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  • Define quantum number.
  • Calculate angle of angular momentum vector with an axis.
  • Define spin quantum number.

Physical characteristics that are quantized—such as energy, charge, and angular momentum—are of such importance that names and symbols are given to them. The values of quantized entities are expressed in terms of quantum numbers    , and the rules governing them are of the utmost importance in determining what nature is and does. This section covers some of the more important quantum numbers and rules—all of which apply in chemistry, material science, and far beyond the realm of atomic physics, where they were first discovered. Once again, we see how physics makes discoveries which enable other fields to grow.

The energy states of bound systems are quantized , because the particle wavelength can fit into the bounds of the system in only certain ways. This was elaborated for the hydrogen atom, for which the allowed energies are expressed as E n 1/ n 2 , where n = 1, 2, 3, ... . We define n to be the principal quantum number that labels the basic states of a system. The lowest-energy state has n = 1 , the first excited state has n = 2 , and so on. Thus the allowed values for the principal quantum number are

n = 1, 2, 3, ... . size 12{n=1, 2, 3, "." "." "." } {}

This is more than just a numbering scheme, since the energy of the system, such as the hydrogen atom, can be expressed as some function of n size 12{n} {} , as can other characteristics (such as the orbital radii of the hydrogen atom).

The fact that the magnitude of angular momentum is quantized was first recognized by Bohr in relation to the hydrogen atom; it is now known to be true in general. With the development of quantum mechanics, it was found that the magnitude of angular momentum L size 12{L} {} can have only the values

L = l l + 1 h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} l = 0, 1, 2, ... , n 1 , size 12{ left (l=0, 1, 2, "." "." "." ,n - 1 right )} {}

where l size 12{l} {} is defined to be the angular momentum quantum number    . The rule for l size 12{l} {} in atoms is given in the parentheses. Given n size 12{n} {} , the value of l size 12{l} {} can be any integer from zero up to n 1 size 12{n - 1} {} . For example, if n = 4 size 12{n=4} {} , then l size 12{l} {} can be 0, 1, 2, or 3.

Note that for n = 1 size 12{n=1} {} , l size 12{l} {} can only be zero. This means that the ground-state angular momentum for hydrogen is actually zero, not h / 2 π as Bohr proposed. The picture of circular orbits is not valid, because there would be angular momentum for any circular orbit. A more valid picture is the cloud of probability shown for the ground state of hydrogen in [link] . The electron actually spends time in and near the nucleus. The reason the electron does not remain in the nucleus is related to Heisenberg’s uncertainty principle—the electron’s energy would have to be much too large to be confined to the small space of the nucleus. Now the first excited state of hydrogen has n = 2 size 12{n=2} {} , so that l size 12{l} {} can be either 0 or 1, according to the rule in L = l l + 1 h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} . Similarly, for n = 3 size 12{n=3} {} , l size 12{l} {} can be 0, 1, or 2. It is often most convenient to state the value of l size 12{l} {} , a simple integer, rather than calculating the value of L size 12{L} {} from L = l l + 1 h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} . For example, for l = 2 size 12{l=2} {} , we see that

L = 2 2 + 1 h = 6 h = 0 . 390 h = 2 . 58 × 10 34 J s . size 12{L= sqrt {2 left (2+1 right )} { {h} over {2π} } = sqrt {6} { {h} over {2π} } =0 "." "390"h=2 "." "58" times "10" rSup { size 8{ - "34"} } " J" cdot s} {}

It is much simpler to state l = 2 size 12{l=2} {} .

As recognized in the Zeeman effect, the direction of angular momentum is quantized . We now know this is true in all circumstances. It is found that the component of angular momentum along one direction in space, usually called the z size 12{z} {} -axis, can have only certain values of L z size 12{L rSub { size 8{z} } } {} . The direction in space must be related to something physical, such as the direction of the magnetic field at that location. This is an aspect of relativity. Direction has no meaning if there is nothing that varies with direction, as does magnetic force. The allowed values of L z size 12{L rSub { size 8{z} } } {} are

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
what is physics
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
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what is viscosity?
David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Mohammed
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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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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