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Δ x × Δ p x = ( Δ x ) ( m Δ v ) 2

This equation allows us to calculate the limit to how precisely we can know both the simultaneous position of an object and its momentum. For example, if we improve our measurement of an electron’s position so that the uncertainty in the position (Δ x ) has a value of, say, 1 pm (10 –12 m, about 1% of the diameter of a hydrogen atom), then our determination of its momentum must have an uncertainty with a value of at least

[ Δ p = m Δ v = h ( 2 Δ x ) ] = ( 1.055 × 10 −34 kg m 2 /s ) ( 2 × 1 × 1 0 −12 m ) = 5 × 1 0 −23 kg m/s .

The value of ħ is not large, so the uncertainty in the position or momentum of a macroscopic object like a baseball is too insignificant to observe. However, the mass of a microscopic object such as an electron is small enough that the uncertainty can be large and significant.

It should be noted that Heisenberg’s uncertainty principle is not just limited to uncertainties in position and momentum, but it also links other dynamical variables. For example, when an atom absorbs a photon and makes a transition from one energy state to another, the uncertainty in the energy and the uncertainty in the time required for the transition are similarly related, as Δ E Δ t 2 . As will be discussed later, even the vector components of angular momentum cannot all be specified exactly simultaneously.

Heisenberg’s principle imposes ultimate limits on what is knowable in science. The uncertainty principle can be shown to be a consequence of wave–particle duality, which lies at the heart of what distinguishes modern quantum theory from classical mechanics. Recall that the equations of motion obtained from classical mechanics are trajectories where, at any given instant in time, both the position and the momentum of a particle can be determined exactly. Heisenberg’s uncertainty principle implies that such a view is untenable in the microscopic domain and that there are fundamental limitations governing the motion of quantum particles. This does not mean that microscopic particles do not move in trajectories, it is just that measurements of trajectories are limited in their precision. In the realm of quantum mechanics, measurements introduce changes into the system that is being observed.

The quantum–mechanical model of an atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, as Bohr had argued, Erwin Schrödinger extended de Broglie’s work by incorporating the de Broglie relation into a wave equation, deriving what is today known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra, and he did so without having to invoke Bohr’s assumptions of stationary states and quantized orbits, angular momenta, and energies; quantization in Schrödinger’s theory was a natural consequence of the underlying mathematics of the wave equation. Like de Broglie, Schrödinger initially viewed the electron in hydrogen as being a physical wave instead of a particle, but where de Broglie thought of the electron in terms of circular stationary waves, Schrödinger properly thought in terms of three-dimensional stationary waves, or wavefunctions , represented by the Greek letter psi, ψ . A few years later, Max Born proposed an interpretation of the wavefunction ψ that is still accepted today: Electrons are still particles, and so the waves represented by ψ are not physical waves but, instead, are complex probability amplitudes. The square of the magnitude of a wavefunction ψ 2 describes the probability of the quantum particle being present near a certain location in space. This means that wavefunctions can be used to determine the distribution of the electron’s density with respect to the nucleus in an atom. In the most general form, the Schrödinger equation can be written as:

Questions & Answers

is methane a molecule
Okologwu Reply
calculations for solubility
malachi Reply
Whats d IUPAC Numenclature of bromine
Emmanuel Reply
The common name is therefore propyl bromide . For the IUPAC name , the prefix for bromine (bromo) is combined with the name for a three-carbon chain (propane), preceded by a number identifying the carbon atom to which the Br atom is attached, so the IUPAC name is 1-bromopropane.
What is Quantum number
Derick Reply
what are the chemical properties of group IIA Element and their atomic structure?
What is mixture
Azeez Reply
A mixture is a mix of substances that can be separated
what is quantum number
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h20 hydrates, nitrogen/dry ice lowers pressure similar to space environment when heated at what location/temp.? +or-, expect location (xyz)
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what's neuron?
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neuron or neutron?
cell of the nerve
prepare a solution of 1m iodine in 250mls of water
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Chemistry is the study of matter
chemistry is the study of matter and changes it undergoes
what is equilibrium
Fatai Reply
what is biology
biology is said to be the science of studying life and living organism including theirs physical structure,chemical processes, molecular interaction development and evolution
atomic number of sodium
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please, how man Bond are present when a methane under goes a complete combustion
moses Reply
Combustion of Methane The reactants are on the left side of the equation and the products are on the right. In the reaction, the bonds in the methane and oxygen come apart, the atoms rearrange and then re-bond to form water and carbon dioxide.
how is ethanol produced using ethene
Ethanol is manufactured by reacting ethene with steam. The reaction is reversible, and the formation of the ethanol is exothermic. Only 5% of the ethene is converted intoethanol at each pass through the reactor
Ethanol can be made by reacting ethene (from cracking crude oil fractions) with steam. A catalyst of phosphoric acid is used to ensure a fast reaction. Notice that ethanol is the only product. The process is continuous – as long as ethene and steam are fed into one end of the reaction vessel, ethano
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what are atoms
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the individual elements of matter.
tiny particles that make up a all matter.
smallest particles of an element
What is the meaning of hybridization
Differentiate between latent heat and specific latent heat of fusion and vaporization
Amos Reply
Ans: The amount of heat energy released or absorbed when a solid changing to liquid at atmospheric pressure at its melting point is known as the latent heat of fusion. while Vaporization of an element or compound is a phase transition from the liquid phase to vapor.

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Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
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