# 6.3 Development of quantum theory  (Page 3/25)

 Page 3 / 25 (a) The interference pattern for electrons passing through very closely spaced slits demonstrates that quantum particles such as electrons can exhibit wavelike behavior. (b) The experimental results illustrated here demonstrate the wave–particle duality in electrons. The electrons pass through very closely spaced slits, forming an interference pattern, with increasing numbers of electrons being recorded from the left image to the right. With only a few electrons recorded, it is clear that the electrons arrive as individual localized “particles,” but in a seemingly random pattern. As more electrons arrive, a wavelike interference pattern begins to emerge. Note that the probability of the final electron location is still governed by the wave-type distribution, even for a single electron, but it can be observed more easily if many electron collisions have been recorded.

## Calculating the wavelength of a particle

If an electron travels at a velocity of 1.000 $×$ 10 7 m s –1 and has a mass of 9.109 $×$ 10 –28 g, what is its wavelength?

## Solution

We can use de Broglie’s equation to solve this problem, but we first must do a unit conversion of Planck’s constant. You learned earlier that 1 J = 1 kg m 2 /s 2 . Thus, we can write h = 6.626 $×$ 10 –34 J s as 6.626 $×$ 10 –34 kg m 2 /s.

$\lambda =\phantom{\rule{0.2em}{0ex}}\frac{h}{mv}$
$\begin{array}{}\\ \\ =\phantom{\rule{0.2em}{0ex}}\frac{6.626\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-34}{\phantom{\rule{0.2em}{0ex}}\text{kg m}}^{\text{2}}\text{/s}}{\left(9.109\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-31}\phantom{\rule{0.2em}{0ex}}\text{kg}\right)\left(1.000\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)}\phantom{\rule{0.2em}{0ex}}\\ =\phantom{\rule{0.2em}{0ex}}7.274\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-11}\text{m}\end{array}$

This is a small value, but it is significantly larger than the size of an electron in the classical (particle) view. This size is the same order of magnitude as the size of an atom. This means that electron wavelike behavior is going to be noticeable in an atom.

Calculate the wavelength of a softball with a mass of 100 g traveling at a velocity of 35 m s –1 , assuming that it can be modeled as a single particle.

1.9 $×$ 10 –34 m.

We never think of a thrown softball having a wavelength, since this wavelength is so small it is impossible for our senses or any known instrument to detect (strictly speaking, the wavelength of a real baseball would correspond to the wavelengths of its constituent atoms and molecules, which, while much larger than this value, would still be microscopically tiny). The de Broglie wavelength is only appreciable for matter that has a very small mass and/or a very high velocity.

Werner Heisenberg considered the limits of how accurately we can measure properties of an electron or other microscopic particles. He determined that there is a fundamental limit to how accurately one can measure both a particle’s position and its momentum simultaneously. The more accurately we measure the momentum of a particle, the less accurately we can determine its position at that time, and vice versa. This is summed up in what we now call the Heisenberg uncertainty principle    : It is fundamentally impossible to determine simultaneously and exactly both the momentum and the position of a particle . For a particle of mass m moving with velocity v x in the x direction (or equivalently with momentum p x ), the product of the uncertainty in the position, Δ x , and the uncertainty in the momentum, Δ p x , must be greater than or equal to $\phantom{\rule{0.2em}{0ex}}\frac{\hslash }{2}$ (recall that $\hslash \phantom{\rule{0.2em}{0ex}}\text{​}=\phantom{\rule{0.2em}{0ex}}\frac{h}{2\pi },$ the value of Planck’s constant divided by 2 π ).

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

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