Weyl quantization

The Weyl quantization is a method in quantum mechanics to systematically map a quantum mechanical Hermitian operator to a classical distribution in phase space . This is why it is also called phase space quantization .

The essential correspondence mapping of phase space functions to operators in Hilbert space on which this quantization method is based is called the Weyl transformation . It was first described by Hermann Weyl in 1927 .

In contrast to Weyl's original intention to find a consistent quantization scheme, this mapping is only a change in representation. It does not have to combine classical and quantum mechanical quantities: the phase space distribution may also depend on Planck's constant h. This is the case in some known cases involving angular momentum.

The reverse of this Weyl transformation is the Wigner function . It maps from the Hilbert space into the phase space representation. This reversible change of representation allows quantum mechanics to be expressed in phase space, as suggested by Groenewold and Moyal in the 1940s .

example

In the following, the Weyl transformation is shown on the 2-dimensional Euclidean phase space. The coordinates of the phase space are ; further be a function that is defined everywhere in phase space. The Weyl transformation of is given by the following operator in Hilbert space (largely analogous to the delta distribution ): ${\displaystyle (q,p)}$ ${\displaystyle f}$ ${\displaystyle f}$

${\displaystyle \Phi [f]={\frac {1}{(2\pi )^{2}}}\iint _{q,\,a}\iint _{p,\,b}f(q,p)\left(e^{i(a(Q-q)+b(P-p))}\right)dq\,dp\,da\,db.}$

Now the operators and are taken as generators of a Lie algebra , the Heisenberg algebra : ${\displaystyle P}$ ${\displaystyle Q}$

${\displaystyle [P,Q]=PQ-QP=-i\hbar ,\,}$

Here is the reduced Planck quantum of action. A general element of a Heisenberg algebra can be written as ${\displaystyle \hbar }$

${\displaystyle aQ+bP-i\hbar z.\,}$

The exponential function of an element of a Lie algebra is then an element of the corresponding Lie group :

${\displaystyle g=e^{aQ+bP-i\hbar z},}$

an element of the Heisenberg group . A special group representation of the Heisenberg group is given, then designated ${\displaystyle \Phi }$

${\displaystyle \Phi \left(e^{aQ+bP-i\hbar z}\right)\,}$

the element of the corresponding representation of the group element . ${\displaystyle g}$

The inverse of the above Weyl function is the Wigner function, which brings the operator back to the phase space function : ${\displaystyle \Phi }$ ${\displaystyle f}$

${\displaystyle f(q,p)=2\int _{-\infty }^{\infty }dy~e^{2ipy/\hbar }~\langle q-y|\Phi [f]|q+y\rangle .}$

In general, the function depends on the Planck constant and can describe quantum mechanical processes well, provided it is put together correctly with the star product listed below . ${\displaystyle f}$ ${\displaystyle h}$

For example, the Wigner function of a quantum mechanical operator for an angular momentum square is not identical to the classical operator, but also contains the term which corresponds to the non-vanishing angular momentum of the ground state of a Bohr orbit. ${\displaystyle (L^{2})}$ ${\displaystyle -{\frac {3}{2}}\hbar ^{2}}$

properties

A typical representation of a Heisenberg group is done by the generators of its Lie algebra: A pair of self-adjoint operators ( Hermitian ) on a Hilbert space , so that its commutator , a central element of the group, gives the identity element on the Hilbert space (the canonical commutation relation ) ${\displaystyle {\mathcal {H}}}$

${\displaystyle [P,Q]=PQ-QP=-i\hbar ~\operatorname {Id} _{\mathcal {H}},}$

The Hilbert space can be assumed as a set of quadratically integrable functions over the real number line (plane waves) or a more restricted set, such as the Schwartz space . Depending on the space involved, different properties follow:

• If is a real-valued function, then the image of the Weyl function is self-adjoint. ${\displaystyle f}$ ${\displaystyle \Phi [f]}$

• If there is an element of Schwartz space , then it is a trace class operator . ${\displaystyle f}$ ${\displaystyle \Phi [f]}$

• More general is an unbounded, densely defined operator. ${\displaystyle \Phi [f]}$

• For the Heisenberg Group's standard representation of the square integrable functions, the function corresponds one-to-one to the Schwartz space (as a subspace of the square integrable functions). ${\displaystyle \Phi [f]}$

Generalizations

Weyl quantization is examined in greater generality in cases where the phase space is a symplectic manifold or possibly a Poisson manifold . Related structures are for example Poisson – Lie groups and the Kac-Moody algebras .

credentials

1. ↑ H.Weyl, "Quantenmechanik und Gruppentheorie", Zeitschrift für Physik , 46 (1927) pp. 1-46, doi: 10.1007 / BF02055756 .
2. ^ HJ Groenewold, "On the Principles of elementary quantum mechanics", Physica , 12 (1946) pp. 405-460. (English)
3. ^ JE Moyal, "Quantum mechanics as a statistical theory", Proceedings of the Cambridge Philosophical Society , 45 (1949) pp. 99-124. (English)
4. ^ R. Kubo , "Wigner Representation of Quantum Operators and Its Applications to Electrons in a Magnetic Field," Jou. Phys. Soc. Japan , 19 (1964) pp. 2127-2139, doi: 10.1143 / JPSJ.19.2127 .