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Δ E = hf . size 12{ΔE = ital "hf"} {}

It might be helpful to mention some macroscopic analogies of this quantization of energy phenomena. This is like a pendulum that has a characteristic oscillation frequency but can swing with only certain amplitudes. Quantization of energy also resembles a standing wave on a string that allows only particular harmonics described by integers. It is also similar to going up and down a hill using discrete stair steps rather than being able to move up and down a continuous slope. Your potential energy takes on discrete values as you move from step to step.

Using the quantization of oscillators, Planck was able to correctly describe the experimentally known shape of the blackbody spectrum. This was the first indication that energy is sometimes quantized on a small scale and earned him the Nobel Prize in Physics in 1918. Although Planck’s theory comes from observations of a macroscopic object, its analysis is based on atoms and molecules. It was such a revolutionary departure from classical physics that Planck himself was reluctant to accept his own idea that energy states are not continuous. The general acceptance of Planck’s energy quantization was greatly enhanced by Einstein’s explanation of the photoelectric effect (discussed in the next section), which took energy quantization a step further. Planck was fully involved in the development of both early quantum mechanics and relativity. He quickly embraced Einstein’s special relativity, published in 1905, and in 1906 Planck was the first to suggest the correct formula for relativistic momentum, p = γmu size 12{p= ital "γmu"} {} .

A photo of German physicist Max Plank is shown.
The German physicist Max Planck had a major influence on the early development of quantum mechanics, being the first to recognize that energy is sometimes quantized. Planck also made important contributions to special relativity and classical physics. (credit: Library of Congress, Prints and Photographs Division via Wikimedia Commons)

Note that Planck’s constant h size 12{h} {} is a very small number. So for an infrared frequency of 10 14 Hz size 12{"10" rSup { size 8{"14"} } `"Hz"} {} being emitted by a blackbody, for example, the difference between energy levels is only Δ E = hf = ( 6 . 63 × 10 –34 J·s ) ( 10 14 Hz ) = 6 . 63 × 10 –20 J, size 12{ΔE = ital "hf""= " \( 6 "." "63 " times " 10" rSup { size 8{"–34"} } " J·s" \) \( "10" rSup { size 8{"14"} } " Hz" \) " = 6" "." "63 " times " 10" rSup { size 8{"–20"} } " J"} {} or about 0.4 eV. This 0.4 eV of energy is significant compared with typical atomic energies, which are on the order of an electron volt, or thermal energies, which are typically fractions of an electron volt. But on a macroscopic or classical scale, energies are typically on the order of joules. Even if macroscopic energies are quantized, the quantum steps are too small to be noticed. This is an example of the correspondence principle. For a large object, quantum mechanics produces results indistinguishable from those of classical physics.

Atomic spectra

Now let us turn our attention to the emission and absorption of EM radiation by gases . The Sun is the most common example of a body containing gases emitting an EM spectrum that includes visible light. We also see examples in neon signs and candle flames. Studies of emissions of hot gases began more than two centuries ago, and it was soon recognized that these emission spectra contained huge amounts of information. The type of gas and its temperature, for example, could be determined. We now know that these EM emissions come from electrons transitioning between energy levels in individual atoms and molecules; thus, they are called atomic spectra    . Atomic spectra remain an important analytical tool today. [link] shows an example of an emission spectrum obtained by passing an electric discharge through a material. One of the most important characteristics of these spectra is that they are discrete. By this we mean that only certain wavelengths, and hence frequencies, are emitted. This is called a line spectrum. If frequency and energy are associated as Δ E = hf , size 12{ΔE = ital "hf"} {} the energies of the electrons in the emitting atoms and molecules are quantized. This is discussed in more detail later in this chapter.

Emission spectrum of oxygen is shown as a band containing all colors with some distinct colors as discrete bold lines.
Emission spectrum of oxygen. When an electrical discharge is passed through a substance, its atoms and molecules absorb energy, which is reemitted as EM radiation. The discrete nature of these emissions implies that the energy states of the atoms and molecules are quantized. Such atomic spectra were used as analytical tools for many decades before it was understood why they are quantized. (credit: Teravolt, Wikimedia Commons)

It was a major puzzle that atomic spectra are quantized. Some of the best minds of 19th-century science failed to explain why this might be. Not until the second decade of the 20th century did an answer based on quantum mechanics begin to emerge. Again a macroscopic or classical body of gas was involved in the studies, but the effect, as we shall see, is due to individual atoms and molecules.

Phet explorations: models of the hydrogen atom

How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental results.

Models of the Hydrogen Atom

Section summary

  • The first indication that energy is sometimes quantized came from blackbody radiation, which is the emission of EM radiation by an object with an emissivity of 1.
  • Planck recognized that the energy levels of the emitting atoms and molecules were quantized, with only the allowed values of E = n + 1 2 hf , size 12{E= left (n+ { { size 8{1} } over { size 8{2} } } right ) ital "hf"} {} where n size 12{n} {} is any non-negative integer (0, 1, 2, 3, …).
  • h size 12{h} {} is Planck’s constant, whose value is h = 6 . 626 × 10 –34 J s. size 12{h = 6 "." "626" times " 10" rSup { size 8{"–34"} } " J " cdot " s"} {}
  • Thus, the oscillatory absorption and emission energies of atoms and molecules in a blackbody could increase or decrease only in steps of size Δ E = hf size 12{ΔE = ital "hf"} {} where f size 12{f} {} is the frequency of the oscillatory nature of the absorption and emission of EM radiation.
  • Another indication of energy levels being quantized in atoms and molecules comes from the lines in atomic spectra, which are the EM emissions of individual atoms and molecules.

