Question 3.1: Show that the wave function,ψ(x) = Ae^i kx , (3.8) represent...
Show that the wave function,
ψ(x) = Ae^{i kx} , (3.8)
represents a state for which the momentum of the particle has the value p = \hbar k. Find the kinetic energy of the particle in this state.
To see whether the wave function ψ(x) corresponds to a state for which the particle has a definite value of the momentum, we multiply the momentum operator (3.2) times the wave function (3.8) to obtain
\hat{p}= – i \hbar \frac{d}{dx} . (3.2)
\left(-i \hbar \frac{d}{dx} \right)A e^{i k x}=\hbar k A e^{i k x}.
This last equation may be written more simply in terms of the momentum operator \hat{p} and the wave function ψ(x)
\hat{p} ψ(x) =\hbar k ψ(x).
which we may identify as the eigenvalue equation for the momentum (3.3). We may thus identify ψ(x) as an eigenfunction of the momentum corresponding to the eigenvalue p = \hbar k.
\hat{p} ψ = p ψ. (3.3)
To see whether the wave function ψ(x) corresponds to a state for which the particle has a definite value of the energy, we use Eq. (3.6) to multiply the energy operator for a free-particle times the wave function (3.8) to obtain
\hat{H} =\frac{-\hbar^{2} }{2m}\frac{d{2}}{dx^{2}}+V(x). (3.6)
\left(\frac{-\hbar ^{2}}{2m}\frac{d{2}}{dx^{2}} \right) A e^{i k x}=\frac{(\hslash k)^{2}}{2m}A e^{i k x}.
This last equation may be written more simply in terms of the energy operator \hat{H} and the wave function ψ(x)
\hat{H}ψ(x)=\frac{(\hbar k)^{2}}{2m} ψ(x) ,
which we may identify as the eigenvalue equation for the energy (3.7). The wave function ψ(x) is thus an eigenfunction of the energy corresponding to the eigenvalue E = (\hbar k)^{2}/2m. Using the fact that the momentum has the value p = \hbar k, the equation for the energy can also be written E = p^{2}/2m as one would expect.
\hat{H} ψ= E ψ. (3.7)