Werner Heisenberg

In the 1930's the German physicist, Werner Heisenberg, articulated the principle that the more precisely the position of a particle is known the less precisely is known its momentum and vice versa. This was called the Uncertainty Principle. More precisely the Uncertainty Principle is that if σ_x is the standard deviation of the location of the particle and σ_p is the standard deviation of its momentum then

σ_x·σ_p ≥ ½h

where h is Planck's constant divided by 2π.

Crude Illustrations of the Uncertainty Principle

Light Diffracted through a Slit

Although for the Uncertainty Principle ithe uncertainty of a variable is properly represented by its standard deviation there are other measures of uncertainty, such as a variable's range, that can be used to illustrate the Uncertainty Principle although the lower limit on the product may be some multiple of ½h rather than ½h itself.

Assume that light of wavelength λ is passing through a slit of width a. By the de Broglie relation the momentum p of a photon in this light is

p = h/λ

where h is Planck's constant.

As a result of passing through the slit the momentum of a photon is not changed in magnitude but it may be changed in direction. The component of momentum in the direction of the slit, say p_y is psin(θ) where θ is the angle of diffraction for the photon. There is a distribution of diffraction angles that involves maxima and minima. There is a peak at θ=0. The first minima occur at θ such that sin(θ)=±λ/a. Since for small θ, sin(θ)≅θ the first minima for the distribution of the diffraction angle are at θ=±(λ/a). Thus λ/a can be taken to be a measure of the uncertainty in sin(θ). Therefore

Δ(p_y) = pΔ(sin(θ)) = p(λ/a)
but
p = h/λ
and hence
Δ(p_y) = (h/λ)(λ/a) = h/a

The uncertainty in the location y of the photon may be taken to be the width of the slit, a; i.e., Δy=a. Thus

Δy·Δp_y = a(h/a) = h ≥ ½h

Observation of a Particle with a Microscope

Consider a particle Q on plate of a microscope where the angle subtended by the objective lens of the microscope is θ. The plate is illuminated from below by light of wavelength λ. A photon of the light has a momentum of p which is equal to h/λ. The particle is observed when a photon is scattered by the particle and then enters the objective lens and is then guided by the optics of the microscope into the eye of the observer.

The direction of the momentum of the photon is changed by its collision with the particle. The momentum of the photon is uncertain but for the photon to be observed by the observer its direction has to be within the angle θ subtended by the objective lens from the viewpoint of the particle. This means that the x-component of the momentum of the photon has to be between +psin(θ/2) and −psin(θ/2).

The particle experiences a recoil from the collision of photon with it that results in the x-component of its momentum, p_x, also being between −psin(θ/2) and +psin(θ/2). The range of the x-component of the momentum of the particle is 2psin·(θ/2). Hence

Δp_x = 2p·sin(θ/2) Δp_x = 2(h/λ)sin(θ/2)

What is observed at the eye-piece of the microscope is the diffraction pattern of the photons scattered by the particle. The width of that diffraction pattern is λ/sin(θ/2). This is called the resolving power of the microscope. Thus

Δx = λ/sin(θ/2)

The product of the ranges of the location x and the x-component of the momentum of the particle is then

Δp_x·Δx = 2(h/λ)sin(θ/2)(λ/sin(θ/2)) = 2h
which of course is greater than ½h

A Proper Derivation of the Uncertainty Principle

A quantum mechanical system is characterized by a complex function defined over space called its wave function. The wave function is such that its squared magnitude is equal to the probability density for the system. That is to say, if ψ(X) is the wave function value for a particle at the point X then the probability density at X is |ψ(X)|²=ψ(X)ψ(X), where ψ(X) is the complex conjugate of ψ(x).

For a system with a wave function ψ(X) the expected value of a variable f(X) for the system is given by

E(f(X)) = ∫f(X)|ψ(X)|²dX = ∫ψ(X)f(X)ψ(X)dX

In quantum mechanics the expected value of a variable f(X) is denoted as <f>.

The Schroedinger Equation

Erwin Schroedinger

The wave function for a system is found as a solution to its Schroedinger equation. The Schroedinger equation for a system is derived from its Hamiltonian function. The Hamiltonian function of a system is its total energy, kinetic plus potential, with the kinetic energy expressed as a function of its momentum.

