# 6.3 Development of quantum theory  (Page 4/25)

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$\text{Δ}x\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{Δ}{p}_{x}=\left(\text{Δ}x\right)\left(m\text{Δ}v\right)\ge \phantom{\rule{0.2em}{0ex}}\frac{\hslash }{2}$

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

$\left[\text{Δ}p=m\text{Δ}v=\phantom{\rule{0.2em}{0ex}}\frac{h}{\left(2\text{Δ}x\right)}\phantom{\rule{0.2em}{0ex}}\right]=\phantom{\rule{0.2em}{0ex}}\frac{\left(1.055\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-34}{\phantom{\rule{0.2em}{0ex}}\text{kg m}}^{\text{2}}\text{/s}\right)}{\left(2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1{0}^{-12}\phantom{\rule{0.2em}{0ex}}\text{m}\right)}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1{0}^{-23}\phantom{\rule{0.2em}{0ex}}\text{kg m/s}.$

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 $\frac{\hslash }{2}.$ 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 ${\text{∣}\psi \text{∣}}^{2}$ 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:

what is electricity
electricity refers to the flow electrons
John
Hi. please can you tell me more on chemical equation
When 1 or 2 chemicals react, they rearrange their atomic composition forming new compounds. The total mass before and after is the same/ constant. Chemical equations of same reactants react in same ratios e.g. 1 Na ion reacts with 1 water molecule or a multipier like 1 mole of Na ions react with 1
Abdelkarim
Mole of water molecules. In that example we multipied by 6.02*10^23 or avogadro constant (L). Or 2 Na+ ions with two water molecules. The arrow means '' to form '' Some times conditions or reactants are written above the arrow like H+ or enzyme or temper, sometimes physical states are written
Abdelkarim
Beside the chemical Aqueous (aq) which means solute dissolved on water. Solid (s) etc Some chemical equations are written next to it ΔH= # which means enthalpy change which describes if the reaction is endothermic (+) or exothermic (-).
Abdelkarim
Some are reversible and have half double arrow sign.
Abdelkarim
what is the meaning of atoma
what is theory
what is chemical compound
Lorrita
A compound is the result of chemical bonding between 2 or more different elements.
Abdelkarim
difference between Amine and amide
what is the difference between alkanal and alkanone
difference between alkanol and alkanal
Michael
whaatt
Saturday
you are not well at all
Ibrahim
is organic compounds used for drying agent
Sulfuric acid is used as a drying agent.
Abdelkarim
what is an atom
An atom is the smallest part of an element, for instance gold atoms are the smallest part of gold that can enter a reaction. An atom must consist protons and electrons of equal number. You can think of those subatomic particles as spheres, but not orbiting randomly they move in specific way in
Abdelkarim
That was partially described mathematically. As a muslim, we believe that god created all matter from nothing. He is the Able, and only who can create matter in the begging in the big bang that was described in the Quran in chapter 17 verse 30, 1400 years ago, you can read it from internet.
Abdelkarim
what is electron affinity
John
do you have a private jet
Ibrahim
what is acid
Proton donor (H+). Like in lemons, oranges and some medicines.
Abdelkarim
what is titration? and how can i get my readings?
what is electrolysis?
what is the formula you use when calculating for gas law?
favor
PV=nrt
Farid
electro-means electricity while lysis-means splitting up so electrolysis simply means splitting up by means of electric current
John
good day. how may I see previous question asked in this chat, please?
Asha
Asha, A person named Favor asked what is meant by the term electrolysis
John
what is a sample
According to google, a speciemen taken for specific testing or analysis.
Abdelkarim
how would you carry out a qualitative analysis for two samples of salt suspected to contain chloride ion, bromide ion,iodide ion and carbonate ion
You can search google for test for these ions or test for anions ingeneral which will help you as it contauns videos too.
Abdelkarim
what is acid
Olubodun
why don't you just search for what is acid
Ibrahim
This is a question you're being tested Abdelkarim stop being rude
John
an element x with electronic configurations as 2s²,2p²,3p6,combines with another element y with electronic configuration as 1s², 2p²,3p²,3p5,derive the chemical formula of the compound
These electrongic configuration are entered wrong, any way, you can use an app called peridoic table 2021 that has advabced electric configuration in your studying. Use it to learn about patterns easier. And use it to solve problem: identify elements, then figure out bonding and ration for example:
Abdelkarim
The elemtents are H and S Then the formula is H2S.
Abdelkarim
an element x with electronic configurations as 2s²,2p²,3p6,combines with another element y with electronic configuration as 1s², 2p²,3p²,3p5,derive the chemical formula of the compound
what is atomic number
Dorcas
It is the number of protons in the nucleus of an atom, which is unique for each element e.g only gold atoms have 79 protons in their nucleus.
Abdelkarim
hmm
favor
the number present in proton in nucleus of an atom
favor