Hamiltonian operator

The hamiltonian $ˆ 𝐻$ is the quantum operator corresponding to the measurement of the total energy of a particle.

ˆ 𝐻 = ˆ 𝑇 + ˆ 𝑉

Position-momentum system

For a typical position-momentum single-particle system with time-independent potential, the hamiltonian is given by

ˆ 𝐻 = ˆ 𝑝 2 2 𝑚 + 𝑉 (ˆ 𝑥)

where $ˆ 𝑥$ and $ˆ 𝑝$ are the position and momentum operators respectively and $𝑉$ is the potential. Thus

ˆ 𝐻 = - ℏ 2 2 𝑚 \nabla 2 + 𝑉 (ˆ 𝑥)

Verification of hermiticity

Let $𝜓, 𝜑 \in 𝐿 2 (𝑀)$ where $𝑀$ is some infinite space. Then
$⟨ 𝜑 | ˆ 𝐻 𝜓 ⟩ = \int 𝑀 𝜑 * (- ℏ 2 𝑚 \nabla 2 𝜓 + 𝑉 𝜓) 𝑑 𝜏 = \int 𝑀 𝜑 * 𝑉 𝜓 𝑑 𝜏 - ℏ 2 𝑚 \int 𝑀 𝜑 * \nabla 2 𝜓 𝑑 𝜏$
Since $𝑉$ is real-valued, applying ^GE1 twice, and using the fact that both wavefunctions and all derivatives are zero at the boundary $𝜕 𝑀$
$⟨ 𝜑 | ˆ 𝐻 𝜓 ⟩ = \int 𝑀 (𝑉 𝜑) * 𝜓 𝑑 𝜏 - ℏ 2 𝑚 (\oint 𝜕 𝑀 𝜑 * \nabla 𝜓 \cdot 𝑑 𝐚 - \int 𝑀 \nabla 𝜑 * \cdot \nabla 𝜓 𝑑 𝜏) = \int 𝑀 (𝑉 𝜑) * 𝜓 𝑑 𝜏 - ℏ 2 𝑚 (\int 𝑀 𝜓 \nabla 2 𝜑 * 𝑑 𝜏 - \oint 𝜕 𝑀 𝜓 \nabla 𝜑 * \cdot 𝑑 𝐚) = \int 𝑀 (𝑉 𝜑) * 𝜓 𝑑 𝜏 - ℏ 2 𝑚 \int 𝑀 (\nabla 2 𝜑) * 𝜓 𝑑 𝜏 = ⟨ ˆ 𝐻 𝜑 | 𝜓 ⟩$
as required.

develop | en | SemBr

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Hamiltonian operator

Position-momentum system

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