nLab semiclassical state

under construction

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 $\mathbb{R}^n$ with its standard kinetic action and some arbitrary force potential

• Of a non-relativistic particle in a potential

Then we discuss the formalization of this in the broader context of symplectic geometry/geometric quantization in

• In symplectic geometry

Semiclassical state of the non-relativistic particle in a potential

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 $V \colon \mathbb{R}^n \to \mathbb{R}$.

As a wave function

The Hamilton operator for this system is the standard

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

where

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

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

Then for

$$\psi \colon \mathbb{R}^n \times \mathbb{R} \to \mathbb{C}$$

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

$$i \hbar \frac{d}{ dt} \psi = \hat H \psi \,,$$

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

$$\psi(x, t) = \phi(x)\exp(- i \omega t)$$

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

$$\phi \colon \mathbb{R}^n \to \mathbb{C}$$

into its modulus and phase by writing it as

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

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

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

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

In terms of this decomposition the Schrödinger equation becomes

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

where $\mathbf{d} S$ 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

$$E \coloneqq \hbar \omega$$

if the phase $S$ and the modulus $a$ satisfy the following two conditions:

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

   $$H(x, \nabla S(x)) = \frac{\vert \mathbf{d} S\vert^2}{2 m } + V = E \,,$$

2. 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.

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

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

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

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

which is a Lagrangian submanifold.

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

$$\mathbf{d}S \colon \mathbb{R}^n \hookrightarrow T^* X$$

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

This is again a Lagrangian submanifold. We write

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

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

The fact that $S$ 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 $E = \hbar \omega$

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

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

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

is equivalent to

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

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

$$\nabla S = \pi_* (v_H)|_{im(\mathbf{d}S)}$$

where $v_{H}$ is the Hamiltonian vector field corresponding to $H$.

This means that the transport equation is equivalently

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

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

1. a Lagrangian submanifold

   • which is a level-set of the Hamiltonian at the given energy

2. such that $\mathbf{a} \coloneqq a \sqrt{vol}$ 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

(…)

Related concepts

	classical mechanics	semiclassical approximation	…	formal deformation quantization	quantum mechanics
order of Planck's constant $\hbar$	$\mathcal{O}(\hbar^0)$	$\mathcal{O}(\hbar^1)$	$\mathcal{O}(\hbar^n)$	$\mathcal{O}(\hbar^\infty)$
states	classical state	semiclassical state			quantum state
observables	classical observable				quantum observable

References

An introduction to the formulation of semiclassical states in symplectic geometry is in the first section of

• Sean Bates, Alan Weinstein, Lectures on the geometry of quantization (pdf)