1 (p,t) = eipx/hu(x, t) dx; 2h 1 4(x,t) = eipx/h (p,t)dp. 27h 8 (3.54) (3.55)...

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1 (p,t) = eipx/hu(x, t) dx; 2h 1 4(x,t) = eipx/h (p,t)dp. 27h 8 (3.54) (3.55) +00 +00 f(x)= F(k) eikx dk F(k) = f(x)e-ikx dx. (2.103) 2TT 27 a) Derive Equations G3.54 and G3.55 from G2.103. b) From G2.103, which uses k= 2/, derive the analogous equations that use the convention k=1/A and have no leading factors of 2. These are the "unitary, ordinary frequency" version of the Fourier transform, and are our preferred version. c)(x,t) = oe2ni(kx-ft) and 2(x,t) = oe2ni(ft-kx) are both plane waves. (Note that these expressions could be written in terms of E and p if we wished. But why would we? Note also that f and k are parameters - not variables - in these expressions.) Show which one "works" with our definitions of the Schrodinger equation and the momentum operator in the position representation, and which one doesn't. Then, using the unitary, ordinary frequency version of the Fourier transform, calculate the full spacetime Fourier transform of the that works. There's a hazard here! Explain what it is, and how the spatial and temporal Fourier transforms have to be different to get around it. Derive the necessary temporal Fourier transform relations from the spatial Fourier transform relations you derived in part (b). 1 (p,t) = eipx/hu(x, t) dx; 2h 1 4(x,t) = eipx/h (p,t)dp. 27h 8 (3.54) (3.55) +00 +00 f(x)= F(k) eikx dk F(k) = f(x)e-ikx dx. (2.103) 2TT 27 a) Derive Equations G3.54 and G3.55 from G2.103. b) From G2.103, which uses k= 2/, derive the analogous equations that use the convention k=1/A and have no leading factors of 2. These are the "unitary, ordinary frequency" version of the Fourier transform, and are our preferred version. c)(x,t) = oe2ni(kx-ft) and 2(x,t) = oe2ni(ft-kx) are both plane waves. (Note that these expressions could be written in terms of E and p if we wished. But why would we? Note also that f and k are parameters - not variables - in these expressions.) Show which one "works" with our definitions of the Schrodinger equation and the momentum operator in the position representation, and which one doesn't. Then, using the unitary, ordinary frequency version of the Fourier transform, calculate the full spacetime Fourier transform of the that works. There's a hazard here! Explain what it is, and how the spatial and temporal Fourier transforms have to be different to get around it. Derive the necessary temporal Fourier transform relations from the spatial Fourier transform relations you derived in part (b).