<< Chapter < Page Chapter >> Page >
L z = m l h size 12{L rSub { size 8{z} } =m rSub { size 8{l} } { {h} over {2π} } } {} m l = l , l + 1, ... , 1, 0, 1, ... l 1, l , size 12{ left (m rSub { size 8{l} } = - l, - l+1, "." "." "." , - 1, 0, 1, "." "." "." l - 1, l right )} {}

where L z size 12{L rSub { size 8{z} } } {} is the z size 12{z} {} -component of the angular momentum and m l size 12{m rSub { size 8{l} } } {} is the angular momentum projection quantum number. The rule in parentheses for the values of m l size 12{m rSub { size 8{l} } } {} is that it can range from l size 12{ - l} {} to l size 12{l} {} in steps of one. For example, if l = 2 size 12{l=2} {} , then m l size 12{m rSub { size 8{l} } } {} can have the five values –2, –1, 0, 1, and 2. Each m l size 12{m rSub { size 8{l} } } {} corresponds to a different energy in the presence of a magnetic field, so that they are related to the splitting of spectral lines into discrete parts, as discussed in the preceding section. If the z size 12{z} {} -component of angular momentum can have only certain values, then the angular momentum can have only certain directions, as illustrated in [link] .

The image shows two possible values of component of a given angular momentum along z-axis. One circular orbit above the original circular orbit is shown for m sub l value of plus one. Another circular orbit below the original circular orbit is shown for m sub l value of minus one. The angular momentum vector for the top circular orbit makes an angle of theta sub one with the vertical axis. The horizontal angular momentum vector at original circular orbit makes an angle of theta sub two with the vertical axis. The angular momentum vector for the bottom circular orbit makes an angle of theta sub three with the vertical axis.
The component of a given angular momentum along the z -axis (defined by the direction of a magnetic field) can have only certain values; these are shown here for l = 1 , for which m l = 1, 0, and +1 . The direction of L is quantized in the sense that it can have only certain angles relative to the z -axis.

What are the allowed directions?

Calculate the angles that the angular momentum vector L size 12{L} {} can make with the z size 12{z} {} -axis for l = 1 size 12{l=1} {} , as illustrated in [link] .

Strategy

[link] represents the vectors L size 12{L} {} and L z size 12{L rSub { size 8{z} } } {} as usual, with arrows proportional to their magnitudes and pointing in the correct directions. L size 12{L} {} and L z size 12{L rSub { size 8{z} } } {} form a right triangle, with L size 12{L} {} being the hypotenuse and L z size 12{L rSub { size 8{z} } } {} the adjacent side. This means that the ratio of L z size 12{L rSub { size 8{z} } } {} to L size 12{L} {} is the cosine of the angle of interest. We can find L size 12{L} {} and L z size 12{L rSub { size 8{z} } } {} using L = l l + 1 h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} and L z = m h size 12{L rSub { size 8{z} } =m { {h} over {2π} } } {} .

Solution

We are given l = 1 size 12{l=1} {} , so that m l size 12{m rSub { size 8{l} } } {} can be +1, 0, or −1. Thus L size 12{L} {} has the value given by L = l l + 1 h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} .

L = l l + 1 h = 2 h size 12{L= { { sqrt {l left (l+1 right )} h} over {2π} } = { { sqrt {2} h} over {2π} } } {}

L z size 12{L rSub { size 8{z} } } {} can have three values, given by L z = m l h size 12{L rSub { size 8{z} } =m rSub { size 8{l} } { {h} over {2π} } } {} .

L z = m l h = { h , m l = + 1 0, m l = 0 h , m l = 1

As can be seen in [link] , cos θ = L z /L, and so for m l =+ 1 size 12{m rSub { size 8{l} } "=+"1} {} , we have

cos θ 1 = L Z L = h 2 h = 1 2 = 0 . 707. size 12{"cos"θ rSub { size 8{1} } = { {L rSub { size 8{Z} } } over {L} } = { { { {h} over {2π} } } over { { { sqrt {2} h} over {2π} } } } = { {1} over { sqrt {2} } } =0 "." "707"} {}

Thus,

θ 1 = cos 1 0.707 = 45 . 0º.

