Lecture #5. Questions you will by able to answer by the end of today s lecture

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Brooke Clark
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1 Today s Program: Lecture #5 1. Review: Fourth postulate discrete spectrum. Fourth postulate cotiuous spectrum 3. Fifth postulate ad discussio of implicatios to time evolutio 4. Average quatities 5. Positio eigevectors 6. Physical iterpretatio of ψ ( ) 7. Commutatio relatios 8. Commutig observables 9. Commutatio of X ad P operators 1. Physical implicatios simultaeous measuremets Questios you will by able to aswer by the ed of today s lecture 1. What are the positio observables eigefuctios ad eigevalues?. Give a wave fuctio for a particle what is the probability of fidig the particle as a fuctio of spatial coordiates. 3. Why ca t we measure the positio ad mometum simultaeously? 4. kow how to check if two operators commute. 5. Why ca t we measure the positio ad mometum simultaeously? Math tools covered today 1. The Dirac Delta fuctio (see hadout).. Commutatio of operators. 3. Properties of operators that commute (commo eigefuctios)

2 Review of last lecture: Eample from vector space: We have a state: ψ = 1 We d like to predict the result of a physical measuremet (eergy) associated with the followig operator: ˆ 1 H = 1 What are the possible outcomes of the eergy measuremet? 1 1 λ1 = 1, u1 = λ = 3, u = 1 P ψ = u1 ψ u1 + u ψ u = c1 c u1 u The probability of obtaiig the measuremet = = + = 1 1 ( λ ) u ψ ( 1, 1) ( 1, 1) 1 1 u1 u1 u

3 Fourth Postulate (cotiuous o-degeerate): Whe the physical quatity a is measured o a system i the ormalized state ψ ( rt, ) the probability ( ) a result betwee α a+ dα is where u ( ) ( ) = ( ) ( ) dp α u ψ dα α dp a of obtaiig α is the ormalized eigevector of A associated with the eigevalue α. Fifth Postulate (discrete o-degeerate): If the measuremet of a physical quatity a o the system i the state ( rt, ) a the state of the system immediately after the measuremet is u ( ). result ψ gives the u ( t, ) ψ ( t, 1) ψ (,) ψ ( t, ) Cosequeces: (i) The state of the system right after a measuremet is always a eigevector correspodig to the specific eigevalue which was the result of the measuremet. (ii) The state of the system is fudametally perturbed by the measuremet process.

4 Average quatities (or what would be the average result of physical measuremet performed over may idetical states) The average of a physical quatity represeted by Â of a system i state ψ is give by- ψ Â ψ A eample: ψ Hˆ ψ = ψ = c u + c u + c u c u where, Hu ˆ = E u ψ ˆ H ψ ( c1 u1 c u c3 u3... c4 u4 ) Hˆ ( c1 u1 c u c3 u3... c u ) ( c1 u1 c u c3 u3... c4 u4 )( ce 1 1 u1 ce u ce 3 3 u3... ce u ) = cce + cce + cce + cce = c E + c = = = = ( ) ( ) ( )... ( ) = Hˆ = E E c E c E = P E E + P E E + P E E + + P E E Eample What are the positio observable s X eigevalues ad eigefuctios i the represetatio? how ca we capture the fact that our particle is at a particular positio? Look back at the defiitio of eigevectors of a operator. ˆXu = u we also kow that the defiitio of the positio operator (observable) is: where the Dirac δ fuctio is defied as: ( ) = ( ) ( ) = ( ) Xˆ ψ ψ u δ ( ) ( ) d= ( ) δ ψ ψ Discuss the physical reasoig behid the choice of the dirac delta fuctio.

5 What are the mometum observables P eigevalues ad eigefuctios i the represetatio? The free particles solutio ( ) = pu( ) ˆPu ħ u i ( ) = pu( ) ik e is a mometum eigestate sice: ħ e = ħ ke = pe i p ik ik ik This leads to the physical iterpretatio of the wavefuctio squared. Whe the positio (correspodig to the observable X) is measured o a system i the ormalized state ψ ( rt, ) the probability dp( ) + dis of obtaiig a result betwee ( ) = ( ) ψ ( ) = δ ( ) ψ ( ) = ψ ( ) dp u d d d which is the probability desity. Commutatio Defiitio: Two observables commute if AB ˆ ˆ = BA ˆˆ AB ˆ, ˆ AB ˆ ˆ BA ˆˆ = Do X ad P commute? Xˆ, Pˆ ħ ħ ψ ( ) = ψ( ) ( ψ( ) ) i i ħ ħ ħ = ψ ψ ψ i i i = iħψ ( ) Xˆ, Pˆ = iħ ( ) ( ) ( )

6 Fudametal theorem (from algebra): If two operators A ad B commute oe ca costruct a basis of the state space with eigevectors commo to A ad B Importat Cosequeces: A. Oe caot measure X, P simultaeously! Eplaatio: Oce oe performs a positio measuremet the system has to be i a positio eigestate. Let s try to ratioalize this by a eample pertaiig to the ifiite well: I this eample the particle is goig to be described by a localized wave fuctio f() described below cetered aroud ad of width f ( ) 1 Note: check if this fuctio satisfies the defiitio of the delta fuctio at the limit δ ( ) ( ) ( ) δ ( ) f d f d The eergy eigefuctios are of the followig form: u k ( ) = 1 by costructio π odd cosk = cos d d d = π eve sik = si d d d me π = = ħ d

7 For simplicity we will assume that the particle is located aroud the origi, ow we ca epad the wavefuctio i terms of eergy eigefuctios: ( ) f = c = 1 π cos d d 1 π π c = f ( ) cos = cos d d d d d 1 d π π d d = si = si = π d d d d A parameter which represets the degree of cofiemet is: let us ow cosider two limitig cases d = 1 low cofiemet, ad << 1 high cofiemet d d π si π d B. The effects of symmetry o the properties of the eigefuctios. Defie a parity operator: ( ) ψ ( ) Π ˆ ψ = Fid the eigevalues ad eigevectors of this operator: ( ) ( ) ( ) ( ) ( ) Π ˆ u = λu ΠΠ ˆˆ u = λλu = u λ =± 1 The eigevectors of the parity operator all have defiite parity either odd or eve. ( ) = ( ) ( ) = ( ) f f f f eve odd Look at the ifiite well problem does H commute with the parity operator? Kow that

8 ħ ħ + V ( ) = + V m m ( ) ( ) Vˆ( Xˆ ) Vˆ( Xˆ ) Vˆ( Xˆ ) ( ) ψ ( ) ( ) ψ ( ) ( ) ψ ( ) ( ) ψ ( ) Π ˆ, =Πˆ Π= ˆ? ΠˆVˆ Xˆ Vˆ Xˆ Π ˆ =ΠˆV V ( ) ψ ( ) ( ) ψ ( ) = V V = Which meas that that oe ca always fid a set of eigefuctios which are commo to both H ad Π.

Lecture #3. Math tools covered today

Lecture #3. Math tools covered today Toay s Program:. Review of previous lecture. QM free particle a particle i a bo. 3. Priciple of spectral ecompositio. 4. Fourth Postulate Math tools covere toay Lecture #3. Lear how to solve separable