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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

derivative of first differential equation
Haruna Reply
why static friction is greater than Kinetic friction
Ali Reply
draw magnetic field pattern for two wire carrying current in the same direction
Ven Reply
An American traveler in New Zealand carries a transformer to convert New Zealand’s standard 240 V to 120 V so that she can use some small appliances on her trip.
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What is the ratio of turns in the primary and secondary coils of her transformer?
nkombo
How electric lines and equipotential surface are mutually perpendicular?
Abid Reply
The potential difference between any two points on the surface is zero that implies È.Ŕ=0, Where R is the distance between two different points &E= Electric field intensity. From which we have cos þ =0, where þ is the angle between the directions of field and distance line, as E andR are zero. Thus
MAHADEV
sorry..E and R are non zero...
MAHADEV
By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?
Elene Reply
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there are seven basic type of wave radio waves, gyamma rays (nuclear energy), microwave,etc you can also search 🔍 on Google :-)
Shravasti
A 20MH coil has a resistance of 50 ohms and us connected in series with a capacitor to a 520MV supply
Musa Reply
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Physics is the branch of science that deals with the study of matter and the interactions it undergoes with energy
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AMIT
A 20MH coil has a resistance of 50 ohms and is connected in series with a capacitor to a 250MV supply if the circuit is to resonate at 100KHZ, Determine 1: the capacitance of the capacitor 2: the working voltage of the circuit, given that pie =3.142
Musa
Physics is the branch of science that deals with the study of matter and the interactions it undergoes with energy
Kelly
Heat is transfered by thermal contact but if it is transfered by conduction or radiation, is it possible to reach in thermal equilibrium?
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Yes, It is possible by conduction if Surface is Adiabatic
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are you asking for qualities or quantities?
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fundamental quantities are, length , mass, time, current, luminous intensity, amount of substance, thermodynamic temperature.
Shravasti
fundamental quantities are quantities that are independent of others and cannot be define in terms of other quantities there is nothing like Qualities we have only fundamental quantities which includes; length,mass,time, electric current, luminous density, temperature, amount of substance etc
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give examples of three dimensional frame of reference
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Universe
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Yes the Universe itself
Astronomy
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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