d dx where k is a spring constant

Transcription

1 Vorlesug IX Harmoic Oscillator 1 Basic efiitios a properties a classical mechaics Oscillator is efie as a particle subject to a liear force fiel The force F ca be epresse i terms of potetial fuctio V F k V, V where k is a sprig costat k Its geeralizatio to higher imesios is straightforwar F k V, V k

2 1 Basic efiitios a properties a classical mechaics The sprig costat k etermies the oscillator frequecy ω ω k m, k mω The motio of a classical particle is give by t ωt b, p t mωa cos t b a si ω A costats a a b have to be etermie from iitial coitios, p The classical motio is a oscillatio with frequecy iepeet o amplitue Classical turig poits ±a, classically allowe regios

3 1 Basic efiitios a properties a classical mechaics Aimatios 5.1 The Hamiltoia fuctio p H, p + m k b quatum mechaics The Hamiltoia operator h H + m k Which acts o the square-itegrable wave fuctio ψ

4 1 Basic efiitios a properties b quatum mechaics The time evolutio is etermie by the time-epeet Schröiger equatio The solutios are oly bou states, o state ca escape to ifiity c scalig trasformatio of the Hamiltoia For the graphical represetatio we ee ew uits of legth, time, a eergy Cosier the scalig trasformatio λ with λ> The scale fuctio is efie ψ ψ

5 1 Basic efiitios a properties c scalig trasformatio of the Hamiltoia The erivative ψ ψ ψ ψ λ For the seco erivative ψ λ ψ The behavior of the positio operator uer scalig ψ λ λ ψ, ψ ψ

6 1 Basic efiitios a properties c scalig trasformatio of the Hamiltoia The actio of H o the scale fuctio 1 k m H ψ λ λ ψ + h Dimesioless uits Because of the opposite scalig behavior of the two summas we ca choose h h h ω λ λ λ λ m km k m, 1 a ω is the oscillator frequecy

7 1 Basic efiitios a properties Dimesioless uits We efie the ew positio by mω h The isplacemet will be measure i multiples of legth The Schröiger equatio the looks ω Hψ h + ψ

8 1 Basic efiitios a properties Dimesioless uits The equatio ca be further simplifie ih if we scale the time variable ψ, t Hψ, t t t ωt ω ψ, t ψ, t t t The equatio becomes after ivisio by i ψ, t t hω 1 + ψ, t

9 1 Basic efiitios a properties Dimesioless uits Scale the eergy such that E E / hω 1 H + New variables a t are imesioless t ωt mω h

10 Eigefuctio Epasio a Eigevalues of the Hamiltoia The eigevalue problem for the eergy operator 1 H + cosists i fiig all umbers E eigevalues for which the statioary equatio Hψ Eψ has a o-zero square-itegrable solutio Aimatios 5. first low-eergy eigestates

11 Eigefuctio Epasio a Eigevalues of the Hamiltoia Whe a eigevalue is kow together with square-itegrable solutio, a solutio of the time-epeet equatio is give by iet ψ, t e The eergy is give by E +1/ 1 1/ 4 1! H e / Where H is the Hermite polyomial of orer.

12 Eigefuctio Epasio a Eigevalues of the Hamiltoia /!!! 1 j j j j j H 4, 1, 1 1 H H H b Epasio ito eigefuctios c + c, Aimatios 5.9 aig the oscillator eigefuctios

13 Eigefuctio Epasio c Compariso with the classical motio The mai ifferece betwee QM a classical motio 1 The sum kietic a potetial terms caot be arbitrarily small ucertaity relatio classical positio probability t si t t cos 1 si t t t t Fractio of time t per semi-perio spet at reagio classical t ρ

14 Eigefuctio Epasio c Compariso with the classical motio t t ρ Sigularity at classical turig poits a E Quatum-mechaically ρ E 1 t Oscillatio arou classical esity Aimatios 5. compariso betwee the classical a q-mech positio probability

15 3 Solutio of the iitial value problem a the time evolutio i t 1 ψ, t + ψ, t, ψ, t ψ The time evolutio of a arbitrary state follows from that of the eigestates Epaig the iitial state ψ Aimatios iet, t e c ψ iet it / it, t c e e c e

16 3 Solutio of the iitial value problem b perioic time epeece, / / e e c e e t i it it i ψ + i e 1 1 t e t i,, / + ψ ψ t t,, ψ ψ +

17 c Fourier trasform 3 Solutio of the iitial value problem E + 1 k E k k k 1 The motio i the mometum space is equivalet to the motio i the -space apart from the phase factor