Question

EXAMPLE 5.7 A Matter Wave Packet (a) Show that the matter wave packet whose amplitude distribution a(k) is a rec tangular pulse of height unity, width Ak, and centered at ko (Fig. 5.23) has the form (x) =-ar sínar. x/2) b) Observe that this wave packet is a complex function. Later in this chapter we hall see how the definition of probability density results in a real function, but for he time being consider only the real part of JC) and make a sketch of its behavior, howing its envelope and the cosine function within. Determine Δχ, and show that n uncertainty relation of the form ΔΧΔk 1 holds

Please explain throughly because I am bad at fourier series.

Answer 1

a(k) = 1 width of packet = delta K center = k_0 because rectangular plate centered at K_0 so wave numbers varies from k_0 - delta k/2 to k_0 + delta k/2 From Fourier integral method f(x) = 1/ squareroot 2 pi integral^+ infinity_-infinity a(k) e^ikx dK = 1/ squareroot 2 pi integral^k_0 + (delta k/2)_k_0 - (delta k/2) 1 middot e^ikx dK 1/squareroot 2 pi 1/ix(e^ikx)^k_0 + delta k/2_k_0 - delta k/2 1/ squareroot 2 pi 1/ix[e^i(k_0 + delta k/2)x - e^i(k_0 - delta k/2)x] 1/ squareroot 2 pi e^i k_0 x/ix [e^i delta k/2 x - e^-i delta k/2 x/2 i] 1/ squareroot 2 pi middot 2 e^i k_0 x/x sin(delta k middot x/2) = delta k/ squareroot 2 pi sin(delta k middot x/2)/(delta k middot x/2) e^ik_0x \r\n Now take fix real point of the wave packet and its sketch looks like full width of main lobe is delta x = delta pi/delta k This gives uncertainty relation delta x middot delta k = 4pi