Baic Quantum Mechanic in Coorinate, Momentum an Phae Space Frank Rioux Department of Chemitry College of St. Beneict St. Johnʹ Univerity The purpoe of thi paper i to ue calculation on the harmonic ocillator to illutrate the relationhip between the coorinate, momentum an phae pace repreentation of quantum mechanic. Firt, the groun tate coorinate pace eigenfunction for the harmonic ocillator i ue for everal traitional quantum mechanical calculation. Then the coorinate wave function i Fourier tranforme into the momentum repreentation, an the calculation repeate howing that the ame reult are obtaine. Next, the coorinate (an ubequently the momentum) wave function i ue to generate the Wigner phae pace itribution function. It i then ue to repeat the quantum mechanical calculation one in the coorinate an momentum repreentation, yieling the ame reult. Finally, a variational calculation i carrie out in all three repreentation for the V = x potential energy function uing a Gauian trial wave function. A might be expecte the three calculation yiel ientical reult. All calculation are carrie out in atomic unit (h = ) with the effective ma an force contant et to unity ( = k = ) for the ake of computational convenience. The firt three harmonic ocillator eigenfunction are given below. While the groun tate eigenfunction i ue in the example calculation, it i eay for the interete reaer to eit the companion Mathca file to repeat the calculation for the other eigenfunction. 0 ( x) exp x ( x) xexp x ( x) x exp x A i well known, in coorinate pace the poition operator i multiplicative an the momentum operator i ifferential. In momentum pace it i the revere, while in phae pace, both poition an momentum are multiplicative operator. In Appenix A Dirac notation i ue to erive the poition an momentum operator in coorinate an momentum pace. Cae () ue the Weyl tranform to how that both the poition an momentum operator are multiplicative in phae pace. In Appenix B a euctive rationalization for the multiplicative character of the poition operator in phae pace i preente. The extenion to the multiplicative character of the momentum operator i traightforwar. Coorinate Space Calculation Coorinate pace integral: Poition operator: x Potential energy operator: x Momentum operator: i Kinetic energy operator:
Diplay the v = 0 coorinate itribution function. 0 ( x) 0 Demontrate that the wave function i normalize an calculate <x>, <x >, <p>, an <p >. Then ue thee reult to emontrate that the uncertainty principle i atifie. x 0 ( x) x ave x 0 ( x) 0 x ave x 0 ( x) p ave 0 ( x) i x 0 ( x) 0 p ave 0 ( x) 0 ( x) In the coorinate repreentation the expectation value involving poition appear to be calculate claically. The average value i the um over each value of x weighte by it probability of occurring, (x). Thi i clearly not the cae for the momentum expectation value in coorinate pace. Quantum weirne i manifet in the momentum calculation an hien in the coorinate calculation. A mentione above, Appenix A how the origin of thi computational ifference between poition an momentum expectation value in coorinate pace. Baically it come own to the fact that in it own (eigen) pace an operator ha pecial privilege, it appear to operate multiplicatively. The uncertainty principle require that xp (in atomic unit). The expectation value from above how that the harmonic ocillator i in compliance. x x ave x ave p p ave p ave xp Demontrate that (x) i an eigenfunction of the energy operator an ue the expectation value from above to calculate the expectation value for energy. 0 ( x) x 0 ( x) = E 0 ( x) olve E p ave x ave
Momentum Space Calculation The energy operator for the harmonic ocillator i, Hˆ pˆ kxˆ m Mot quantum mechanical problem are eaier to olve in coorinate pace. Becaue of it ymmetry, the harmonic ocillator i a eay to olve in momentum pace a it i in coorinate pace. However, we generate the momentum wave function by Fourier tranform of the coorinate pace wave function. It i then hown that it give the ame reult a the wave function in the poition bai. 0 ( exp( ipx) 0 ( x) p e Firt, we emontrate that the Fourier tranform of thi momentum wave function return the coorinate pace wave function. exp( ipx) 0 ( x e Diplay the v = 0 momentum itribution function. 0 ( 0 Notice that the coorinate an momentum itribution function are ientical given the parameterization of the calculation ( = k = ). p Momentum pace integral: Momentum operator: p Kinetic energy operator: p Poition operator: i Potential energy operator:
Demontrate that the wave function i normalize an calculate <x>, <x >, <p>, an <p >. Then ue thee reult to emontrate that the uncertainty principle i atifie. 0 ( p ave p 0 ( 0 p ave p 0 ( x ave 0 ( i p 0 ( 0 x ave 0 ( 0 ( In momentum pace, it i the momentum operator that appear to behave claically, an the poition operator that manifet quantum weirne. Thee momentum pace calculation are in compliance with the uncertainty principle. x x ave x ave p p ave p ave xp Demontrate that ( i an eigenfunction of the energy operator an ue the expectation value from above to calculate the expectation value for energy. p 0 ( 0 ( = E 0 ( olve E p ave x ave In ummary, we ee that coorinate an momentum pace calculation give the ame reult. However, the coorinate wave function oe not tell u anything about the itribution of momentum tate, only the average value. Likewie, the momentum wave function oe not provie etail on the patial itribution of the particle it repreent, only the average poition. Phae Space Calculation Phae pace calculation require a phae pace itribution, uch a the Wigner function. Becaue thi approach to quantum mechanic i not a familiar a the Schröinger formulation, everal important equation will be econtructe uing Dirac notation. Expree in Dirac notation, the Wigner function reemble a claical trajectory. W( x, x x p p x x The four Dirac bracket are rea from right to left a follow: () i the amplitue that a particle in the tate ha poition (x /); () i the amplitue that a particle with poition (x /) ha momentum p; (3) i the amplitue that a particle with momentum p ha poition (x + /); () i the amplitue that a particle with poition (x + /) i (till) in the tate. The Wigner function i the integral of the prouct of thee probability amplitue over all value of.
