Wavefunction in position space

For a system in which position is the only degree of freedom, we may choose eigenfunctions of the position operator $ˆ 𝑥$ as a basis. Thus $Ψ : ℝ 3 \times ℝ \to ℂ$ is a wavefunction in position space. The Hamiltonian operator may then be represented as

ˆ 𝐻 (𝑡) = ˆ 𝐩 \cdot ˆ 𝐩 2 𝑚 + 𝑉 (ˆ 𝐱, 𝑡) = - ℏ 2 2 𝑚 \nabla 2 + 𝑉 (ˆ 𝐱, 𝑡)

wherefore the Schrödinger equation becomes

𝑖 ℏ 𝜕 𝜕 𝑡 Ψ (⃗ 𝐱, 𝑡) = - ℏ 2 𝑚 \nabla 2 Ψ (⃗ 𝐱, 𝑡) + 𝑉 (⃗ 𝐱, 𝑡) Ψ (⃗ 𝐱, 𝑡)

Wavefunction in position space

See also