Similarly, for m l = 0 size 12{m rSub { size 8{l} } =0} {} , we find cos θ 2 = 0 size 12{"cos"θ rSub { size 8{2} } =0} {} ; thus,

θ 2 = cos 1 0 = 90 . 0º. size 12{θ rSub { size 8{2} } ="cos" rSup { size 8{ - 1} } 0="90" "." 0°} {}

And for m l = 1 size 12{m rSub { size 8{l} } = - 1} {} ,

cos θ 3 = L Z L = h 2 h = 1 2 = 0 . 707, size 12{"cos"θ rSub { size 8{3} } = { {L rSub { size 8{Z} } } over {L} } = { { - { {h} over {2π} } } over { { { sqrt {2} h} over {2π} } } } = - { {1} over { sqrt {2} } } = - 0 "." "707"} {}

so that

θ 3 = cos 1 0 . 707 = 135 . 0º. size 12{θ rSub { size 8{3} } ="cos" rSup { size 8{ - 1} } left ( - 0 "." "707" right )="135" "." 0°} {}

Discussion

The angles are consistent with the figure. Only the angle relative to the z size 12{z} {} -axis is quantized. L size 12{L} {} can point in any direction as long as it makes the proper angle with the z size 12{z} {} -axis. Thus the angular momentum vectors lie on cones as illustrated. This behavior is not observed on the large scale. To see how the correspondence principle holds here, consider that the smallest angle ( θ 1 in the example) is for the maximum value of m l = 0 , namely m l = l . For that smallest angle,

cos θ = L z L = l l l + 1 , size 12{"cos"θ= { {L rSub { size 8{z} } } over {L} } = { {l} over { sqrt {l left (l+1 right )} } } } {}

which approaches 1 as l size 12{l} {} becomes very large. If cos θ = 1 size 12{"cos"θ=1} {} , then θ = . Furthermore, for large l , there are many values of m l , so that all angles become possible as l gets very large.

Got questions? Get instant answers now!

Intrinsic spin angular momentum is quantized in magnitude and direction

There are two more quantum numbers of immediate concern. Both were first discovered for electrons in conjunction with fine structure in atomic spectra. It is now well established that electrons and other fundamental particles have intrinsic spin , roughly analogous to a planet spinning on its axis. This spin is a fundamental characteristic of particles, and only one magnitude of intrinsic spin is allowed for a given type of particle. Intrinsic angular momentum is quantized independently of orbital angular momentum. Additionally, the direction of the spin is also quantized. It has been found that the magnitude of the intrinsic (internal) spin angular momentum    , S size 12{S} {} , of an electron is given by