Conceptual questions

Give an example of a physical entity that is quantized. State specifically what the entity is and what the limits are on its values.

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Give an example of a physical entity that is not quantized, in that it is continuous and may have a continuous range of values.

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What aspect of the blackbody spectrum forced Planck to propose quantization of energy levels in its atoms and molecules?

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If Planck’s constant were large, say 10 34 size 12{"10" rSup { size 8{"34"} } } {} times greater than it is, we would observe macroscopic entities to be quantized. Describe the motions of a child’s swing under such circumstances.

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Why don’t we notice quantization in everyday events?

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Problems&Exercises

A LiBr molecule oscillates with a frequency of 1 . 7 × 10 13 Hz. size 12{1 "." 7 times "10" rSup { size 8{"13"} } " Hz"} {} (a) What is the difference in energy in eV between allowed oscillator states? (b) What is the approximate value of n size 12{n} {} for a state having an energy of 1.0 eV?

(a) 0.070 eV

(b) 14

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The difference in energy between allowed oscillator states in HBr molecules is 0.330 eV. What is the oscillation frequency of this molecule?

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A physicist is watching a 15-kg orangutan at a zoo swing lazily in a tire at the end of a rope. He (the physicist) notices that each oscillation takes 3.00 s and hypothesizes that the energy is quantized. (a) What is the difference in energy in joules between allowed oscillator states? (b) What is the value of n size 12{n} {} for a state where the energy is 5.00 J? (c) Can the quantization be observed?

(a) 2 . 21 × 10 34 J size 12{2 "." "21" times "10" rSup { size 8{"34"} } " J"} {}

(b) 2 . 26 × 10 34 size 12{2 "." "26" times "10" rSup { size 8{"34"} } } {}

(c) No

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Questions & Answers

Suppose that a grandfather clock is running slowly; that is, the time it takes to complete each cycle is longer than it should be. Should you (@) shorten or (b) lengthen the pendulam to make the clock keep attain the preferred time?
Aj Reply
I think you shorten am not sure
Uche
shorten it, since that is practice able using the simple pendulum as experiment
Silvia
it'll always give the results needed no need to adjust the length, it is always measured by the starting time and ending time by the clock
Paul
it's not in relation to other clocks
Paul
wat is d formular for newton's third principle
Silvia
okay
Silvia
discuss under damped
Prince Reply
resistance of thermometer in relation to temperature
Ifeanyi Reply
how
Bernard
that resistance is not measured yet, it may be probably in the next generation of scientists
Paul
Is fundamental quantities under physical quantities?
Igwe Reply
please I didn't not understand the concept of the physical therapy
John Reply
physiotherapy - it's a practice of exercising for healthy living.
Paul
what chapter is this?
Anderson
this is not in this book, it's from other experiences.
Paul
please I have probably with calculate please can you please and help me out
John Reply
Sure
Gift
What is Boyce law
Sly Reply
how to convert meter per second to kilometers per hour
grace Reply
Divide with 3.6
Mateo
multiply by (km/1000m) x (3600 s/h) -> 3.6
Muhammad
2 how heat loss is prevented in a vacuum flask
Abdullah Reply
what is science
Helen
logical reasoning for a particular phenomenon.
Ajay
I don't know anything about it 😔. I'm sorry, please forgive 😔
Adarsh
due to non in contact mean no conduction and no convection bec of non conducting base and walls and also their is a grape between the layer like to take the example of thermo flask
Abdul
dimensions v²=u²+2at
Lagben Reply
what if time is not given in finding the average velocity?
Alan Reply
the magnetic circuit of a certain of the flux paths in each of the long and short sides being 25cm and 20cm reprectielectrove. there is an air gap of 2mm long in one the long sides if a flux density of 0.8weber/m is to produce in the magnet of 1500 turns..
Daniel Reply
How do you calculate precision
Sacky Reply
what module is that?
Fillemon
Chemisty 1A?
Fillemon
No it has something to do with measurements bro... What we did today in class
Sacky
Tah bra honestly I didn't understand a thing in that class..when re your Tutorials?
Fillemon
Friday bro... But the topics we did are in this app... Just try to master them quickly before the test dates... Are you done with the Maths sheet
Sacky
I eat ass
Anderson
I'll work on the maths sheet tomorrow bra @Sacky Malyenge but I'll try mastering them
Fillemon
I'll eat your mom's ass with a side of tendies
Anderson
@Fillemon Nanwaapo
Anderson
lol, hush
Emi
There are very large numbers of charged particles in most objects. Why, then, don’t most objects exhibit static electricity?
Bilkisu Reply
Because there's an equal number of negative and positive charges... objects are neutral in nature
NELSON
when a ball rolls on a smooth level ground,the motion of its centre is?
Mary Reply
what is electro magnetic field?
Mary
electromagnetic field is a special type of field been produced by electric charges..!!! like the word electro from Electricity and the word magnetic from Magnetism.. so it is more of a join field..!!!
NELSON
Electromagnetic field is caused by moving electric charge
Muhammad
when a ball rolls on a smooth level ground,the motion of its centre is?
Mumeh
Practice Key Terms 4

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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