QM in 3D position-space

Consider the Hilbert space $𝐿 2 (ℝ 3)$ with vectors represented in the position basis $Ψ (𝐫, 𝑡) = ⟨ 𝐫 | 𝜓 (𝑡) ⟩$ . The momentum operators are given by

ˆ 𝑝 𝑗 = - 𝑖 ℏ 𝜕 𝑗 ˆ 𝐩 = - 𝑖 ℏ \nabla

and thus the Hamiltonian operator by

ˆ 𝐻 (𝑡) = - ℏ 2 2 𝑚 \nabla 2 + 𝑉 (ˆ 𝐫, 𝑡)

and the Schrödinger equation becomes

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

Time independent Schrödinger equation

If $𝑉$ is time-independent the stationary states are given by the time-independent Schrödinger equation

Ψ 𝑛 (𝐫, 𝑡) = 𝜓 𝑛 (𝐫) 𝑒 - 𝑖 𝐸 𝑛 𝑡 / ℏ 𝐸 𝑛 𝜓 𝑛 = - ℏ 2 2 𝑚 \nabla 2 𝜓 𝑛 + 𝑉 (ˆ 𝐫) 𝜓 𝑛

and thus general solutions are given by

Ψ (𝐫, 𝑡) = \sum 𝑐 𝑛 𝜓 𝑛 (𝐫) 𝑒 - 𝑖 𝐸 𝑛 𝑡 / ℏ

Properties

The canonical commutation relations are

[ˆ 𝑟 𝑗, ˆ 𝑝 𝑘] = 𝑖 ℏ 𝛿 𝑗 𝑘 [ˆ 𝑟 𝑗, ˆ 𝑟 𝑘] = 0 [ˆ 𝑝 𝑗, ˆ 𝑝 𝑘] = 0

an example of the Standard Heisenberg algebra for QM. 2. $[𝑓 (ˆ 𝐫), ˆ 𝑝 𝑗] = 𝑖 ℏ 𝜕 𝑗 𝑓 (ˆ 𝐫)$ 3. The energy of a normalizable solution must exceed the infimum of the potential.

Proof of 1–2

For any $| 𝜓 ⟩ \in H$ with $𝜓 (𝐫) = ⟨ 𝐫 | 𝜓 ⟩$

[ˆ 𝑟 𝑗, ˆ 𝑝 𝑘] | 𝜓 ⟩ = (- 𝑟 𝑗 𝑖 ℏ 𝜕 𝑘 + 𝑖 ℏ 𝜕 𝑘 𝑟 𝑗) 𝜓 = - 𝑟 𝑗 𝑖 ℏ 𝜕 𝑘 𝜓 + 𝑟 𝑗 𝑖 ℏ 𝜕 𝑘 𝜓 + 𝑖 ℏ 𝜓 𝜕 𝑘 𝑟 𝑗 = 𝑖 ℏ 𝛿 𝑗 𝑘 | 𝜓 ⟩ [ˆ 𝑟 𝑗, ˆ 𝑟 𝑘] | 𝜓 ⟩ = (𝑟 𝑗 𝑟 𝑘) 𝜓 = 0 [ˆ 𝑝 𝑗, ˆ 𝑝 𝑘] | 𝜓 ⟩ = (- ℏ 2 𝜕 𝑗 𝜕 𝑘 + ℏ 2 𝜕 𝑘 𝜕 𝑗) 𝜓 = 0

as required

Since any normalizable solution is a linear combination of stationary states, it is sufficient to show all stationary states have definite energy greater than this infimum. According to the Time independent Schrödinger equation

\nabla 2 𝜓 = 2 𝑚 ℏ 2 (𝑉 (𝐫) - 𝐸) 𝜓

If $𝐸 \leq 𝑉 (𝐫)$ for all $𝐫 \in ℝ 3$ then $\nabla 2 𝜓 (𝐫)$ never has the opposite sign to $𝜓 (𝐫)$ . If $𝜓 (𝐫)$ is positive then $𝜓$ is concave up, and if $𝜓 (𝐫)$ is negative then $𝜓$ is concave down. Hence $𝜓 (𝐫)$ never approaches zero as $𝐫 \to \infty$ .

Spherical coördinates

In Spherical coördinates the Hamiltonian is

ˆ 𝐻 = - ℏ 2 2 𝑚 (1 𝑟 2 𝜕 𝜕 𝑟 (𝑟 2 𝜕 𝜕 𝑟) + 1 𝑟 2 s i n 𝜃 𝜕 𝜕 𝜃 (s i n 𝜃 𝜕 𝜕 𝜃) + 1 𝑟 2 s i n 2 𝜃 𝜕 2 𝜕 𝜙 2) + ˆ 𝑉 = 1 2 𝑚 𝑟 2 (- ℏ 2 𝜕 𝜕 𝑟 (𝑟 2 𝜕 𝜕 𝑟) + ˆ 𝐿 2) + ˆ 𝑉

QM in 3D position-space

Time independent Schrödinger equation

Properties

Spherical coördinates

Examples

See also