An Almanack

Complex Time in Quantum Tunneling

Perhaps the most spectacular early prediction of quantum mechanics was tunneling: that particles can do things that are forbidden in Newton’s mechanics, although with a small probability. A particle of mass $m$ and energy $E$ moving in a potential $V(x)$
must satisfy the conservation of energy

${p^2\over 2m}+V(x)=E.$

In Newtonian mechanics, kinetic energy is always positive, being the square of a real quantity $p$ , the momentum divided by a positive quantity $m$ . Thus only values of $x$ where $E>V(x)$ are allowed. As a particle approaches the boundary of this region, where $E=V(x)$ , its momentum will vanish. The normal component of momentum will then change sign, meaning that the particle is reflected back into the region with $E>V(x)$ . Thus the boundary consists of turning points . But in quantum mechanics, the particle can continue into the forbidden region. The normal component of momentum becomes imaginary (allowing the kinetic energy to become negative). An imaginary momentum has a physical meaning in quantum mechanics: the probability of finding the particle in this region decays exponentially with the distance from the boundary.

The Classical Limit
We can derive this in the approximation where Plank’s constant $\hbar\to 0$ from the Schrodinger equation
$-{\hbar^2\over 2m}\nabla^2\psi+V(x)\psi=E\psi.$
This is a tricky approximation because the wave-function itself has an essential singularity in the limit $\hbar\to 0$ . For example, for a free particle
$\psi(x)=e^{{i\over \hbar}px}$ : there is no sensible limit as $\hbar\to 0$ . The change of variable
$\psi(x)=e^{{i\over \hbar}W(x)}$
gives a quantity $W$ (` the eikonal’) that does have a limit as
$\hbar\to 0$ . It satisfies
${(\nabla W)^2\over 2m}+V(x)+{i\hbar \over 2m}\nabla^2 W=E.$
In the limit
$\hbar\to 0$ we get the first order equation
${({\nabla W})^2\over 2m}+V(x)=E.$
Clearly $\nabla W$ has the physical meaning of momentum.

Since it no longer has any $\hbar$ in it, this last equation ought to have a meaning in classical mechanics. Indeed, it is the celebrated Hamilton-Jacobi equation, derived long before quantum mechanics was invented. So how is this related to Newton’s equation of motion?:

$m{d^2x\over dt^2}=-\nabla V$

Mass times acceleration equals the force, which is the negative gradient of potential energy.

The point is that every first order PDE of the type

$H\left(x,{\partial W\over \partial x}\right)=E$
is equivalent to a system of ODEs, whose solutions determine its characteristic curves. This is a beautiful piece of mathematical physics from the era of steam engines : second half of nineteenth century. The systematic derivation uses Poisson brackets and Hamilton-Jacobi theory. The way to get the path of a particle from a solution to the Hamilton-Jacobi equation is to use
$p={\partial W\over \partial x},\quad t={\partial W\over \partial E}.$
We can then eliminate $E$ and express $x,p$ as functions of time $t$ . It turns out that this is equivalent to solving the differential equations
${dx\over dt}={\partial H\over\partial p}\quad {dp\over dt}=-{\partial H\over \partial x}.$

These are for us
${dx\over dt}={p\over m}\quad {dp\over dt}=-\nabla V$
which is Hamilton’s way of writing Newton’s second law.

So why go through this familiar story here? After all, it is in every respectable quantum mechanics textbook¹.

In the Forbidden Region
We want to look now at the classically forbidden region in the same approximation $\hbar\to 0$ . We will see some things that I have never seen in any quantum mechanics textbook and are only hinted at even in research papers. ( e.g., Son and Rubakov, Ezra)

When
$E<V(x)$ , it is clear that $W$ can no longer be real. In the famous example of tunneling across a one dimensional potential barrier, $W'$ is purely imaginary. an imaginary value of momentum is perfectly sensible physically: it just means that the wavefunction is exponentially decreasing in this region, instead of oscillating as in the classically allowed region. Suppose $x_1$ is a turning point such that $E>V(x)$ for $x<x_1$ and $E<V(x)$ for $x>x_1$ . Then the wavefunction in the forbidden region is

$\psi(x)\approx e^{-{1\over \hbar}\int_{x_1}^x\sqrt{2m[V(x)-E]}}dx.$

If we put $W=i\tilde W$ , the Hamilton-Jacobi equation becomes in the forbidden region
$-{\tilde W^\prime^2\over 2m}+V(x)=E.$
It can be thought of as having the hamiltonian $\tilde W^\prime\to \tilde p$
$\tilde H(\tilde p,x)=-{{\tilde p}^2\over 2m}+V(x)$
Just like any other first order PDE, we can turn this into a system of ODEs,
${dx\over d\tilde t}=-{\tilde p\over m},\quad {d\tilde p\over d\tilde t}=-\nabla V.$
These lead to a Newton-like law in the forbidden region
$-m{d^2 x\over d{\tilde t}^2}=-\nabla V.$
This can be interpreted as Newton’s laws in {\em imaginary time}. A particle in classically forbidden motion of this kind is called an instanton: an idea invented by Gerard ‘t Hooft ( independently by Polyakov and also by Gribov).

Thus it looks like momentum is either real, in classically allowed regions, or purely imaginary, in forbidden regions. Momentum space including tunneling effects is thus a pair of lines that intersect at the point of zero momentum: not a manifold, but a real algebraic variety with a mild singularity at the origin. It has the shape of a cross. Phase space is the product of this `cross’ by the real line, leading to a pair of planes that intersect along a line.

