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# SPTP ULTRAFAST LASER PHY I PHYS 689

Texas A&M

GPA 3.73

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This 2 page Class Notes was uploaded by Alba Ankunding on Wednesday October 21, 2015. The Class Notes belongs to PHYS 689 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 32 views. For similar materials see /class/225906/phys-689-texas-a-m-university in Physics 2 at Texas A&M University.

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Date Created: 10/21/15

The motion of a quantum particle due to a constant force Lecture Notes The potential of the constant force F reads V iFac The stationary state is determined by the Schrodinger equation 2 121 if E F as 0 1 2 M lt w lt gt At positive F the classically permitted region extends to the right from the turning point 10 7 By a change of variables E Q 13 awf 2m 5 2 the Schrodinger equation is reduced to a standard dimensionless form d2 4 w0 3 Its solution will be sought by a complex modi cation of the Laplace transfore mation as a contour integral 1MB WWW lt4 1quot The only reuirements to the contour T in the complex plane A are that it can not be deformed to a point and that the value of the Laplace transform q5 at the contour ends must be zero Since differentiation over 5 commutes with the integration over A the equation can be rewritten as follows e A2 5 MAMA 0 lt5 The second term in this equation can be represented as follows 1 A ga qmwx 7e dx 6 r r lndeed after the integration by parts of the integral in the rrh s of equation 6 keeping in mind that q5 is zero at the contour ends we obtain the expression in the leftehand side of the same equation Thus equation will be satis ed if ab A satis es the rst order differential equation 1amp5 A T AZQWA 0 7 The general solution of this equation reads MA oexp 8 where C is an arbitrasry constant and according to 4 the general solution of Airy equation reads 5 C ef w 9 It seems that it is determined by only one constant C in contradiction with the general fact that there are two independent solutions of the second order differential equation However this contradiction is ctitious since we still did not choose the contour T We will see that there are only two possible contours which can not be reduced each to other The function ab A de ned by equation 8 does not turn into zero at any nite A Therefore the ends of the contour T must be at in nity The exponent in equation 8 decreases at in nity if Re A3 lt 0 Using the polar representation of the complex number A lAl 5w one can see that Re A3 is negative if cos 3go lt 0 It happens if 3go is con ned to one of 3 intervals lt 3go lt 3i 7377 lt 3go lt 7 or 577 lt 3go lt This means that go is con ned in the following 3 sectors 1 g lt go lt 2 7r 7r 77f 5 lt go lt 75 or 3 5 lt go lt i Each of this sectors has angular width They are shown in Fig 1 Possible contours must start in one of the three sectors and end in another The contours which start and end in the same sector can be deformed into a point It means that the integral along such I a contour is zero There are only two independent solutions One of them corresponds to a contour which starts in the sector 1 and ends in the sector 2 the contour corresponding to the second solution starts in 1 and ends in The remaining choice of the contour going from 2 to 3 gives the solution which is the difference of the second and the rst solutions as it is clearly seen on Fig 1 We are looking for a solution which exponentially decreases in classically forbidden region 5 lt 0 Therefore it is described by a real wave function as any state with complete re ection The contour T which corresponds to the real solution must be deformable into a contour symmetric with respect to real axis A Then upper and lower parts of this contour pass through complex conjugated values of A at which the inegrand and the differentials dA accept complex conjugated values at real Only the contour passing through the regions 1 and 2 has this property The solution 9 with the constant C 2gEfl is called the Airy function and denoted Ai7 The contour T can be deformed to the imaginary axis Then by the change of variable A iu the integral 9 can be transformed to the following form u Ai7 70m 7 ugt du 10

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