these ames will be see below. From the above defiitios we read the relatios betwee a, ˆ â ad xˆ,pˆ: ipˆ aˆ = xˆ+, â ipˆ = xˆ. 2 The iverse relatios are may times useful as well, xˆ = â+â, 1.12 pˆ = i â ˆ a. 2 While either â or â is hermitia they are hermitia cojugates of each other, the above equatios are cosistet with the hermiticity of xˆ ad pˆ. We ca ow write the Hamiltoia i terms of the â ad â operators. Usig 1.9 we have ad therefore back i 1.6 we get V 2 V = â â, 1.13 Ĥ = ω â â+ 1 = ω Nˆ + 1, Nˆ â 2 2 â ˆ The above form of the Hamiltoia is factorized: up to a additive costat H is the product of a positive costattimes theoperator productâ â. Iherewehave droppedtheidetity operator, which 2

3 is usually uderstood. We have also itroduced the umber operator N. ˆ This is, by costructio, a hermitia operator ad it is, up to a scale ad a additive costat, equal to the Hamiltoia. A eigestate of Ĥ is also a eigestate of Nˆ ad it follows from the above relatio that the respective eigevalues E ad N are related by E = ω N Let us ow show the powerful coclusios that arise from the factorized Hamiltoia. O ay state ψ that is ormalized we have ad movig the â to the first iput, we get Ĥ ψ = ψ,ĥψ = ω ψ,â aψ ˆ ω ψ,ψ, 1.16 Ĥ ψ = ω aψ ˆ,âψ ω 1 2 ω The iequality follows because ay expressio of the form ϕ,ϕ is greater tha or equal to zero. This ˆ shows that for ay eergy eigestate with eergy E: Hψ = Eψ we have Eergy eigestates: E 1 2 ω This importat result about the spectrum followed directly from the factorizatio of the Hamiltoia. But we also get the iformatio required to fid the groud state wavefuctio. The miimum eergy 1 2 ω will be realized for a state ψ if the term aψ ˆ,aψ ˆ i 1.17 vaishes. For this to vaish aψ ˆ must vaish. Therefore, the groud state wavefuctio ϕ 0 must satisfy âϕ 0 = The operator â aihilates the groud state ad this why â is called the aihilatio operator. Usig the defiitio of â i 1.11 ad the positio space represetatio of pˆ, this becomes i d d x+ ϕ 0 x = 0 x+ ϕ 0 x = i dx dx Remarkably, this is a first order differetial equatio for the groud state. Not a secod order equatio, like the Schrödiger equatio that determies the geeral eergy eigestates. This is a dramatic simplificatio afforded by the factorizatio of the Hamiltoia ito a product of first-order differetial operators. The above equatio is rearraged as Solvig this differetial equatio yields dϕ 0 dx = xϕ ϕ x = e 2 0 x2, 1.22 π where we icluded a ormalizatio costat to guaratee that ϕ 0,ϕ 0 = 1. Note that ϕ 0 is ideed a eergy eigestate with eergy E 0 : Ĥϕ 0 = ω â â+ 1 ϕ 0 = 1 ωϕ 0 E = 2 ω

