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Single Photon in a Gaussian Wavepacket. Consider a plane-wave wavepacket (Sec. 2.6A) containing a single photon traveling in the z direction, with complex wavefunction U(r, t) = a(t – ) (13.1-18) where a(t) = exp(-) oxp(j2runt). (13.1-19) (a) Show that the uncertainties in its time and z position are o, = r and o, = cơ,, respectively. (b) Show that the uncertainties in its energy and momentum satisfy the minimum uncertainty rela- tions: (13.1-20) 0,0p = h/2. (13.1-21) Equation (13.1-21) is the minimum-uncertainty limit of the Heisenberg position-momentum uncertainty relation provided in (A.2-7) of Appendix A.

Single Photon in a Gaussian Wavepacket. Consider a plane-wave wavepacket (Sec. 2.6A) containing a single photon traveling in the z direction, with complex wavefunction U(r, t) = a(t – ) (13.1-18) where a(t) = exp(-) oxp(j2runt). (13.1-19) (a) Show that the uncertainties in its time and z position are o, = r and o, = cơ,, respectively. (b) Show that the uncertainties in its energy and momentum satisfy the minimum uncertainty rela- tions: (13.1-20) 0,0p = h/2. (13.1-21) Equation (13.1-21) is the minimum-uncertainty limit of the Heisenberg position-momentum uncertainty relation provided in (A.2-7) of Appendix A.

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Single Photon in a Gaussian Wavepacket. Consider a plane-wave wavepacket (Sec. 2.6A) containing a single photon traveling in the z direction, with complex wavefunction U(r, t) = a(t -) (13.1-18) where a(t) = exp(-) exp(j27unt). (13.1-19) (a) Show that the uncertainties in its time and z position are o, = 7 and o, = cơ, respectively. (b) Show that the uncertainties in its energy and momentum satisfy the minimum uncertainty rela- tions: (13.1-20) 0,0p = h/2. (13.1-21) Equation (13.1-21) is the minimum-uncertainty limit of the Heisenberg position-momentum uncertainty relation provided in (A.2-7) of Appendix A. (A.2-7) Heisenberg Uncertainty Relation 2.6 A. Temporal and Spectral Description Although a polychromatic wave is described by a wavefunction e(r, t) with nonhar- monie time dependence, it may be expanded as a superposition of harmonie func- tions, each of which represents a monochromatic wave. Since we already know how monochromatic waves propagate in free space and through various optical components, we can determine the effect of optical systems on polychromatic light by using the principle of superposition. Fourier methods permit the expansion of an arbitrary function of time u(t), repre- senting the wavefunction u(r, t) at a fixed position r. as a superposition integral of harmonic functions of different frequencies, amplitudes, and phases: u(t) = v(v) exp(j2xvt) dv, (2.6-1) where v(1) is determined by carrying out the Fourier transform v(v) = u(t) exp(-j2xvt) dt . (2.6-2)
Transcribed Image Text:Single Photon in a Gaussian Wavepacket. Consider a plane-wave wavepacket (Sec. 2.6A) containing a single photon traveling in the z direction, with complex wavefunction U(r, t) = a(t -) (13.1-18) where a(t) = exp(-) exp(j27unt). (13.1-19) (a) Show that the uncertainties in its time and z position are o, = 7 and o, = cơ, respectively. (b) Show that the uncertainties in its energy and momentum satisfy the minimum uncertainty rela- tions: (13.1-20) 0,0p = h/2. (13.1-21) Equation (13.1-21) is the minimum-uncertainty limit of the Heisenberg position-momentum uncertainty relation provided in (A.2-7) of Appendix A. (A.2-7) Heisenberg Uncertainty Relation 2.6 A. Temporal and Spectral Description Although a polychromatic wave is described by a wavefunction e(r, t) with nonhar- monie time dependence, it may be expanded as a superposition of harmonie func- tions, each of which represents a monochromatic wave. Since we already know how monochromatic waves propagate in free space and through various optical components, we can determine the effect of optical systems on polychromatic light by using the principle of superposition. Fourier methods permit the expansion of an arbitrary function of time u(t), repre- senting the wavefunction u(r, t) at a fixed position r. as a superposition integral of harmonic functions of different frequencies, amplitudes, and phases: u(t) = v(v) exp(j2xvt) dv, (2.6-1) where v(1) is determined by carrying out the Fourier transform v(v) = u(t) exp(-j2xvt) dt . (2.6-2)
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