<Dd> Ψ (x) = 1 2 π ħ ∫ Φ (p) e i ħ p x d p . (\ displaystyle \ Psi (x) = (\ frac (1) (\ sqrt (2 \ pi \ hbar))) \ int \ Phi (p) e ^ ((\ frac (i) (\ hbar)) px) dp \, .) </Dd> <P> The position - space and momentum - space wave functions are thus found to be Fourier transforms of each other . The two wave functions contain the same information, and either one alone is sufficient to calculate any property of the particle . As representatives of elements of abstract physical Hilbert space, whose elements are the possible states of the system under consideration, they represent the same state vector, hence identical physical states, but they are not generally equal when viewed as square - integrable functions . </P> <P> In practice, the position - space wave function is used much more often than the momentum - space wave function . The potential entering the relevant equation (Schrödinger, Dirac, etc .) determines in which basis the description is easiest . For the harmonic oscillator, x and p enter symmetrically, so there it doesn't matter which description one uses . The same equation (modulo constants) results . From this follows, with a little bit of afterthought, a factoid: The solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in L . </P> <P> Following are the general forms of the wave function for systems in higher dimensions and more particles, as well as including other degrees of freedom than position coordinates or momentum components . </P>

State the condition that must be met by a wave function for it to be well behaved