Bayesian programming

In Quantum Mechanics, universality is the observation that there are properties for a large class of systems that are independent of the exact structural details of the system. The notion of Universality (dynamical systems) is familiar in the study and application of statistical mechanics to various physcial systems since its introduction in a very precise fashion by Leo Kadanoff. Although not quite the same, it is closely related to universality as applied to quantum systems. This concept links to the essence of renormalization and scaling in many problems. Renormalization is based on the notion that a measurement device of wavelength $\lambda$ is insensitive to details of structure at distances much smaller than $\lambda$. An important consequence of universality is that we can mimic the real short-structure distance of the measurement device and the system to be measured by simple short-distance structure. Even though it is seen that scaling, universality and renormalization are closely related, they are not to be used interchangeably.

Universal Characteristics of Shapes of Potential Steps

For low energy particles, the exact detail of the potential step is irrelevant. This means that the intermediate change in the potential between $0$ and $V_{0}$ is irrelevant for particles with extremely low energies. An important consequence of this is that theoretical calculations made for a Heaviside step function potential work perfectly well for potentials that have a finite spatial rate of change in potential: finite ∆$V$ over ∆$x$, which is the case with physical situations, for e.g. - Quantum confinement systems of nano dots, etc.

For a general potential barrier:

$V(x)={\begin{cases}0&x<-a\\V(x)&-a<x<a\\V_{0}&a<x\end{cases}},$

The time-dependent Schrödinger's equation in one-dimension is:

$i\hbar {\frac {\partial }{\partial t}}\Psi (x,t)=\left[{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+V(x,t)\right]\Psi (x,t)$

It is instructive to analyse this equation after expanding $\Psi (x,t)$ in the plane wave basis $exp[i(px-Et)/\hbar ]$. $p$, then, is real for $E$ > $V$. For when $E$ < $V$, the relevant length scale in this problem is the range of the wavefunction under the barrier which is given by: $r={\frac {\hbar }{\sqrt {2m(V_{0}-E)}}}$ ≃ $r={\frac {\hbar }{\sqrt {2mV_{0}}}}$ for low energy particles, i.e. - $E$ → $0$. When $r$ ≫ $a$, the wave cannot resolve the detailed shape of the potential and one must have $R=1$. The wavelength of a particle in the region of zero potential (essentially, a free particle) with energy $E$ is given as: $\lambda ={\frac {\hbar }{\sqrt {2mE}}}$. For low-energy waves incident on this potential from the left: $\lambda$ ≫ $a$. Essentially, for low energy, or large wavelength, particles, the exact detail of the potential barrier is irrelevant. This is a feature that is universal to different $V(x)$ for a well behaved function $V(x)$.

Universal Characteristics of Potential Barriers

Potentials display universality in their structural aspects and their resonance phenomena. For scattering at a finite potential barrier of height $V_{0}$, the time-independent Schrödinger equation for the wave function $\psi (x)$ reads:

$H\psi (x)=\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)\right]\psi (x)=E\psi (x)$

where $H$ is the Hamiltonian, $\hbar$ is the (reduced) Planck constant, $m$ is the mass, $E$ the energy of the particle and

$V(x)=V_{0}[\Theta (a-x)-\Theta (a+x)]$

is the barrier potential with height $V_{0}>0$ and width $2a$. $\Theta (x)=0,\;x<0;\;\Theta (x)=1,\;x>0$ is the Heaviside step function.

In terms of dimensionless parameters $y=x/a$, $z=E/V_{0}$, $\rho ={\frac {a{\sqrt {2mE}}}{\hbar }}$ and $\rho '=\rho {\sqrt {\frac {z-1}{z}}}$

$\psi _{I}(x)=A_{I}e^{i\rho y}+A_{II}e^{-i\rho y}\quad y<-1$

$\psi _{II}(x)=B_{I}e^{i\rho 'y}+B_{II}e^{-i\rho 'y}\quad |y|<1$

$\psi _{III}(x)=C_{I}e^{i\rho y}+C_{II}e^{-i\rho y}\quad y>1$

With the subscripts I, II and III to $\psi$ given by the regions of y mentioned alongside their expressions.

For a wave that is incident from the left, we have $C_{II}=0$ and one finds the reflection coefficient to be:

$R={\frac {(\rho ^{2}-\rho '^{2})^{2}(\sin(2\rho ))^{2}}{4\rho ^{2}\rho '^{2}+(\rho ^{2}-\rho '^{2})^{2}(\sin(2\rho ))^{2}}}$

The wavelength of the incident wave is an exact multiple of the barrier width whenever $2\rho =2n\pi$. For such energies, one finds that $R=0$ and $T=1$. In terms of the dimensionless variables, these Resonances in scattering from potentials occur at energies, $E_{n}^{*}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2ma^{2}}}$.

The quantity $r^{2}$ defined as:

$r^{2}:=\rho ^{2}-\rho '^{2}={\frac {2mV_{0}a^{2}}{\hbar ^{2}}}$

is a shape property of the potential, and is independent of the energy. Two different potentials (for $V_{0}$ and $a$) with the same $r$ have the same physics. This can be heuristically understood as a relation to the relevant length scales in the problem: $a$ is the length of the potential and this identifies a momentum scale for this problem as, setting $\hbar =1$, $1/a$. In mass weighted scales, we identify an energy scale as $1/a^{2}$ and form a dimensionless quantity using the height of the potential and the energy scale $1/a^{2}$ as $V_{0}a^{2}$. Reintroducing $\hbar$ and $m$, we discover: $2mV_{0}a^{2}/\hbar ^{2}$, which is precisely our shape parameter $r$.

Using $r$, the reflection coefficient becomes:

$R={\frac {r^{4}(\sin(2\rho ))^{2}}{4\rho ^{2}\rho '^{2}+r^{4}(\sin(2\rho ))^{2}}}$

For $2\rho =2n\pi +\delta$, one finds $R$ ≅ ${\frac {r^{4}\delta ^{2}}{16n^{4}\pi ^{4}}}$ ∝ $\delta ^{2}$. The power of $\delta$ is universal in the sense that it is independent of $n$. However, the constant of proportionality depends on $n$.

Other Systems

A complex current source J(r,t) of size d that generates radiation with wavelengths $\lambda >>d$ is accurately mimicked by a sum of point-like multipole currents. The Multipole Expansion is a simple example of a renormalization analysis. In thinking about the long-wavelength radiation it is generally much easier to treat the source as a sum of multipoles than to deal with the true current directly. This is particularly true since usually only one or two multipoles are needed for sufficient accuracy.

In a Quantum Field Theory, a Quantum Fluctuation probes arbitrarily short distances. This is evident when one computes radiative corrections in perturbation theory which seem infinitely sensitive to short-distance behavior. Universality tells us that to look at the low momentum physics, or the large wavelength limit, we do not need to know what happens at large momenta. As in the multipole expansion, we can replicate the highly non-trivial high-energy by a generic set of effective point-like interactions.

References

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