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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

following processes: Solid phosphorus pentachloride decomposes to liquid phosphorus trichloride and chlorine gas b. Deep blue solid copper(II) sulfate pentahydrate is heated to drive off water vapor to form white solid copper(II) sulfate
Hisham Reply
How to know periodic table oftend
Ahmed Reply
how to get atomic number of an element
Ogunleye Reply
how do you solve the examples in a much more explanatory way
Ogunleye
The reaction of aceto nitrile with propane in the presence of the acid
Explain this paragraph in short
Manish Reply
What is solid state?
Manish Reply
What is chemical reaction
Manish
transforming reactants to product(s)
Andre
process
Andre
Example of Lewis acid
Chidera Reply
Example of Lewis acid
Chidera
Chlorine
Mikidad
Anything with an empty orbital... the hydrogen ion is the most common example. BH3 is the typical example, but any metal in a coordination complex can be considered a Lewis acid.
Eszter
okay thanks
Jovial
aluminium and sulphur react to give aluminium sulfide.How many grams of Al are required to produce 100g of aluminium sulphide
Soni Reply
aluminium and sulphur react to give aluminium sulphide how many grams of Al are required to produce 100g of aluminium sulphide?
Soni
aluminium and sulphur react to give aluminium sulphide how many grams of Al are required to produce 100g of aluminium sulphide?
Soni
2Al+3S=Al2S3
galina
m(Al)=100×27×2/150=36g
galina
150 comes from?
Soni
thank you very much
Soni
molar mass of Al2S3
galina
150.158
thiru
Why can't atom be created or destroyed
Jacaranda Reply
matter simply converts to pure energy
that's nice
Meshach
explain how to distinguish ethanol from a sample of ethanoic acid by chemical test
Alice Reply
explain how ethanol can be distinguished from ethanoic acid by chemical test
Alice
Using a suitable experiment, describe how diffusion occurs in gases.
Melody Reply
when the excited energy which are in gaseous state collides with another to liberate from one place to another
Meshach
what is electrolytes?
charity Reply
substance which splits into ions during melting or dissolving
galina
on passing electric current though electrode
Kv
what is a radical
Jacob Reply
State that use law of partial pressure in a gas jar containing a gas and water what is the total pressure composed of 272cm^3 of carbon (iv) oxide were collected over water at15°c and 782mmHg pressure. calculate the volume of the dry gas at stp(SVP of water at 15°c is 12mmHg)
Aminat Reply
was Dalton's second postulate"atoms of the same kind have have similar/same mass and size" Or " the one mentioned in B here?
Maureen Reply

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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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