What happens with more degrees of freedom? It is not hard to think of situations where one component of momentum is imaginary and the others are real: think of a potential barrier shaped like a slab, constant in two directions and a step function in the other. The momentum normal to the boundary can become imaginary while the remaining components remain real even in the forbidden region. Burgess has shown by additional examples that in general, an imaginary time isonly part of the story. Time can become a complex variable with both real and imaginary parts. But that is not the whole story either.

A complex $W$ is perfectly sensible in quantum mechanics. Thus we should expect that the momentum and time also become complex
$p={\partial W\over \partial x},\quad t={\partial W\over \partial E}.$
The position $x$ and energy $E$ remain real in this picture.
Also, $p^2$ is still real, since
$p^2=2m[E-V(x)]$ :
${\rm Im}\; p^2=0$
In other words, the real and imaginary parts of $p$ are always orthogonal. Momentum space is the set of pairs of orthogonal vectors
$V_n=\{p=k+i\tilde k|k\cdot \tilde k=0\}.$
When $n=1$ this variety is shaped like a cross, but is a kind of cone in more general situations. The subset where $|k|>|\tilde k|$ has positive kinetic energy while the tunneling region has $|k|<|\tilde k|$ . The hamiltonian is
$H(k,\tilde k,x)={k^2-\tilde k^2\over 2m}+V(x)$
What are the charactersitic curves? Introduce a complex time variable
$t=\tau+i\tilde\tau$
and write Hamilton's equations
${dx\over dt}={p\over m},\quad {dp\over dt}=-\nabla V$
separated into real and imaginary parts:
${\partial x\over \partial \tau}={k\over m}\quad {\partial x\over \partial \tilde\tau}=-{\tilde k\over m}$
${\partial k\over \partial \tau}+{\partial \tilde k\over \partial \tilde \tau}=-\nabla V.$
The imaginary part $dp\over dt$ is identically zero because of the first equation. These are to be supplemented by the condition
$k\cdot \tilde k=0.$

A Semi-classical Equation of Motion

Newton’s equations are then replaced by the PDE
$m\left[{\partial^2 x\over \partial \tau^2}-{\partial^2 x\over \partial{\tilde\tau}^2}<br /> \right]=-\nabla V,\quad {\partial x\over \partial\tau}\cdot{\partial x\over\partial \tilde\tau}=0.$
If $x$ has only one component this splits into cases where it depends on $\tau$ alone or $\tilde\tau$ alone.

This is a system of hyperbolic Partial Differential equations, much like an inhomogenous wave equation. The imaginary part of time acts like a spatial co-ordinate in this wave equation. There is in addition a condition on the initial data, that the spatial and time derivatives be orthogonal. Can we find some explicit solutions with just two degrees of freedom? No one seems to have explored this terrain. The case of a quadratic $V$ leads to linear equations, so is the simplest case to study first. ( Unless you are tired of the harmonic oscillator.)

There is an analogy here with the Virasoro condition of string theory but I don’t know what to make of that.

Evanescent Waves

An optical analogy (see for example, Born and Wolf ) helps to visualize what is going on. Consider a plane boundary between two regions of refractive indexes $n_1>n_2$ . If a ray of light arrives at the surface with a small angle of incidence $\theta$ it is refracted as it passes into the second medium through an angle $r$ given by
$\sin r={n_1\over n_2}\sin \theta.$
As we increase $\theta$ , $r$ will increase as well, until at a critical angle of incidence $\theta_c=\arcsin {n_2\over n_1}$ , the outgoing ray grazes the surface, $r={\pi \over 2}$ . What happens if if we increase $\theta>\theta_c$ is that the light is reflected back into the denser medium. But there is an exponentially decaying field in the lighter medium, whose wavenumber vector is $(kn_2\sin \theta,ik\sqrt{n_2^2\sin^2\theta-n_1^2})$ . (We are assuming that the boundary is the $x^1-x^3$ -plane and that the incoming light ray lies in the $x^1-x^2$ plane.) Thus there is a real component parallel to the boundary and an imaginary normal component. So the ray can be thought of as spread along the $x^1-x^2$ plane, with an oscillation in the $x^1$ direction and a decay in the $x^2$ direction.

This plane is the one containing the normal to the surface and the incoming light ray, extended into the lighter medium.A detector placed in the second medium out to see these photons, concentrated on this plane. Thus a particle can be thought of as occupying a plane during tunneling. The initial condition for the solution of the hyperbolic PDE above would be specified on $\tau=0$ .

1. L. Landau and Lifshitz Quantum Mechanics ; For those who find this classic too intimidating, there is the delightful Principles of Quantum Mechanics by R. Shankar

One Response to “Complex Time in Quantum Tunneling”

Sunday mid-morning links! « Entertaining Research Says:
October 21st, 2007 at 4:18 pm
[…] Rajeev at his Almanack has a nice and detailed (albeit a bit mathematical) essay on quantum tunnelin…: Perhaps the most spectacular early prediction of quantum mechanics was tunneling: that particles can do things that are forbidden in Newton’s mechanics, although with a small probability. […]

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Complex Time in Quantum Tunneling

One Response to “Complex Time in Quantum Tunneling”

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