The time-dependent Schrödinger equation for a free particle (with no potential energy) is given by:
where \( \phi(x) \) is the wave function, \(E\) is the energy, \( \hbar \) is the Planck constant, and \( m \) is the mass of the particle. We need to find \( E \) and \( \phi(x) \).
The spatial part \( \phi(x) \) satisfies the equation:
Let \( k^2 = \frac{2mE}{\hbar^2} \). Then the differential equation is:
The general solution to this differential equation is:
where \( A \) and \( B \) are constants that can be determined by normalisation, and \( k \) is the wave number related to the momentum \( p \) by \( p = \hbar k \). Finally, the solution is
\[ E = \frac{\hbar^2 k^2}{2m}, \;\;\; {\rm and} \;\;\; \phi(x) = A e^{ikx} + B e^{-ikx} \]This represents a superposition of plane waves with wave numbers \( \pm k \). Note that in this problem there are no other restrictions on the solution (except for normalisation). Hence, energy \( E > 0 \) can take any real value.