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 $\lambda=h / p .$ A reasonable first guess for the time-dependent wave function for such a particle is $\Psi(x, t)=A \cos (k x-\omega t),$ where $A$ is a constant, $\omega=2 \pi f$ is the angular frequency, and $k=2 \pi / \lambda$ is the wave number. This is the same function we used to describe a mechanical wave [see Eq. (15.7)$]$ or an electromagnetic wave propagating in the $x$ direction Isee Eq. (32.16)]. (a) Show that $\omega=E h, k=p / \hbar,$ and $\omega=\hbar k^{2} / 2 m .$ (Hint: The energy is purely kinetic, so $\left.E=p^{2} / 2 m .\right)$
(b) To check this guess for the time-dependent wave function, substitute $\Psi(x, t)=A \cos (k x-\omega t)$ into the time-dependent Schrödinger equation (see Problem 39.63 ) with $U(x)=0$ (so the particle is free). Show that this guess for $\Psi(x, t)$ 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, $\Psi(x, t)=A \sin (k x-\omega t),$ is also not a suitable wave function for a free particle. (d) Consider a combination of the functions proposed in parts (b) and (c):$$\Psi(x, t)=A \cos (k x-\omega t)+B \sin (k x-\omega t)$$
By using the procedure described in part (b), show that this wave function is a solution to the time-dependent Schrödinger equation with $U(x)=0,$ but only if $B=i$. (Hint: To satisfy the timedependent Schrödinger equation for all $x$ and $t,$ the coefficients of $\cos (k x-\omega t)$ on both sides of the equation must be equal. The same is true for the coefficients of $\sin (k x-\omega t)$ on both sides of the equation.) This is an example of the general result that timedependent wave functions always have both a real part and an imaginary part.