<< Chapter < Page Chapter >> Page >
λ = h m v = h p

This is called the de Broglie wavelength . Unlike the other values of λ discussed in this chapter, the de Broglie wavelength is a characteristic of particles and other bodies, not electromagnetic radiation (note that this equation involves velocity [ v , m/s], not frequency [ ν , Hz]. Although these two symbols are identical, they mean very different things). Where Bohr had postulated the electron as being a particle orbiting the nucleus in quantized orbits, de Broglie argued that Bohr’s assumption of quantization can be explained if the electron is considered not as a particle, but rather as a circular standing wave such that only an integer number of wavelengths could fit exactly within the orbit ( [link] ).

This figure includes a circle formed from a dashed line. A sinusoidal wave pattern indicated with a solid red line is wrapped around the circle, centered about the edge of the circle. Line segments extend outward from the circle extending through 2 wave crests along the circle. A double ended arrow is drawn between these segments and is labeled, “wavelength, lambda.” A dashed double headed arrow is drawn from the center to the edge of the circle and is labeled, “radius r.”
If an electron is viewed as a wave circling around the nucleus, an integer number of wavelengths must fit into the orbit for this standing wave behavior to be possible.

For a circular orbit of radius r , the circumference is 2 πr , and so de Broglie’s condition is:

2 π r = n λ , n = 1 , 2 , 3 ,

Since the de Broglie expression relates the wavelength to the momentum and, hence, velocity, this implies:

2 π r = n λ = n h p = n h m v = n h r m v r = n h r L

This expression can be rearranged to give Bohr’s formula for the quantization of the angular momentum:

L = n h 2 π = n

Classical angular momentum L for a circular motion is equal to the product of the radius of the circle and the momentum of the moving particle p .

L = r p = r m v (for a circular motion)
The diagram shows a blue circle. At the center, there is an arrow labeled, “L,” which points upward. Another arrow labeled, “r,” points from the center to the edge of the circle. Another arrow labeled, “m times v” extends from the point where the r-labeled arrow reaches the edge of the circle.
The diagram shows angular momentum for a circular motion.

Shortly after de Broglie proposed the wave nature of matter, two scientists at Bell Laboratories, C. J. Davisson and L. H. Germer , demonstrated experimentally that electrons can exhibit wavelike behavior by showing an interference pattern for electrons travelling through a regular atomic pattern in a crystal. The regularly spaced atomic layers served as slits, as used in other interference experiments. Since the spacing between the layers serving as slits needs to be similar in size to the wavelength of the tested wave for an interference pattern to form, Davisson and Germer used a crystalline nickel target for their “slits,” since the spacing of the atoms within the lattice was approximately the same as the de Broglie wavelengths of the electrons that they used. [link] shows an interference pattern. It is strikingly similar to the interference patterns for light shown in [link] . The wave–particle duality of matter can be seen in [link] by observing what happens if electron collisions are recorded over a long period of time. Initially, when only a few electrons have been recorded, they show clear particle-like behavior, having arrived in small localized packets that appear to be random. As more and more electrons arrived and were recorded, a clear interference pattern that is the hallmark of wavelike behavior emerged. Thus, it appears that while electrons are small localized particles, their motion does not follow the equations of motion implied by classical mechanics, but instead it is governed by some type of a wave equation that governs a probability distribution even for a single electron’s motion. Thus the wave–particle duality first observed with photons is actually a fundamental behavior intrinsic to all quantum particles.