4 Before proceedig with the aalysis of excited states, let us view the properties of factorizatio more geerally. Factorizig a Hamiltoia meas fidig a operator Aˆ such that we ca rewrite the Hamiltoia as A ˆ A up to a additive costat. Here A ˆ is the Hermitia cojugate of A, ˆ a operator that is defied by ψ,â ϕ = Âψ,ϕ We say that we have factorized a Hamiltoia Hˆ if we ca fid a Aˆ for which Ĥ ˆ = A Â + E 0 1, 1.25 where E 0 is a costat with uits of eergy that multiplies the idetity operator. This costat does ot complicate our task of fidig the eigestates of the Hamiltoia, or their eergies: ay eigefuctio of A ˆ Aˆ is a eigefuctio of H. ˆ Two key properties follow from the factorizatio Ay eergy eigestate must have eergy greater tha or equal to E 0. First ote that for a arbitrary ormalized ψx we have ψ,ĥψ = ψ,â Âψ +E 0 ψ,ψ = Âψ,Âψ +E0, 1.26 ˆ Sice the overlap Aψ,Âψ is greater tha or equal to zero, we have show that ψ,ĥψ E If we take ψ ˆ to be a eergy eigestate of eergy E: Hψ = Eψ, the above relatio gives E E This shows, as claimed, that all possible eergies are greater tha or equal to E A wavefuctio ψ 0 that satisfies is a eergy eigestate that saturates the iequality Ideed, Âψ 0 = 0, 1.29 Ĥψ0 = Â Aψ ˆ 0 +E 0 ψ 0 = Aˆ Aψ ˆ 0 +E 0 ψ 0 = E 0 ψ The state ψ 0 satisfyig Aψ ˆ 0 = 0 is the groud state. For covetioal Hamiltoias this is a first order differetial equatio for ψ 0 ad much easier to solve tha the Schrödiger equatio. 2 Operator maipulatio ad the spectrum We have see that all eergy eigestates are eigestates of the Hermitia umber operator N ˆ = â â. This is because H ˆ = ωn ˆ Note that sice aϕ ˆ 0 = 0 we also have Nˆϕ 0 =

5 We ca quickly check that ] Nˆ,â = â a, ˆ â] = â,â]â = â, Nˆ,â ] = 2.2 â a, ˆ â ] = â a, ˆ â ] = â, which we summarize as Nˆ, â ] = â, Nˆ, â ] 2.3 = â. Usig these idetities ad iductio you should be able to show that: ] Nˆ,â k = kâ k, 2.4 Nˆ, â k ] = kâ k. These relatios suggest why Nˆ is called the umber operator. Actig o powers of creatio or aihilatio operators by commutatio it gives the same object multiplied by plus or mius the umber of creatio or aihilatio operators, k i the above. Closely related commutators are also useful: â,â k ] = kâ k â, â k ] = kâ k 1. These commutators are aalogous to pˆ,xˆ k ] ad xˆ,pˆ k ]. We will also make use of the followig Lemma which helps i evaluatios where we have a operator Aˆ that kills a state ψ ad we aim to simplify the actio of AB, ˆˆ where Bˆ is aother operator, actig o ψ. Here is the result If Aψ ˆ = 0, the ABψ ˆ ˆ = A, ˆ B ˆ ] ψ. 2.6 This is easily proved. First ote that ÂBˆ = A,B ˆ ˆ ]+BÂ, ˆ 2.7 as ca be quickly checked expadig the right-had side. It the follows that ÂBˆψ = A,Bˆ ˆ ]+B ˆ ψ = Â, Bˆ]ψ, 2.8 because BÂψ ˆ = Bˆ Aψ ˆ = 0. This is what we wated to show. This is all we eed to kow about commutators ad we ca ow proceed to costruct the states of the harmoic oscillator. Sice â aihilates ϕ 0 cosider actig o the groud state with â. It is clear that â caot also aihilate ϕ 0. If that would happe actig with both sides of the commutator idetity â,â ] = 1 o ϕ 0 would lead to a cotradictio: the left-had side would vaish but the right-had side would ot. Thus cosider the wavefuctio ϕ 1 â ϕ We are goig to show that this is a eergy eigestate. For this purpose we act o it with the umber operator: Nˆϕ 1 = Na ˆˆ ϕ 0 = Nˆ,â ]ϕ 0,

