Answer
The time-dependent wave function $\Psi(x, t)$ is given by
$\Psi(x, t)=\psi(x)e^{-i\omega t}$
Substituting, $\psi(x)=Ae^{ikx}$, we get
$\Psi(x, t)=Ae^{ikx}e^{-i\omega t}$
or, $\Psi(x, t)=Ae^{i(kx-\omega t)}$
where $A$ is a constant
Using Euler's equation, we get
$\Psi(x,t)=A\{\cos (kx-\omega t)+i\sin (kx-\omega t)\}$
or, $\Psi (x,t)=a+ib$
where, $a=A\cos (kx-\omega t)$, and $b=A\sin (kx-\omega t)$ are real quantities.
Work Step by Step
The time-dependent wave function $\Psi(x, t)$ is given by
$\Psi(x, t)=\psi(x)e^{-i\omega t}$
Substituting, $\psi(x)=Ae^{ikx}$, we get
$\Psi(x, t)=Ae^{ikx}e^{-i\omega t}$
or, $\Psi(x, t)=Ae^{i(kx-\omega t)}$
where $A$ is a constant
Using Euler's equation, we get
$\Psi(x,t)=A\{\cos (kx-\omega t)+i\sin (kx-\omega t)\}$
or, $\Psi (x,t)=a+ib$
where, $a=A\cos (kx-\omega t)$, and $b=A\sin (kx-\omega t)$ are real quantities.