Questions & Answers

what is zero gravity
Blessing Reply
every object is that zero gravity
Rabiu
Probably when an object is in space and there are no nearby masses that pull her, and exert gravity
Abdelkarim
Alright. .good job
Rabiu
And all majesty to God, (وَهُوَ ٱلَّذِی خَلَقَ ٱلَّیۡلَ وَٱلنَّهَارَ وَٱلشَّمۡسَ وَٱلۡقَمَرَۖ كُلࣱّ فِی فَلَكࣲ یَسۡبَحُونَ) [سورة الأنبياء 33 And it is He who created the night and the day and the sun and the moon; all [heavenly bodies] in an orbit are swimming. General theory of relativity in Qur
Abdelkarim
what is lattice energy
Getrude Reply
why is CO a neutral oxide and CO2 an acidic oxide
Emmanuel Reply
Because when CO2 dissolves in water forming a weak acid. CO does not dissolve in water as it has strong triple bond.
Abdelkarim
What is acid
Progress Reply
which donate H+ or accept lone pair of electron
Kajal
kinetic theory of matter and gas law
Victoria Reply
hi
Victoria
pls explain
Victoria
what is clay
Thankgod Reply
material containing clay minerals. Clays develop plasticity when wet, due to a molecular film of water surrounding the clay particles, but become hard, brittle and non–plastic upon drying or firing. Most pure clay minerals are white or light-coloured, but natural clays show a variety of colours
Abdelkarim
due iron oxide. The four types of clay are Earthenware clay, Stoneware clay, Ball clay, and Porcelain. All of them can be used to make pottery, but the end result would differ a lot thanks to their different textures, colors, and flexibilities.
Abdelkarim
And do you know that god has created human from clay (وَلَقَدۡ خَلَقۡنَا ٱلۡإِنسَـٰنَ مِن صَلۡصَـٰلࣲ مِّنۡ حَمَإࣲ مَّسۡنُونࣲ) [سورة الحجر 26] And We did certainly create man out of clay from an altered black mud. You can install Quran from paly store for free with translations.
Abdelkarim
darw a periodic table
Hazard Reply
draw a periodic table
Hazard
You will arrange the elements into row and coloumns according to increasing proton number. You may want to use symbols or their names. Hydrogen, Helium, etc. God has created all these elements from nothing, in Islam we know God is the creator.
Abdelkarim
why are you drawing a periodic table? why not just print one from the internet and use as a reference
Jakhari
Great thought
Bright
how are you?
Abel Reply
alright , how about you
Marina
am fine
Agbo
your name is Agbo?
Marina
my name is amel
Farid
l use the email of my husband
Farid
Define organic chemistry
Edward Reply
It is the chemistry concerning molecules that have Carbon skeletons and hydrogen atoms. We find organic molecules like in plants, living derivatives, etc.
Abdelkarim
what's matter
Joshua Reply
Anything that can be to cutting from all dimensions to halve. So you end up with 4 cubes of 5 cm side. Repeat with one of the cubes. 10, 5, 2.5, .., 0 1st 2nd 3rd Nth Un= a(r) ^ n-1
Abdelkarim
Anything that has mass and can reflect or absorb waves. GOD created everything from nothing only he can destroy it as prooved.
Abdelkarim
Suppose you have a cube of side 10 cm. Then you start cutting from all dimensions to halve. So you end up with 4 cubes of 5 cm side. Repeat with one of the cubes. 10, 5, 2.5, .., 0 1st 2nd 3rd Nth Un= a(r) ^ n-1 0= 10 (1/2)^n-1 0= (1/2) ^ n-1 Log0= (n-1) Log(1/2) - infinity =( n-1)
Abdelkarim
matter is anything that has mass,volume and can occupy space
Getrude
what is electrolysis
Rabi Reply
good equation
Aliyu
differenciate between fat and oil
Mustapha Reply
what is the meaning of coordinate bond
Asmaa Reply
It is the alternative for dative which is a covalent bond but both electrons of the pair are from shared from the same (one) atom.
Abdelkarim
can someone please tell me what does an Entropy means
Afiwape Reply
what is chemistry?
Archie Reply
what is chemistry
Afiwape
chemistry is a brach of science which deal with the study of the nature, composition structure and with the force that hold the structure together and the change matter will undergo undedifferent conditions
Afiwape
And god has created everything from nothing
Abdelkarim

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Chemistry' conversation and receive update notifications?

Ask