The Quantum Theory of Atoms and Molecules: The Schrodinger equation. Hilary Term 2008 Dr Grant Ritchie

Transcription

1 e Quanum eory of Aoms and Molecules: e Scrodinger equaion Hilary erm 008 Dr Gran Ricie

2 An equaion for maer waves? De Broglie posulaed a every paricles as an associaed wave of waveleng: / p Wave naure of maer confirmed by elecron diffracion sudies ec see earlier. If maer as wave-like properies en ere mus be a maemaical funcion a is e soluion o a differenial equaion a describes elecrons aoms and molecules. e differenial equaion is called e Scrödinger equaion and is soluion is called e wavefuncion Ψ.

3 e classical wave equaion We ave seen previously a e wave equaion in 1 d is: 1 v Can is be used for maer waves in free space? ry a soluion: e.g. i k e No correc For a free paricle we know a Ep /m.

4 An alernaive. ry a modified wave equaion of e following ype: α is a consan Now ry same soluion as before: e.g. i k e Hence e equaion for maer waves in free space is: i m For i k e en we ave k m wic as e form: KE wavefuncion oal energy wavefuncion

5 e ime-dependen Scrödinger equaion For a paricle in a poenial en p E m and we ave KE PE wavefuncion oal energy wavefuncion i m DSE Poins of noe: 1. e DSE is one of e posulaes of quanum mecanics. oug e SE canno be derived i as been sown o be consisen wi all eperimens.. SE is firs order wi respec o ime cf. classical wave equaion. 3. SE involves e comple number i and so is soluions are essenially comple. is is differen from classical waves were comple numbers are used imply for convenience see laer.

6 e Hamilonian operaor $ % % & ' H m m ˆ ˆ ˆ ˆ m p m H $ $ % & ' ' i p ˆ LHS of DSE can be wrien as: were Ĥ is called e Hamilonian operaor wic is e differenial operaor a represens e oal energy of e paricle. us were e momenum operaor is us sorand for DSE is: i H ˆ

7 Solving e DSE m i Suppose e poenial is independen of ime i.e. en DSE is: LHS involves variaion of ψ wi wile RHS involves variaion of ψ wi. Hence look for a separaed soluion: i m en Now divide by ψ : i m 1 1 LHS depends only upon RHS only on. rue for all and so bo sides mus equal a consan E E separaion consan. us we ave: E m E i 1 1

8 ime-independen Scrödinger equaion Solving e ime equaion: i 1 d d ie ie / d E d Ae is is eacly like a wave e -iω wi E ћω. erefore depends upon e energy E. o find ou wa e energy acually is we mus solve e space par of e problem... e space equaion becomes E or Hˆ E m is is e ime independen Scrödinger equaion ISE. e ISE can be very difficul o solve i depends upon

9 Eigenvalue equaions e Scrödinger Equaion is e form of an Eigenvalue Equaion: H E were Ĥ is e Hamilonian operaor ψ is e wavefuncion and is an eigenfuncion of Ĥ; ˆ ˆ ˆ d H m d E is e oal energy and an eigenvalue of Ĥ. E is jus a consan ˆ Laer in e course we will see a e eigenvalues of an operaor give e possible resuls a can be obained wen e corresponding pysical quaniy is measured.

10 ISE for a free-paricle For a free paricle 0 and ISE is: and as soluions e ik or e E m ik were E k m us e full soluion o e full DSE is: i ± k E / e Corresponds o waves ravelling in eier ± direcion wi: i an angular frequency ω E / ћ E ћ ω ii a wavevecor k me 1/ / ћ p / ћ p / λ WAE-PARICLE DUALIY

11 Inerpreaion of Ψ As menioned previously e DSE as soluions a are inerenly comple Ψ canno be a pysical wave e.g. elecromagneic waves. erefore ow can Ψ relae o real pysical measuremens on a sysem? e Born Inerpreaion Probabiliy of finding a paricle in a small leng d a posiion and ime is equal o * d d P d Ψ*Ψ is real as required for a probabiliy disribuion and is e probabiliy per uni leng or volume in 3d. e Born inerpreaion erefore calls Ψ e probabiliy ampliude Ψ*Ψ P e probabiliy densiy and Ψ*Ψ d e probabiliy.

12 Epecaion values us if we know Ψ a soluion of DSE en knowledge of Ψ*Ψ d allows e average posiion o be calculaed: i P i In e limi a δ 0 en e summaion becomes: i P d $ d e average is also know as e epecaion value and are very imporan in quanum mecanics as ey provide us wi e average values of pysical properies because in many cases precise values canno even in principle be deermined see laer. Similarly P d $ d

13 Normalisaion oal probabiliy of finding a paricle anywere mus be 1: P d $ d 1 is requiremen is known as e Normalisaion condiion. is condiion arises because e SE is linear in Ψ and erefore if Ψ is a soluion of DSE en so is cψ were c is a consan. Hence if original unnormalised wavefuncion is Ψ en e normalisaion inegral is: N $ d And e re-scaled normalised wavefuncion Ψ norm 1/N Ψ. Eample 1: Wa value of N normalises e funcion N - L of 0 L? Eample : Find e probabiliy a a sysem described by e funcion 1/ sin π were 0 1 is found anywere in e inerval

14 Boundary condiions for Ψ In order for ψ o be a soluion of e Scrödinger equaion o represen a pysically observable sysem ψ mus saisfy cerain consrains: 1. Mus be a single-valued funcion of and ;. Mus be normalisable; is implies a e ψ 0 as ; 3. ψ mus be a coninuous funcion of ; 4. e slope of ψ mus be coninuous specifically dψ /d mus be coninuous ecep a poins were poenial is infinie. Ψ Ψ Ψ Ψ

15 Saionary saes Earlier in e lecure we saw a even wen e poenial is independen of ime e wavefuncion sill oscillaes in ime: Soluion o e full DSE is: e ie / Bu probabiliy disribuion is saic: ie / ie / P * e e us a soluion of e ISE is known as a Saionary Sae.

16 Summary i m d P d d * $ 1 d d P DSE: Born inerpreaion: Normalisaion: ISE: E H E m ˆ or / ie e Boundary condiions on wavefuncion: single-valued coninuous normalisable coninuous firs derivaive.

17 I s never as bad as i seems.