nLab
semiclassical state

under construction

Context

Geometric quantum theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

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

(…)

classical mechanicssemiclassical approximationformal deformation quantizationquantum mechanics
order of Planck's constant \hbar𝒪( 0)\mathcal{O}(\hbar^0)𝒪( 1)\mathcal{O}(\hbar^1)𝒪( n)\mathcal{O}(\hbar^n)𝒪( )\mathcal{O}(\hbar^\infty)
statesclassical statesemiclassical statequantum state
observablesclassical observablequantum observable

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.