Wave function
Relations between position and momentum representations

The x and p representations are | Ψ ⟩ = I | Ψ ⟩ = ∫ | x ⟩ ⟨ x | Ψ ⟩ d x = ∫ Ψ (x) | x ⟩ d x, | Ψ ⟩ = I | Ψ ⟩ = ∫ | p ⟩ ⟨ p | Ψ ⟩ d p = ∫ Φ (p) | p ⟩ d p. {\displaystyle {\begin{aligned}|\Psi \rangle =I|\Psi \rangle &=\int |x\rangle \langle x|\Psi \rangle dx=\int \Psi (x)|x\rangle dx,\\|\Psi \rangle =I|\Psi \rangle &=\int |p\rangle \langle p|\Psi \rangle dp=\int \Phi (p)|p\rangle dp.\end{aligned}}}

Now take the projection of the state Ψ onto eigenfunctions of momentum using the last expression in the two equations, ∫ Ψ (x) ⟨ p | x ⟩ d x = ∫ Φ (p ′) ⟨ p | p ′ ⟩ d p ′ = ∫ Φ (p ′) δ (p − p ′) d p ′ = Φ (p). {\displaystyle \int \Psi (x)\langle p|x\rangle dx=\int \Phi (p')\langle p|p'\rangle dp'=\int \Phi (p')\delta (p-p')dp'=\Phi (p).}

Then utilizing the known expression for suitably normalized eigenstates of momentum in the position representation solutions of the free Schrödinger equation ⟨ x | p ⟩ = p (x) = 1 2 π ℏ e i ℏ p x ⇒ ⟨ p | x ⟩ = 1 2 π ℏ e − i ℏ p x, {\displaystyle \langle x|p\rangle =p(x)={\frac {1}{\sqrt {2\pi \hbar }}}e^{{\frac {i}{\hbar }}px}\Rightarrow \langle p|x\rangle ={\frac {1}{\sqrt {2\pi \hbar }}}e^{-{\frac {i}{\hbar }}px},} one obtains Φ (p) = 1 2 π ℏ ∫ Ψ (x) e − i ℏ p x d x. {\displaystyle \Phi (p)={\frac {1}{\sqrt {2\pi \hbar }}}\int \Psi (x)e^{-{\frac {i}{\hbar }}px}dx\,.}

Likewise, using eigenfunctions of position, 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\,.}

The position-space and momentum-space wave functions are thus found to be Fourier transforms of each other. They are two representations of the same state; containing the same information, and either one is sufficient to calculate any property of the particle.

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 does not matter which description one uses. The same equation (modulo constants) results. From this, with a little bit of afterthought, it follows that solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in L.
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