#position operator

[peeterjoot]

Position operator in momentum space representation

Position operator in momentum space representation

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A derivation of the position space representation of the momentum operator \( -i \Hbar \partial_x \) is made in [1], starting with the position-momentum commutator. Here I’ll repeat that argument for the momentum space representation of the position operator.

The we want to do is expand the matrix element of the commutator. First using…

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[peeterjoot]

Heisenberg picture position commutator

Heisenberg picture position commutator

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Question: Heisenberg picture position commutator ([1] pr. 2.5)

Evaluate

\begin{equation}\label{eqn:positionCommutator:20} \antisymmetric{x(t)}{x(0)}, \end{equation}

for a Heisenberg picture operator \( x(t) \) for a free particle.

Answer

The free particle Hamiltonian is

\begin{equation}\label{eqn:positionCommutator:40} H = \frac{p^2}{2m},

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