We get the traitional form of the Wigner itribution function by recognizing that the mile bracket, which function a a propagator between the initial an final poitional tate, can be combine a follow, ( ) ( ) ip x ip x ip e e e x p p x Now we can generate the Wigner function for the v = 0 harmonic ocillator tate uing the coorinate eigenfunction. W 0 ( x 0 x exp( i 0 x p e x In coorinate pace, momentum i repreente by a ifferential operator, the firt erivative with repect to poition. In momentum pace, poition i repreente by the firt erivative with repect to momentum. Part of the appeal of the phae pace approach to quantum mechanic i that both poition an momentum are repreente by multiplicative operator (). Thu phae pace quantum mechanic, at firt glance, appear to more cloely reemble claical mechanic than the traitional Schröinger formulation with it ifferential operator. Phae pace integral: Poition operator: x Potential energy operator: x Momentum operator: p Kinetic energy operator: p Demontrate that the Wigner function i normalize over phae pace an calculate <x>, <x >, <p>, an <p >. Then ue thee reult to emontrate that the uncertainty principle i atifie. In Appenix B Dirac notation i ue to econtruct (unpack) the firt two phae pace calculation below an how that they are equivalent to the traitional quantum mechanical calculation carrie out previouly. W 0 ( x x ave W 0 ( x x 0 x ave W 0 ( x x p ave W 0 ( x p 0 p ave W 0 ( x p x x ave x ave p p ave p ave xp
Calculate the expectation value for the total energy. W 0 ( x p x In ummary, the phae pace calculation bae on the Wigner function give the ame reult a the calculation carrie out in coorinate an momentum pace. Next, we emontrate that integrating the Wigner function over momentum pace yiel the coorinate itribution function. (See the Appenix C for a econtruction of thi integral uing Dirac notation.) W 0 ( x e x 0 ( x) e x Likewie, integrating the Wigner function over coorinate pace yiel the momentum itribution function. (See the Appenix C for a econtruction of thi integral uing Dirac notation.) W 0 ( x e p 0 ( e p Jut a we have previou graphe the coorinate an momentum itribution function, we now iplay the Wigner itribution function. 6i N 60 i 0 N x i 3 j 0 N p j 5 N 0 j N Wigner i j W 0 x i p j Wigner
In thee phae pace calculation W(x, appear to behave like a claical probability function. By eliminating the nee for ifferential operator, it eem to have remove ome of the weirne from quantum mechanic. However, we will now ee that the Wigner function, phae pace approach only temporarily hie the weirne. Thi houlnʹt come a a urprie becaue, after all, the Wigner function wa generate uing a Schröinger wave function. To ee how the weirne i hien we generate the Wigner function for the v = harmonic ocillator tate. W ( x p x exp( i x x x p e Next, it i emontrate that the Wigner function for the groun an excite harmonic ocillator tate are orthogonal over phae pace. W 0 ( x W ( x 0 Thi reult inicate that W (x, mut be negative over ome part of phae pace, becaue the graph of W 0 (x, how that it i poitive for all value of poition an momentum. To explore further we iplay the Wigner itribution for the v = harmonic ocillator tate. Wigner i j W x i p j Wigner Wigner Thi graphic how that the Wigner function i inee negative for certain region of phae pace. Thi make it impoible to interpret it a a probability itribution function. For thi reaon the Wigner function i frequently referre to a a quaiprobability itribution.
