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: Lagrangians and Action functionals + Geometric Quantization
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In physics, a semiclassical state is the approximation to a quantum state in semiclassical approximation.
In the original sense of the WKB approximation, in the Schrödinger picture a semiclassical state is a wave function which solves the Schrödinger equation to first order in Planck's constant $\hbar$.
In the broader formalization of quantum physics in symplectic geometry/geometric quantization one finds that such WKB semiclassical states are formalized as being Lagrangian submanifolds of the given phase space symplectic manifold equipped with with a half-density.
We first give the traditional definition of semiclassical states according to the WKB method for a non-relativistic particle propagating on the Euclidean space $\mathbb{R}^n$ with its standard kinetic action and some arbitrary force potential
Then we discuss the formalization of this in the broader context of symplectic geometry/geometric quantization in
Consider the physical system given by a non-relativistic particle of mass $m$ propagating on the Cartesian space $\mathbb{R}^n$ with standard kinetic action and sunbject to a force induced by a given potential smooth function $V \colon \mathbb{R}^n \to \mathbb{R}$.
The Hamilton operator for this system is the standard
where
is the Laplace operator on $\mathbb{R}^n$ regarded as a Riemannian manifold with its canonical flat metric ($\mathbf{d}$ is the de Rham differential).
Then for
a smooth 1-parameter collection of smooth functions (of wave functions), the Schrödinger equation is
where $\frac{d}{d t}$ is the differentiation with respect to the additional parameter (time).
We say that $\psi$ is a stationary solution to the Schrödinger equation if it is a solution of the form
for some $\omega \in \mathbb{R}$. For the following it is useful to decompose the remaining complex-valued smooth function
into its modulus and phase by writing it as
for two smooth functions $S \colon \mathbb{R}^n \to \mathbb{R}$ and $a \colon \mathbb{R}^n \to \mathbb{R}_{\geq 0}$.
In fact it is often useful (such as in the symplecto-geometric interpretation that we turn to below) to restrict attention to non-vanishing solutions (or else to solutions restricted to their support) in which case we can regard $\phi$ as a function of the form
and then this decomposition is unique up to a global global offset of $S$ by $2\pi i \cdot n$ for $n \in \mathbb{Z}$.
In terms of this decomposition the Schrödinger equation becomes
where $\mathbf{d} S$ is the gradient covector field of $S$, where $\mathbf{d}^\dagger ( a^2 \mathbf{d}S)$ is the divergence of $a ^2 \mathbf{d}S$, and where $\mathcal{O}(\hbar^2)$ denotes all further terms that are non-linear in $\hbar$.
This means that $\psi(-,t) = \exp(i S / \hbar) a \exp(- i \omega )$ is a semiclassical stationary state with energy
if the phase $S$ and the modulus $a$ satisfy the following two conditions:
The phase function $S$ satisfies the Hamilton-Jacobi equation or eikonal? equation
The modulus $a$ is such that $a^2 \mathbf{d} S$ satisfies the homogeneous transport equation? in that it is a divergence-free vector field.
The above characterization of semiclassical wave functions of the non-relativistic particle in a potential has a natural equivalent reformulation in terms of symplectic geometry/geometric quantization.
The phase space is
Into this space is canonically embedded as the 0-section:
which is a Lagrangian submanifold.
Now every phase function $S \colon \mathbb{R}^n \to \mathbb{R}$ as above induces a deformation of this by regarding the de Rham differential $\mathbf{d}S$ as a section of the cotangent bundle
(This is what related phase and phase space in physics.)
This is again a Lagrangian submanifold. We write
for the restriction of the cotangent bundle projection to this Lagrangian submanifold.
The fact that $S$ satisfies the Hamilton-Jacobi equation means equivalently that this Lagrangian submanifold is the level-set of the Hamiltonian $H \colon \mathbb{R}^n \to \mathbb{R}$ at energy $E = \hbar \omega$
For the interpretation of the modulus function $a$ in this reformulation, first notice that for $vol$ the canonical volume form on $\mathbb{R}^n$, the homogeneous transport equation
is equivalent to
where on the left we have the Lie derivative along the gradient of $S$. Next observe that
where $v_{H}$ is the Hamiltonian vector field corresponding to $H$.
This means that the transport equation is equivalently
Hence this says that $\pi^* a^2 vol$ is a volume form on $im(\mathbf{d}S)$ which is invariant with respect to the Hamiltonian flow of time evolution.
Finally, if instead of a volume form we choose a half-density $\sqrt{vol}$, then $a \sqrt{vol}$ is another half-density and the condition is that this be invariant under the Hamiltonian flow.
In summary then, the semiclassical wave fuction is equivalently
such that $\mathbf{a} \coloneqq a \sqrt{vol}$ is a half-density on the Lagranian submaifold
This formulation now suggests a more general definition of semiclassical states in symplectic geometry/geometric quantization.
(…)
abstracting the above we have that
(…)
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order of $\hbar$ | $\mathcal{O}(\hbar^0)$ | $\mathcal{O}(\hbar^1)$ | $\mathcal{O}(\hbar^n)$ | $\mathcal{O}(\hbar^\infty)$ | |
An introduction to the formulation of semiclassical states in symplectic geometry is in the first section of
Last revised on March 22, 2013 at 14:12:46. See the history of this page for a list of all contributions to it.