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Time-Dependent Wave Function for a Free Particle One example of a time-dependent wave function is that of a free particle [one for which U(x) = 0 for all x] of energy E and x-component of momentum p. From the de Broglie relationships (see Section 39.1), such a particle has associated with it a frequency f = E/h and a wavelength A = h/p. A reasonable first guess for the time-dependent wave function for such a particle is ?(x, t) = A cos (kx ? wt), where A is a constant, w = 2?/? is the angular frequency, and k = 2?/? is the wave number. This is the same function we used to describe a mechanical wave or an electromagnetic wave propagating in the x-direction [see Eq. (32.16)].
(a) Show that w = Eh, k = p/h, and w = hk2/2m.
(b) To check this guess for the time-dependent wave function, substitute ?(x, t) = A cos (kx ? wt) into the time-dependent Schrodinger equation (see Problem 39.63) with U(x) = 0 (so the particle is free). Show that this guess for ? (x, t) does not satisfy this equation and so is not a suitable wave function for a free particle.
(c) Use the procedure described in part (b) to show that a second guess, ? (x, t) = A sin (kx ? wt), is also not a suitable wave function for a free particle. (d) Consider a combination of the functions proposed in parts (b) and (c): By using the procedure described in part (b), show that this wave function is a solution to the time-dependent Schrodinger equation with U(x) = 0, but only if B = iA. This is an example of the general result that time dependent wave functions always have both a real part and an imaginary part.