The Variation Metho A a final example of the equivalence of the three approache to quantum mechanic preente, we look at a variation metho calculation on a potential function that reemble (omewhat) the harmonic ocillator. Vx ( ) x A Gauian trial wave function i choen for the coorinate pace calculation. x exp x The variational energy integral i evaluate. E x Vx ( ) x x aume 0 The momentum wave function i obtaine from the coorinate wave function by Fourier tranform. p exp( ipx) x aume 0 p 3 e The momentum pace variational energy integral i evaluate. E p p p p i p p p aume 0 3 The Wigner function i calculate uing the coorinate wave function (the momentum wave function will yiel the ame reult). Wxp x exp i ( x aume 0 x p e
The variational energy integral in phae pace i evaluate. E Wxp p Vx ( ) aume 0 3 A i to be expecte, the three metho yiel the ame expreion for the variational energy. Minimization of the energy with repect to the variational parameter,, yiel = 0.5 an E() = 0.83. Thi i in goo agreement with the reult obtaine by numerical integration of Schröingerʹ equation for thi potential, E = 0.809. Appenix A In coorinate pace, the poition eigenvalue equation can be written in two equivalent form. In both cae the operator, in it home pace, extract the eigenvalue an return the eigenfunction. It thu appear to be a multiplicative operator. xˆ x x x x xˆ x x Uing the form on the right we emontrate that the poition operator in the coorinate repreentation operate multiplicatively on an arbitrary tate function. x xˆ x x x( x) Making ue of the coorinate pace completene relation, x x we can illuminate the poition expectation value in the coorinate repreentation. x xˆ x x xˆ x x x x ( x) In momentum pace, the momentum eigenvalue equation can alo be written in two equivalent form. A in coorinate pace, the home pace operator extract the eigenvalue an return the eigenfunction. pˆ p p p p pˆ p p Uing the form on the right we emontrate that the momentum operator in momentum pace operate multiplicatively on an arbitrary tate function. p pˆ p p p( To procee to a jutification of the ifferential form of the momentum operator in coorinate pace an the ifferential form of the poition operator in momentum pace require the following Dirac bracket between poition an momentum.
xp * px exp ipx Thi relation i obtaine by ubtitution of the ebroglie wave equation into the Euler equation for a plane wave (). h h mv p exp x i x The Dirac bracket, <x p> an <p x> are ubiquitou in quantum mechanic an are, eentially, a ictionary for tranlating from momentum language to poition language an vice vera. In other wor, they are momentum poition Fourier tranform. The momentum operator in coorinate pace i obtaine by projecting the momentum eigenvalue expreion onto coorinate pace. ˆ x pp pxp pexp ipx xp i Comparing the firt an the lat term, give the momentum operator in poition pace. x pˆ i Uing thi form we emontrate the momentum operator in coorinate pace operating on an arbitrary tate function. x pˆ x ( x) i i Uing the coorinate completene relation, we erive the mathematical tructure of the calculation of the momentum expectation value in the coorinate repreentation. x x * ( x) p pˆ x x pˆ x ( x) i i The poition operator in momentum pace i obtaine by projecting the poition eigenvalue expreion onto momentum pace. ˆ ipx p xx x px xexp px i Comparing the firt an the lat term, give the poition operator in momentum pace. pxˆ i p
Uing thi form we emontrate the poition operator in momentum pace operating on an arbitrary tate function. pxˆ p ( i i Illutrating the origin of the quantum mechanical calculation for the expectation value for momentum an poition in momentum pace are imilar to the calculation in coorinate pace, except that the completene relation in momentum pace i require. p p In the text it wa note that in phae pace, both poition an momentum are multiplicative operator (). The following table ummarize the operator notation for the three pace. Operator poition momentum CoorinateSpace x i MomentumSpace i p PhaeSpace x p Appenix B In Dirac notation the phae pace normalization conition i, W( x, x x p p x x Utilizing the momentum completene relation, yiel, p p x x x x However, the integral over i zero unle = 0. Thu, for a normalize coorinate wave function we arrive at, x x
Uing imilar argument it i eay to how that, W ( x, x x x x x Appenix C To how that integrating the Wigner function over momentum pace yiel the coorinate itribution function, we procee a hown below. W( xp, ) x x p px x Uing the momentum completene relation (ee above) on the right ie give, W( x, x x x x However, the right ie i zero unle = 0, yieling W x p x x x x * (, ) ( ) ( ) To facilitate the emontration that integrating the Wigner function over coorinate pace yiel the momentum itribution function, we firt how that the Wigner function can alo be generate uing the momentum wave function. Wp 0 ( x 0 p exp( i x) 0 p p e x W( x, p p x x p p Employing the coorinate completene relation yiel x x W( x, p p p p
However, the right ie i zero unle = 0, yieling W x p p p p p * (, ) ( ) ( ) Reference:.. Cae, W. B. Wigner function an Weyl tranform for peetrian. Am. J. Phy. 008, 76 937 96. Rioux, F.; Johnon, B. J. Uing Optical Tranform to Teach Quantum Mechanic. Chem. Eucator. 00, 9, 6.