In mathematics an operator is a function defined over a set of functions and whose values are functions. Such a function of functions is also called a functional. The Schroedinger equation for a system is derived from its Hamitonian Operator, which is its Hamiltonian function with momentum being replaced by ih∇ where i is the imaginary unit, h is Planck's constant divided by 2π and ∇ is the gradient operator for the sytem. For a one dimensional system in which the spatial variable is x the gradient operator is ∂/∂x.

If h is the Hamiltonian operator for the system and t is time then the Schroedinger equation for the system is

hψ = ih∂(ψ/∂t)

Deviations from the Expected Values

The deviation of a variable f(X) from its expected value is expressed

Δf = f(X) − <f>

The expected value of the squared deviations of a variable is called its variance. Let V(f) be the variance of the variable f(X). Then

V(f) = <(Δf)²)> = ∫f(X)|ψ(X)|²dX = ∫ψ(X)(Δf)⊃ψ(X)dX

A One Dimensional System

Let x be the spatial dimension and p the linear momentum for a system. Consider the deviations Δx and Δp. The operator corresponding to Δp is denoted ΔP.

The Schwartz Inequality

Let f and g be two complex functions over the variable x. The Schwartz Inequality is then

[∫|f|²dx]·[∫|g|²dx] ≥ |∫fgdx|²

In the Schwartz Inequality let f be Δx·ψ and g be ΔP·ψ. Then, since Δx is real and thus the same as its conjugate,

V(x)·V(p) ≥ |∫ψΔx·ΔPψ|²

The integral ∫ψΔx·ΔPψ is equivalent to <Δx·Δp>. Thus

V(x)·V(p) ≥ |<Δx·Δp>|²

The commutator of two operators, R and Q, is defined as

[R, Q] = RQ − QR

Thus an operator RQ can be expressed

RQ = RQ − ½QR + ½QR = ½RQ − ½QR + ½RQ + ½QR
which is equivalent to
RQ = ½[R, Q] + ½(RQ + QR)

For the case of the operator ΔxΔP this means

ΔxΔP = ½[Δx, ΔP] + ½(ΔxΔP + ΔPΔx)

The commutator of Δx and ΔP is equal to ih. Thus

ΔxΔP = ½ih + ½(ΔxΔP + ΔPΔx)

The expected value of ΔxΔP is the same as the expected value of ΔxΔp. And likewise the expected values of ΔxΔP and ΔPΔx are the same as those of ΔxΔp and ΔpΔx. Thus

<ΔxΔp> = ½ih + ½<ΔxΔp + ΔpΔx>

Thus on the RHS of the above equation the term ½ih is purely imaginary and the other term is purely real. Thus the magnitude of their sum is

|<ΔxΔp>|² = ¼h² + ¼<ΔxΔp + ΔpΔx>²

Since the second term on the RHS is nonnegative

|<ΔxΔp>|² ≥ ¼h²
and since
V(x)·V(p) ≥ |∫ψΔx·ΔPψ|²
it follows that
V(x)·V(p) ≥ ¼h²

This is the Uncertainty Principle. Stated in terms of the standard deviations σ_x and σ_p, which are the square roots of the variances,

σ_x·σ_p ≥ ½h

Thus the product of the uncertainties of the location and of the momentum of a particle must be at least as great as ½h.

The Uncertainty Principle and Chaos Theory

Newtonian mechanics indicates that if the locations and velocities of the particles of a system are exactly known at any time then the future locations and velocities can be computed with any desired degree of precision. However Edward N. Lorenz discovered that the solutions to systems of nonlinear dynamic equations can be infinitely sensitive to initial conditions. The deviations resulting from slight deviations in initial conditions can grow at enormous rates such that after a period of time the deviations are substantial. Lorenz was concerned with meteorological prediction and the implication of his discovery is that meteorologist would not be able to make useful forecasts of the weather beyond ten days or so, but it applied to any dynamic system involving nonlinearity. In the case of meteorology the limit of the accuracy of the measurement of initial conditions was the technological problem of the accuracy of the measuring instruments. In the case of the dynamics of systems of particles the limit of accuracy is intrinsic due to the Uncertainty Principle.