Position operator

In quantum mechanics, the position operator belongs to the position measurement of particles .

The physical state of a particle is in quantum mechanics given mathematically by the associated vector of a Hilbert space H . This state is thus described in Bra-Ket notation by the vector . The observables are self-adjoint operator to  H shown. ${\displaystyle \Psi }$  ${\displaystyle |\Psi \rangle }$

In particular, the position operator is the combination of the three observables such that ${\displaystyle {\hat {\mathbf {x} }}=({\hat {x}}_{1},{\hat {x}}_{2},{\hat {x}}_{3})}$

${\displaystyle E({\hat {x}}_{j})={\langle {\hat {x}}_{j}\,\Psi ,\Psi \rangle }_{\mathrm {H} }\ ,\quad j=1,2,3}$

is the mean value ( expected value ) of the measurement results of the j-th position coordinate of the particle in the state . ${\displaystyle \Psi }$

Definition and characteristics

• The three position operators are self-adjoint operators which, together with the self-adjoint momentum operators, satisfy the following canonical commutation relations: ${\displaystyle {\hat {x}}_{j}}$ ${\displaystyle {\hat {p}}_{k}}$

${\displaystyle [{\hat {x}}_{j},{\hat {p}}_{k}]=\mathrm {i} \,\hbar \,\delta _{jk}\ ,\quad [{\hat {x}}_{j},{\hat {x}}_{k}]=0=[{\hat {p}}_{j},{\hat {p}}_{k}]\ ,\quad j,k\in \{1,2,3\}}$

• It follows from this that the three position coordinates can be measured together and that their spectrum (range of possible measured values ) consists of the entire room . The possible locations are therefore not quantized, but rather continuous . ${\displaystyle \mathbb {R} ^{3}}$

Location representation

The location representation is defined by the spectral representation of the location operator. The Hilbert space is the space of the square integrable complex functions of the spatial space , each state is given by a spatial wave function. ${\displaystyle H=L^{2}(\mathbb {R} ^{3};\mathbb {C} )}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \Psi }$ ${\displaystyle \psi (\mathbf {x} )}$

The position operators are the multiplication operators with the coordinate functions, i.e. H. the position operator acts on position wave functions by multiplying the wave function by the coordinate function ${\displaystyle {\hat {\mathbf {x} }}=({\hat {x}}_{1},{\hat {x}}_{2},{\hat {x}}_{3})}$ ${\displaystyle {\hat {x}}_{j}}$ ${\displaystyle \psi (\mathbf {x} )}$ ${\displaystyle x_{j}}$

${\displaystyle ({\hat {x}}_{j}\,\psi )(\mathbf {x} )=x_{j}\cdot \psi (\mathbf {x} )}$

As a multiplication operator , this operator is a tightly defined operator and is closed . It is defined on the subspace that is dense in H. ${\displaystyle {\hat {x}}_{j}}$ ${\displaystyle D=\{\psi \in H\,|\,x\cdot \psi \in H\}}$

The expected value is

${\displaystyle E({\hat {x}}_{j})={\langle {\hat {x}}_{j}\,\Psi ,\Psi \rangle }_{L^{2}}=\int _{\mathbb {R} ^{3}}x_{j}\,\psi (\mathbf {x} )\,{\overline {\psi (\mathbf {x} )}}\,\mathrm {d} x=\int _{\mathbb {R} ^{3}}x_{j}\,|\psi (\mathbf {x} )|^{2}\mathrm {d} x}$

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

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

Eigenfunctions

The eigenfunctions of the position operator must have the eigenvalue equation

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

fulfill, where represents the eigenfunction of the position operator for the eigenvalue . ${\displaystyle \psi _{\mathbf {x_{0}} }(\mathbf {x} )}$ ${\displaystyle \mathbf {x_{0}} }$

The eigenfunctions for the position operator correspond to delta distributions : ${\displaystyle \psi (\mathbf {x_{0}} )}$ ${\displaystyle {\hat {\mathbf {x} }}\delta (\mathbf {x} -\mathbf {x_{0}} )=\mathbf {x_{0}} \delta (\mathbf {x} -\mathbf {x_{0}} )}$

with the identity: ${\displaystyle f(x)\delta (x-x_{0})=f(x_{0})\delta (x-x_{0})}$

Momentum display

In the momentum representation, the momentum operator has a multiplicative effect on momentum wave functions ${\displaystyle {\tilde {\psi }}(\mathbf {p} )}$

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

and the position operator as a differential operator:

${\displaystyle ({\hat {x}}_{j}\,{\tilde {\psi }})(\mathbf {p} )=\mathrm {i} \,\hbar \,{\frac {\partial }{\partial p_{j}}}{\tilde {\psi }}(\mathbf {p} )}$

literature

• Jochen Pade: Quantum Mechanics on Foot 1 . Springer, Berlin, Heidelberg 2012, ISBN 978-3-642-25226-6 , doi : 10.1007 / 978-3-642-25227-3 .