Questions & Answers

Is earth is an inertial frame?
Sahim Reply
The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system
Sahim
thanks
Irungu
Most welcome
Sahim
Hey.. I've a question.
Sahim Reply
?
Shii
Is earth inertia frame?
Sahim
only the center
Shii
What is an abucus?
Irungu
what would be the correct interrogation "what is time?" or "how much has your watch ticked?"
prakash Reply
a load of 20N on a wire of cross sectional area 8×10^-7m produces an extension of 10.4m. calculate the young modules of the material of the wire is of length 5m
Ebenezer Reply
Young's modulus = stress/strain strain = extension/length (x/l) stress = force/area (F/A) stress/strain is F l/A x
El
so solve it
Ebenezer
please
Ebenezer
two bodies x and y start from rest and move with uniform acceleration of a and 4a respectively. if the bodies cover the same distance in terms of tx and ty what is the ratio of tx to ty
Oluwatola Reply
what is cesium atoms?
prakash Reply
The atoms which form the element Cesium are known as Cesium atoms.
Naman
A material that combines with and removes trace gases from vacuum tubes.
Shankar
what is difference between entropy and heat capacity
Varun
Heat capacity can be defined as the amount of thermal energy required to warm the sample by 1°C. entropy is the disorder of the system. heat capacity is high when the disorder is high.
Chathu
I want learn physics
Vinodhini Reply
sir how to understanding clearly
Vinodhini
try to imagine everything you study in 3d
revolutionary
pls give me one title
Vinodhini
displacement acceleration how understand
Vinodhini
vernier caliper usage practically
Vinodhini
karthik sir is there
Vinodhini
what are the solution to all the exercise..?
What is realm
Vinodhini Reply
The quantum realm, also called the quantum scale, is a term of art inphysics referring to scales where quantum mechanical effects become important when studied as an isolated system. Typically, this means distances of 100 nanometers (10−9meters) or less or at very low temperature.
revolutionary
How to understand physics
Vinodhini Reply
i like physics very much
Vinodhini
i want know physics practically where used in daily life
Vinodhini
I want to teach physics very interesting to studentd
Vinodhini
hy
Aderibigbe
sir
Vinodhini
how can you build interest in physics
Prince
by reading it
Austin
understanding difficult
Vinodhini
vinodhini mam, physics is used in our day to day life in all events..... everything happening around us can be explained in the base of physics..... saying simple stories happening in our daily life and relating it to physics and questioning students about how or why its happening like that can make
revolutionary
your class more interesting
revolutionary
anything send about physics daily life
Vinodhini
How to understand easily
Vinodhini
check out "LMES" youtube channel
revolutionary
even when you see this message in your phone...it works accord to a physics principle. you touch screen works based on physics, your internet works based on physics, etc....... check out google and search for it
revolutionary
hi vinodhini
Karthi
hi
Dada
yea
Solomon
Hi guys
poojaa
hello
Abednego
what is mean by Newtonian principle of Relativity? definition and explanation with example
revolutionary Reply
what is art physics
Akinbulejo Reply
I've been trying to download a good and comprehensive textbook for physics, pls can somebody help me out?
Olanrewaju
try COLLEGE PHYSICS!! I think it will give you an edge.
Lawal
c=1/c1+c2/1+c3 what is the answer
Akinbulejo Reply
got on answers bro
smith
This may seem like a really stupid question, but is mechanical energy the same as potential energy? If not, what is the difference?
Nikki Reply
what is c=1\c1,c=2\c2,c=3\c3
Akinbulejo
mechanical energy is of two types 1: kinetic energy 2: potential energy,so, potential energy is actually the type of mechanical energy ,the mechanical due to position is designated as potential energy
Iram
Thank you!!!!!
Nikki
Can someone possibly walk me through this problem? " A worker drives a 0.500 kg spike into a rail tie with a 2.50 kg sledgehammer. The hammer hits the spike with a speed of 65.0 m/s. If one-third Of the hammer's kinetic energy is converted to the internal energy of rhe hammer and spike.
Nikki
how much does the total internal energy increase
Nikki
you know the mass and the velocity of the hammer. therefore using the equation (mv^2)/2 you can find the kinetic energy. then take one third of this value and that will be your change in internal energy. here, the important thing is that spike is stationary so it does not contribute to initial Energ
Chathu
Thabk you! :)
Nikki
what is the formula for finding the to total capacitance in series arrangement
Austin Reply
Don't know
Ugwu
C = 1/C1+1/C2+1/C3
Nangbun
c=1/c1+1/c2+1/c3
Akinbulejo
what is heat capacity?
smith Reply
Amount of heat that increases the temperature of 1 kg of matter by 1 degree(either celsius or kalvin)
Mateo
it is the ratio between the amount of heat added to an object and the temperature change
Giovanni
what is the different between specific heat capacity and latent heat capacity?
smith
Practice Key Terms 7

Get the best College physics course in your pocket!





Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask