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Mormonism and Physics, Astronomy, Quantum Mechanics, String Theory, M-theory, etc.
This article attempts to accumulate general Mormon views regarding physics, astronomy, quantum mechanics, string theory, m-theory, the omniverse, as well as philosophy.

Mormon Apologetics and Studies

 * Neal A. Maxwell Institute for Religious Scholarship
 * Neal A. Maxwell Institute for Religious Scholarship
 * Maxwell Institute Blog

Heisenberg's Uncertainty Principle
In quantum mechanics, the uncertainty principle, also known as Heisenberg's uncertainty principle, is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as position x and momentum p, can be known simultaneously. Introduced first in 1927, by the German physicist Werner Heisenberg, it states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa. The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928:

(ħ is the reduced Planck constant).

Historically, the uncertainty principle has been confused with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting the systems. Heisenberg offered such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty. It has since become clear, however, that the uncertainty principle is inherent in the properties of all wave-like systems, and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems, and is not a statement about the observational success of current technology. It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.

Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their main research program. These include, for example, tests of number–phase uncertainty relations in superconducting or quantum optics systems. Applications dependent on the uncertainty principle for their operation include extremely low noise technology such as that required in gravitational-wave interferometers.

Introduction




As a fundamental constraint, higher level descriptions of the universe must supervene on quantum mechanical descriptions which includes Heisenberg's uncertainty relationship. However, humans do not form an intuitive understanding of this uncertainty principle in everyday life. This is because the constraint is not readily apparent on the macroscopic scales of everyday experience. So it may be helpful to demonstrate how it is integral to more easily understood physical situations. Two alternative conceptualizations of quantum physics can be examined with the goal of demonstrating the key role the uncertainty principle plays. A wave mechanics picture of the uncertainty principle provides for a more visually intuitive demonstration, and the somewhat more abstract matrix mechanics picture provides for a demonstration of the uncertainty principle that is more easily generalized to cover a multitude of physical contexts.

Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). A nonzero function and its Fourier transform cannot both be sharply localized. A similar tradeoff between the variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a sharp spike at a single frequency, while its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave. In quantum mechanics, the two key points are that the position of the particle takes the form of a matter wave, and momentum is its Fourier conjugate, assured by the de Broglie relation $p = ħk$, where $k$ is the wavenumber.

In matrix mechanics, the mathematical formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable $A$ is performed, then the system is in a particular eigenstate $Ψ$ of that observable. However, the particular eigenstate of the observable $A$ need not be an eigenstate of another observable $B$: If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.

===Wave mechanics interpretation ===

According to the de Broglie hypothesis, every object in the universe is a wave, a situation which gives rise to this phenomenon. The position of the particle is described by a wave function $$\Psi(x,t)$$. The time-independent wave function of a single-moded plane wave of wavenumber k0 or momentum p0 is


 * $$\psi(x) \propto e^{ik_0 x} = e^{ip_0 x/\hbar} ~.$$

The Born rule states that this should be interpreted as a probability density function in the sense that the probability of finding the particle between a and b is


 * $$ \operatorname P [a \leq X \leq b] = \int_a^b |\psi(x)|^2 \, \mathrm{d}x ~.$$

In the case of the single-moded plane wave, $$|\psi(x)|^2$$ is a uniform distribution. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. Consider a wave function that is a sum of many waves, however, we may write this as


 * $$\psi(x) \propto \sum_{n} A_n e^{i p_n x/\hbar}~, $$

where An represents the relative contribution of the mode pn to the overall total. The figures to the right show how with the addition of many plane waves, the wave packet can become more localized. We may take this a step further to the continuum limit, where the wave function is an integral over all possible modes


 * $$\psi(x) = \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{\infty} \phi(p) \cdot e^{i p x/\hbar}\, dp ~, $$

with $$\phi(p)$$ representing the amplitude of these modes and is called the wave function in momentum space. In mathematical terms, we say that $$\phi(p)$$ is the Fourier transform of $$\psi(x)$$ and that x and p are conjugate variables. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, having become a mixture of waves of many different momenta.

One way to quantify the precision of the position and momentum is the standard deviation σ. Since $$|\psi(x)|^2$$ is a probability density function for position, we calculate its standard deviation.

The precision of the position is improved, i.e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i.e. increased σp. Another way of stating this is that σx and σp have an inverse relationship or are at least bounded from below. This is the uncertainty principle, the exact limit of which is the Kennard bound. Click the show button below to see a semi-formal derivation of the Kennard inequality using wave mechanics.

United States
For a breakdown of membership statistics by state in the United States, see The Church of Jesus Christ of Latter-day Saints membership statistics (United States)

This page shows membership statistics of The Church of Jesus Christ of Latter-day Saints within the United States.
 * Official LDS Membership - Membership count on record provided by the Church of Jesus Christ of Latter-day Saints. These records include adults and children, and also include both active and less active members.
 * From religious surveys - General religious surveys conducted within the United States. These surveyed U.S. adults about their religious beliefs.

Official LDS Membership
Membership reported by The Church of Jesus Christ of Latter-day Saints on January 1, 2012 was used to determine the number of members in each state. The United States Census Bureau 2012 Census population estimates was used as the basis for the general population.

Table
Each state link gives a brief history and additional membership information for that state.

From religious surveys
The American Religious Identification Survey (ARIS) 2001 was based on a random digit-dialed telephone survey of 50,281 American adults in the continental U.S. Its findings are found on the map below on the left.

The Pew Forum on Religion & Public Life published a survey of 35,556 adults living in the United States that was conducted in 2007. These results are found on the map below on the right. Note: some less populated states were combined in this survey. These include:Montana-Wyoming,D.C.-Maryland, North & South Dakota, New Hampshire-Vermont, and Connecticut-Rhode Island.

2007 Pew Forum on Religion & Public Life
The 2007 Pew Forum on Religion & Public Life survey, conducted by Princeton Survey Research Associates International (PSRAI), found 1.7% of the U.S. adult population self identified themselves as Mormon. The table below lists a few significant findings, from the survey, about Mormons.

Canada
For a breakdown of membership statistics by province in Canada, see The Church of Jesus Christ of Latter-day Saints membership statistics (Canada)

The general population was taken from Statistics Canada using the first quarter 2011 population estimates. The official membership statistics as of Jan 1, 2012 by The Church of Jesus Christ of Latter-day Saints was used for all other data.

This table is sortable. The link under the names of each province corresponds to brief LDS history and other information for that particular area.