Ortsoperator – Wikipedia

The Orts operator In quantum mechanics belong to the local measurement of particles.

The physical condition

${displaystyle Psi }$

A particle is mathematically given in quantum mechanics by the associated vector of a Hilbert dream H . This condition is consequently in the BRA-KET notation by the vector

${displaystyle |Psi rangle }$

described. The observables are opened by self -cubs H shown.

The local operator in particular is the summary of the three observable

${displaystyle {hat {mathbf {x} }}=({hat {x}}_{1},{hat {x}}_{2},{hat {x}}_{3})}$

, so that

${Displastyle E ({Hat {x}} _ {j}) = {langle {Hat {x}} _ {j}, psi, psi, psi row} _ {mathrm {h}}, quad j = 1,2,3 }$

The mean (expectation value) of the measurement results of the J-Ten location coordinate of the particle in the state

${displaystyle Psi }$

is.

${Displaystyle [{has {x}} _ {j}, {has {p}} _ {k}] = mathrm {i}, hbar, delta _ {jk}, quad [{has {x}} _ {j }, {has {x} _ {k}] = 0 = [{has {p}} _ {j}, {has {p}} _ {k}], quad j, kin {1,2,3 }}$

• It follows that the three local coordinates are measurable together and that their spectrum (area of ​​the possible Measured values ) from all over the room
  ${displaystyle mathbb {R} ^{3}}$

  consists. The possible places are not quantized, but continuously.

Local representation [ Edit | Edit the source text ]

The local presentation is defined by the spectral presentation of the local operator. The Hilbertraum

${displaystyle H=L^{2}(mathbb {R} ^{3};mathbb {C} )}$

Is the space of the square integrable complex functions of the local area

${displaystyle mathbb {R} ^{3}}$

, every condition

${displaystyle Psi }$

Is through a local wave function

${displaystyle psi (mathbf {x} )}$

given.

The local operators

${displaystyle {hat {mathbf {x} }}=({hat {x}}_{1},{hat {x}}_{2},{hat {x}}_{3})}$

are the The multiplication operator With the coordinate functions, i.e. H. The local operator

${displaystyle {hat {x}}_{j}}$

acts on local wave functions

${displaystyle psi (mathbf {x} )}$

By multiplication of the wave function to the coordinate function

${displaystyle x_{j}}$

${DisplayStyle ({Hat {x}} _ {j}, psi) (mathbf {x}) = x_ {j} cdot psi (mathbf {x})}$

This operator

${displaystyle {hat {x}}_{j}}$

Is a multiplication operator
A densely defined operator and completed.
He is on the underground

${displaystyle D={psi in H,|,xcdot psi in H}}$

defined that lies in h.

The expectation value is

${Displastyle e ({Hat {x}} _ {j}) = {langle {Hat {x}} _ {j}, psi, psi, psi, psi, psi, psi rusle} _ {l^{2}} = int _ {mathbb {r} ^{3}} x_ {j}, psi (mathbf {x}), {overline {psi (mathbf {x})}}, mathrm {d} x = int _ {mathbb {r} ^{3}} x_ {J}, | Psi (Mathbf {x}) |^{2} Mathrm {d} x}$

The impulse operator acts as a differential operator on local wave functions (if the phases are suitable):

${displaystyle {bigl (}{hat {p}}_{k}psi {bigr )}(mathbf {x} )=-mathrm {i} ,hbar ,{frac {partial }{partial x_{k}}}psi (mathbf {x} )}$

Self -functions [ Edit | Edit the source text ]

The own functions of the local operator must

${displaystyle ({hat {x}}, psi _ {mathbf {x_ {0}}}) (mathbf {x}) = mathbf {x_ {0}} cdot psi _ {mathbf {x_ {0}}} (mathbf {X})}$

fulfill, whereby

${displaystyle psi _{mathbf {x_{0}} }(mathbf {x} )}$

The own function of the local operator to the eigenvalon

${displaystyle mathbf {x_{0}} }$

represent.

The own functions

${displaystyle psi (mathbf {x_{0}} )}$

Delta distributions correspond to the local operator:

${displaystyle {hat {mathbf {x}}} delta (mathbf {x} -mathbf {x_ {0}}) = mathbf {x_ {0}} delta (mathbf {x} -mathbf {x_ {0}})}}}}$

With identity:

${displaystyle f(x)delta (x-x_{0})=f(x_{0})delta (x-x_{0})}$

Impulse presentation [ Edit | Edit the source text ]

In the impulse presentation, the pulse operator has a multiplicative effect on impulse wave functions

${Displaystyle {tilde {psi}} (mathbf {p})}$

${displaystyle ({hat {p}} _ {k}, {tilde {psi}}) (mathbf {p}) = p_ {k} cdot {tilde {psi}} (mathbf {p})}$

and the local operator as a differential operator:

${hat {x}} _ {{j} {tilde {psi}}) (Math} {psi) = {PSI) = Mathrm {}, HBAR, {frac {partial} {partial p_ {}}} {tilde {}} (MathBF {P})}$