6 where we oted that Nˆ ϕ 0 = 0 ad used Lemma 2.6. Give that N,â ˆ ] = â, we get Nˆϕ 1 = â ϕ 0 = ϕ Thus ϕ 1 is a eigestate of the operator Nˆ with eigevalue N = 1. Sice ϕ 0 has Nˆ eigevalue zero, the effect of actig o ϕ 0 with â was to icrease the eigevalue of the umber operator by oe uit. The operator â is called the creatio operator because it creates a state out of the groud state. Alteratively, it is called the raisig operator, because it raises by oe uit the eigevalue of N. ˆ Sice N = 1 for ϕ 1 it follows that ϕ 1 is a eergy eigestate with eergy E 1 give by It also turs out that ϕ 1 is properly ormalized: 3 E 1 = ω = 2 ω ϕ 1,ϕ 1 = â ϕ0,â ϕ 0 = ϕ 0,ââ ϕ 0, 2.13 where we used the Hermitia cojugatio property to move the â actig o the left iput ito the right iput, where it goes as â = â. We the have ϕ 1,ϕ 1 = ϕ0,ââ ϕ 0 = ϕ 0,a, ˆ â ]ϕ 0 = ϕ 0,ϕ 0 = 1, 2.14 where we used 2.6 i the evaluatio of ââ ψ 0. Ideed the state ϕ 1 is correctly ormalized. Next cosider the state This has Nˆϕ Nˆ 2 = â â ϕ 0 = ϕ â â 2 ϕ Nˆ,â â ] ϕ0 = 2â â ϕ 0 = 2ϕ 2, 2.16 so ϕ 2 is a state with umber N = 2 ad eergy E 2 = 5 2 ω. Is it properly ormalized? We fid ϕ2,ϕ 2 = â â ϕ 0,â â ϕ 0 = ϕ 0,âââ â ϕ a, ˆ 0 = ϕ 0,â â â ] ϕ = ϕ0,2ââ ϕ 0 = ϕ0, ϕ 0 = 2. The properly ormalized wavefuctio is therefore 1 ϕ 2 â â ϕ We ow claim that the -th excited state of the simple harmoic oscillator is Exercise: Verify that this state has Nˆ eigevalue. Exercise: Verify that the state ϕ is properly ormalized. Sice the Nˆ eigevalue of ϕ is, its eergy E is give by 1 1 ϕ â â ϕ = }{ â ϕ. 2.19! {} 0 0! E = ω

7 Sice the various states ϕ are eigestates of a Hermitia operator the Hamiltoia H ˆ with differet eigevalues, they are orthoormal ϕ, ϕ m = δ m, We ow ote that aϕ ˆ is a state with 1 operators â actig o ϕ 0 because the â elimiates oe of the creatio operators i ϕ. Thus we expect aϕ ˆ ϕ 1. We ca make this precise 1 1 âϕ = â â ϕ 0 = â, â ] ϕ0 =!! At this poit we use 2.19 with set equal to 1 ad thus get By the actio of â o ϕ we get Collectig the results, we have 1 a ˆ ϕ ! âϕ = 1!ϕ 1 = ϕ ! â 1 ϕ = â +1 1 ϕ 0 = +1!ϕ +1 = +1ϕ !! âϕ = ϕ 1, â ϕ = ϕ +1. These relatios make it clear that â lowers the umber of ay eergy eigestate by oe uit, except for the vacuum ϕ 0 which it kills. The raisig operator â icreases the umber of ay eigestate by oe uit. Exercise: Calculate the ucertaity x of positio i the -th eergy eigestate. Solutio: By defiitio, x 2 = xˆ2 ϕ xˆ 2 ϕ The expectatio value xˆ vaishes for ay eergy eigestate sice we are itegratig x, which is odd, agaist ϕ x 2, which is always eve. Still, it is istructive to see how this happes explicitly: xˆ ϕ = ϕ,xˆϕ = ϕ, â+â ϕ, 2.27 usig the formula for xˆ i terms of â ad â. The above overlap vaishes because aϕ ˆ ϕ 1 ad â ϕ ϕ +1 ad both ϕ 1 ad ϕ +1 are orthogoal to ϕ. Now we compute the expectatio value of xˆ2 xˆ2 ϕ = ϕ,xˆ2ϕ = ϕ, â+â â+â ϕ = ϕ, ââ+ââ +â â+â â ϕ Sice âaϕ ˆ ϕ 2 ad â â ϕ ϕ +2 ad both ϕ 2 ad ϕ +2 are orthogoal to ϕ, the ââ ad â â terms do ot cotribute. We are left with xˆ2 ϕ = ϕ, ââ +â âϕ

8 At this poit we recogize that â â = Nˆ ad that ââ = â,â ]+â â = 1+N. ˆ As a result x 2 ˆ ϕ = ϕ, 1+2 Nˆ ϕ = We therefore have x 2 = The positio ucertaity grows liearly with the umber. Sarah Geller ad Adrew Turer trascribed Zwiebach s hadwritte otes to create the first LaTeX versio of this documet. 8

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