nLab
semiclassical state

under construction

Context

Geometric quantum theory

: Lagrangians and Action functionals + Geometric Quantization

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Prequantum field theory

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Geometric quantization

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Physics

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Contents

Idea

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.

Definition

We first give the traditional definition of semiclassical states according to the WKB method for a non-relativistic particle propagating on the Euclidean space n\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

Semiclassical state of the non-relativistic particle in a potential

Consider the physical system given by a non-relativistic particle of mass mm propagating on the Cartesian space n\mathbb{R}^n with standard kinetic action and sunbject to a force induced by a given potential smooth function V: nV \colon \mathbb{R}^n \to \mathbb{R}.

As a wave function

The Hamilton operator for this system is the standard

H^ 22mΔ+V, \hat H \coloneqq - \frac{\hbar^2}{2 m} \Delta + V \,,

where

Δ=d d= i=1 nx ix i \Delta = \mathbf{d}^\dagger \mathbf{d} = \sum_{i = 1}^n \frac{\partial}{\partial x^i} \frac{\partial}{\partial x^i}

is the Laplace operator on n\mathbb{R}^n regarded as a Riemannian manifold with its canonical flat metric (d\mathbf{d} is the de Rham differential).

Then for

ψ: n× \psi \colon \mathbb{R}^n \times \mathbb{R} \to \mathbb{C}

a smooth 1-parameter collection of smooth functions (of wave functions), the Schrödinger equation is

iddtψ=H^ψ, i \hbar \frac{d}{ dt} \psi = \hat H \psi \,,

where ddt\frac{d}{d t} is the differentiation with respect to the additional parameter (time).

We say that ψ\psi is a stationary solution to the Schrödinger equation if it is a solution of the form

ψ(x,t)=ϕ(x)exp(iωt) \psi(x, t) = \phi(x)\exp(- i \omega t)

for some ω\omega \in \mathbb{R}. For the following it is useful to decompose the remaining complex-valued smooth function

ϕ: n \phi \colon \mathbb{R}^n \to \mathbb{C}

into its modulus and phase by writing it as

ϕ(x)=exp(iS(x)/)a(x) \phi(x) = \exp(i S(x)/\hbar) a(x)

for two smooth functions S: nS \colon \mathbb{R}^n \to \mathbb{R} and a: n 0a \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 ϕ\phi as a function of the form

ϕ: n ×U(1)× >0 \phi \colon \mathbb{R}^n \to \mathbb{C}^\times \simeq U(1) \times \mathbb{R}_{\gt 0}

and then this decomposition is unique up to a global global offset of SS by 2πin2\pi i \cdot n for nn \in \mathbb{Z}.

In terms of this decomposition the Schrödinger equation becomes

0 =(iddtH^)ψ =((|dS| 22m+(Vω))i2mad (a 2dS))exp(iS/)a+𝒪( 2), \begin{aligned} 0 &= \left(i \hbar \frac{d}{dt} - \hat H\right) \psi \\ & = \left( \left( \frac{{\vert \mathbf{d} S \vert}^2 }{2 m } + (V - \hbar \omega) \right) - \frac{i \hbar}{ 2m a} \mathbf{d}^\dagger \left(a^2 \mathbf{d} S\right) \right) \exp( i S / \hbar ) a + \mathcal{O}(\hbar^2 ) \end{aligned} \,,

where dS\mathbf{d} S is the gradient covector field of SS, where d (a 2dS)\mathbf{d}^\dagger ( a^2 \mathbf{d}S) is the divergence of a 2dSa ^2 \mathbf{d}S, and where 𝒪( 2)\mathcal{O}(\hbar^2) denotes all further terms that are non-linear in \hbar.

This means that ψ(,t)=exp(iS/)aexp(iω)\psi(-,t) = \exp(i S / \hbar) a \exp(- i \omega ) is a semiclassical stationary state with energy

Eω E \coloneqq \hbar \omega

if the phase SS and the modulus aa satisfy the following two conditions:

  1. The phase function SS satisfies the Hamilton-Jacobi equation or eikonal? equation

    H(x,S(x))=|dS| 22m+V=E, H(x, \nabla S(x)) = \frac{\vert \mathbf{d} S\vert^2}{2 m } + V = E \,,
  2. The modulus aa is such that a 2dSa^2 \mathbf{d} S satisfies the homogeneous transport equation? in that it is a divergence-free vector field.

As a Lagrangian submanifold of phase space equipped with a half-density

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

T * n 2n. T^* \mathbb{R}^n \simeq \mathbb{R}^{2 n} \,.

Into this space is canonically embedded as the 0-section:

0=(x(x,p=0)): nT * n 0 = (x \mapsto (x, p = 0)) \; \colon \; \mathbb{R}^n \hookrightarrow T^* \mathbb{R}^n

which is a Lagrangian submanifold.

Now every phase function S: nS \colon \mathbb{R}^n \to \mathbb{R} as above induces a deformation of this by regarding the de Rham differential dS\mathbf{d}S as a section of the cotangent bundle

dS: nT *X \mathbf{d}S \colon \mathbb{R}^n \hookrightarrow T^* X

(This is what related phase and phase space in physics.)

This is again a Lagrangian submanifold. We write

π:im(dS) n \pi \colon im(\mathbf{d}S) \to \mathbb{R}^n

for the restriction of the cotangent bundle projection to this Lagrangian submanifold.

The fact that SS satisfies the Hamilton-Jacobi equation means equivalently that this Lagrangian submanifold is the level-set of the Hamiltonian H: nH \colon \mathbb{R}^n \to \mathbb{R} at energy E=ωE = \hbar \omega

im(dS)=H 1(E). im(\mathbf{d}S) = H^{-1}(E) \,.

For the interpretation of the modulus function aa in this reformulation, first notice that for volvol the canonical volume form on n\mathbb{R}^n, the homogeneous transport equation

div(a 2dS)=0 div( a^2 \mathbf{d}S) = 0

is equivalent to

S(a 2vol)=0 \mathcal{L}_{\nabla S} ( a^2 vol ) = 0

where on the left we have the Lie derivative along the gradient of SS. Next observe that

S=π *(v H)| im(dS) \nabla S = \pi_* (v_H)|_{im(\mathbf{d}S)}

where v Hv_{H} is the Hamiltonian vector field corresponding to HH.

This means that the transport equation is equivalently

(v H)π *(a 2vol)=0. \mathcal{L}_{(v_H)} \pi^* (a^2 vol) = 0 \,.

Hence this says that π *a 2vol\pi^* a^2 vol is a volume form on im(dS)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 vol\sqrt{vol}, then avola \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

  1. a Lagrangian submanifold

  2. such that aavol\mathbf{a} \coloneqq a \sqrt{vol} is a half-density on the Lagranian submaifold

    • which in addition is invariant under the Hamiltonian flow.

This formulation now suggests a more general definition of semiclassical states in symplectic geometry/geometric quantization.

In symplectic geometry / geometric quantum theory

(…)

abstracting the above we have that

(…)

order of \hbar𝒪( 0)\mathcal{O}(\hbar^0)𝒪( 1)\mathcal{O}(\hbar^1)𝒪( n)\mathcal{O}(\hbar^n)𝒪( )\mathcal{O}(\hbar^\infty)

References

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.