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by: Vernice Schuster

QuantumMechanicsI PHYS326

Marketplace > Drexel University > Physics 2 > PHYS326 > QuantumMechanicsI
Vernice Schuster
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This 5 page Class Notes was uploaded by Vernice Schuster on Wednesday September 23, 2015. The Class Notes belongs to PHYS326 at Drexel University taught by Staff in Fall. Since its upload, it has received 33 views. For similar materials see /class/212532/phys326-drexel-university in Physics 2 at Drexel University.


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Date Created: 09/23/15
Physics 326 Quantum Mechanics I Prof Michael S Vogeley Fall 2007 The Fourier Transform and Free Particle Wave Functions 1 The Fourier Transform 11 Fourier transform of a periodic function A function f that is periodic with period 2L7 f95 f962L can be expanded in a Fourier Series over the interval 7L7L7 mm mm An 7 B 7 fx n2 cos L 2 s1n L Noting that the coef cients may be complex we never said f was real7 and recalling that 619 0050 z39sint97 the series may be written as Z aneimrzL The can are the Fourier coef cients Each an speci es the contribution to f of a wave expz n7rxL with wavenumber k mrL The Fourier modes are orthonormal7 1 L 1 L MumL lmeL ianmL lmeL 72117de lt6 6 ifQLidee 6 76m where 6m is the Kronecker delta function To obtain the values of the Fourier coef cients an7 integrate both sides of the Fourier series expansion above 1 L 1 L 7 d eiwurmL 7 d eiwurmL a elmeL 2L L f L 7L mg m and use the orthonormality of the plane waves note that only terms with m n survive on the right hand side to obtain the Fourier coef cients L an iLL d feim7rzL 12 Fourier integral To proceed to the Fourier transform integral rst note that we can rewrite the Fourier series above as 00 f Z anemmLATL TLOO where An 1 is the spacing between successive integers If we de ne and then the Fourier series may be written as k Now take the limit L a 00 so that Ak becomes in nitesimal Now the discrete sum becomes an integral 1 00 z 7 A k emdk f gt 7 00 lt gt Following the discussion above that yielded the coef cients an the coef cients Ak of the continuous transform are obtained by Ak V A fze mdk These two equations de ne the Fourier transform relations The latter is usually referred to as a forward transform decomposing the spatial function fx into Fourier modes represented by their coef cients Ak while the former is the inverse transform which reconstructs the spatial function Note that here l7ve put factors of Um in front of both integrals some conventions leave out the factor in front of the forward transform and put 127T in front of the inverse transform or vice versa The functions f and Ak are a Fourier transform pair Complete knowledge of one of the pair yields the other through an analytic transform thus they have identical information content They are just di ferent representations of the same function Note carefully that both f and Ak are in general complex In the case where f is a real function then Ak must be Hermitian Ak A7k The common pairs of transform spaces includes position wavenumber and time frequency 13 Dirac Delta Function If we combine the two equations above for f and Ak plugging the second into the rst we obtain 1 0 I 00 we g 00 dkem 00 dy MW Now interchange the order of integration 1 co co we g Lmdyfyl mdk The term in brackets is the Dirac Delta function 1 00 6 i 7 dk WWI 96 y 2W 700 6 This function obviously has the property that mom 7 y M 14 Fourier transform pairs lf f is very narrow then its Fourier transform Ak is a very broad function and vice versa The Dirac delta function provides the most extreme example of this property If the Fourier transform is a constant say Ak Um then the spatial function is exactly the function f In this case the spatial function is precisely localized and its Fourier transform is completely delocalized If this sounds like the uncertainty relation to you you7re on the right track To lead into discussion of another Fourier transform pair let us begin with another form ofthe Dirac delta function Any properly normalized meaning that the integral over all space is unity peaked function approaches a delta function in the limit of in nitesimal width so a Gaussian will work 7 i 704 6a 7 0113 We The Fourier transform of this delta function is obviously a constant function which is the same as a Gaussian of in nite width We can see this by considering the Fourier transform of a 1 d Gaussian which can be shown to be another Gaussian 1 2 2 A k 76 H4 7 Now we clearly see the relationship between the dispersion in the spatial function and its Fourier transform Narrowing the extent of the function in one domain broadens its extent in the other domain For a Gaussian the dispersions are in the relation 010k 1 3 2 The Schrodinger Equation and Free Particle Wave Functions 21 Wave Packets The wave particle duality problem can be somewhat reconciled by thinking about parti cles as localized wave packets Waves of different frequencies are superposed so that they interfere completely or nearly so outside of a small spatial region Clearly both the amplitudes magnitude of waves of different frequencies and phases relative shifting of the waves are required to achieve such a superposition A plane wave which varies only in the x but not in y or 2 has the form eikmiiwt where k is the wavenumber k 27rA and w is the angular frequency of the wave In general to A superposition of such waves can be used to represent the wave packet m WMAMW MW If the wave packet is quite localized in kispace with gk narrowly centered at k k0 such as M 0lt 6 k k02 with corresponding to wk0 then we can associate this wave packet with a particle that has energy E ha and momentum p hk This suggests that in general we can rewrite any wave packet as a superposition of plane waves that have a distribution of momenta p 6ipmiEth wew wam 22 Solutions to the Schrodinger equation The wave packet 7Lt above is a general solution to the differential equation ngwii pwam Z 3t 7 2m 3x2 This is the time dependent Schrodinger equation for a free particle7 ie7 where V 07 written in the position representation This shows that plane waves expz kx 7 at are eigenfunctions of the free particle Hamiltonian The probability of nding a particle at position x is proportional to the squared mod ulus lwxtl2 Likewise7 the probability of nding this same particle with momentum p is proportional to l pl2 These two representations are related by the Fourier transform relation above The Fourier transform relation between the position and momentum representations immediately suggests the Heisenberg uncertainty relation We showed above that the dispersions of a spatial Gaussian and its Fourier transform are in the relation 010k 1 The product of dispersions is minimized by a Gaussian7 thus in general AzAk 2 1 Using p hk this suggests that the uncertainty relation must be of order AxAp 2 h 3 References More about Fourier transforms can be found in the classic text The Fourier Transform and Its Application7 RN Bracewell McGraw Hill The discussion above closely follows the development in Quantum Physich7 S Gasiorowicz Wiley


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