Position operator

The position operator in quantum mechanics belongs to the local measurement of particles.

The physical state of a particle in quantum mechanics is given mathematically by an associated vector of a Hilbert space H. This state is thus described in the Bra- Ket notation by the vector. The observables are represented by self-adjoint operators on H. Specifically, the position operator is the summary of the three observables, so that

The average value ( expected value ) of the measurement results of the j-th position coordinate of the particle in the state.

Definition and properties

• The three local operators are self-adjoint operators that satisfy the following canonical commutation relations with the equally self-adjoint momentum operators,

• It follows that the three location coordinates are jointly measured and that its spectrum is composed ( the range of possible values) from the entire room. The possible locations are therefore not quantized, but continuously.
• The spatial representation is defined by the spectral representation of the position operator. The Hilbert space H = is the space of square integrable complex functions of the local space, each state is given by a spatial wave function.

The local operators, the operators of the coordinates multiplication functions, that is, the local operator acts on local wave functions by multiplying the wave function of the coordinate function

The expectation value

The momentum operator acts on spatial wave functions ( with a suitable choice of phases) as a differential operator

• In the momentum representation of the momentum operator has a multiplicative effect on pulse wave functions

And the position operator acts as a differential operator

• Quantum mechanics
• Quantum Chemistry