The original Uncertainty Principle as formulated by Werner Heisenberg in 1926 was for a particle with a trajectory. According to that principle the uncertainties of the particle's location and momentum could not be established to arbitrary precision. Specifically it said the product of the uncertainties of location and momentum had to be greater than or equal to Planck's constant divided by 4π. The uncertainties are measured by the standard deviations of the probability distributions of position x and momentum p. These are denoted as σ_x and σ_p, respectively. Thus

σ_x·σ_p ≥ h/(4π)

There is a closely related concept called the Observer Effect which says that the properties of a physical system cannot be measured without perturbing the properties of that system. The following does not pertain to the Observer Effect; instead it deals with the probability distributions of undisturbed systems.

The Generalized Uncertainty Principle

Consider a system involving a set of variables {q₁, q₂, … q_n} and their associated momenta {p₁, p₂, … p_n}. Let H be the Hamiltonian function for the system.

Before proceeding further it is necessary to create the mathematical setting for the analysis. Let W be the space of the system. This generally would be the real space R²ⁿ. Let U be the set of complex-valued functions on W. The set U included the set of wave functions Ψ. A wave function ψ is such that it squared value is a probability density function and a probability density function is a nonnegative real-valued function such that its integral over W is equal to unity.

Furthermore there is a complex-valued binary function defined over U, called the inner product; i.e.,

(·, ·): U×U → C

where C denotes the set of complex numbers.

The inner-product function is linear in its arguments; i.e.,

(ax+by, z) = a·(x, z) + b·(y, z)

Let an underscore of a variable denote its complex conjugate. The inner-product function must be such that for all x and y in U an interchange of the argument produces the complex conjugate

(y, x) = (x, y)

This implies that for all x in U, (x, x) is real. It is required that (x, x) be nonnegative and that if (x, x) is equal to zero then x must be the zero vector. It can then be shown that (ax, y) is equal to a·(x, y) and that (x, by) is equal to b·(x, y).

Operators

An operator is merely a function from U to U. The adjoint of an operator S is the operator S* such that

(Sx, y) = (x, S*y)

There are special operators such that S=S*. These are called self-adjoint or Hermitian. For two self-adjoint operators A and B

(x, ABy) = (Ax, By)

Statistical Notation

In the following use is made of statistical notation because physics notation places too much of a burden on brackets of various sorts. The statistical notation includes:

The expected value of x: E{x} = ∫xP(x)dx
The deviations from the expected value: Δx = x − E{x}
The variance: σ_x² = E{(Δx)²}
The standard deviation: σ_x = [E{(Δx)²}]^½

The expected value of x could be represented in terms of the inner product as (x, P(x)).

In physics the probability density function is usually represented by a wave function ψ such that

P = |ψ|² = ψψ

Thus the expected value of a variable x can be represented as

E{x} = (x, P(x)) = ( x, ψ(x)ψ(x)) = (ψ(x)x, ψ(x))

The Bracket Notation

For two operators A and B, BA is not necessarily the same as AB. A binary function [·, ·] is defined for operators such that

[A, B] = AB − BA

The Operators for a Physical System

For a state variable q the associated operator is just multiplication by q. For a momentum variable p associated with q the operator is −;hi(∂/∂q), where h is Planck's constant divided by 2π and i is the square root of −1. Generally physical properties of a system are associated with self-adjoint (Hermitian) operators.

The Generalized Uncertainty Principle

For two self-adjoint operators A and B and any wave function ψ

σ_Aσ_B ≥ ½|(1/i)E{[A, B]}|

Proof:

Let the wave functions for the deviations from the means of A and B, be denoted as

ψ_A = ΔA·ψ
ψ_B = ΔB·ψ

The standard deviations of A and B can then be represented as the magnitudes of ψ_A and ψ_B, respectively. By Schwarz's Inequality

||ψ_A||·||ψ_B|| ≥ |(ψ_A, ψ_B)|

Let

(ψ_A, ψ_B) = c + id
so |(ψ_A, ψ_B)| = (c + id)^½

Since (c + id)^½ ≥ |d|

|(ψ_A, ψ_B)| ≥ |d|

Now consider d, the imaginary component of (ψ_A, ψ_B).

(ψ_A, ψ_B) − (ψ_A, ψ_B) = 2id

But

(ψ_A, ψ_B) = (ψ_B, ψ_A)
and hence
d = (1/2i)[(ψ_A, ψ_B) − (ψ_B, ψ_A)]

But ψ_A is equal to ΔA·ψ and likewise for ψ_B. Thus the above formula for d can be expressed as

d = (1/2i)[(ΔA·ψ, ΔB·ψ) − (ΔB·ψ, ΔA·ψ)]
which reduces to
d = (1/2i)[(ψ, (AB−BA)ψ)]
which, utilizing the bracket notation
for AB−BA is
d = (1/2i)(ψ, [A, B]ψ)

Since this form of the inner product is equivalent to the expected value of [A, B]

d = (1/2i)E{[A, B]}
or, equivalently
d = ½[(1/i)E{[A, B]}]

It was establish initially in the proof that

σ_Aσ_B ≥ |d|
and therefore
σ_Aσ_B ≥ ½|[(1/i)E{[A, B]}]|

Application

For a physical system

[q_j, p_k] = q_j(−hi)(∂/∂q_k) − (−hi)(∂/∂q_k)(q_j)
which reduces to
hi(∂q_j/∂q_k) = hiδ_jk

Therefore

σ_qj·σ_pk ≥ ½hδ_jk

If j≠k then q_j and p_k can be measured simultaneously to any degree of accuracy.