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What are the allowed directions?

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

The image shows three possible values of a component of a given angular momentum along z-axis. The upper circular orbit is shown for m sub t = 1 at a distance L sub z above the origin. The vector L makes an angle of theta one with the z axis. The radius of the orbit is the component of L perpendicular to the z axis. The middle circular orbit is shown for m sub t = 0. It is in the x y plane. The vector L makes an angle of theta two of 90 degrees with the z axis. The radius of the orbit is L. The lower circular orbit is shown for m sub t = -1 at a distance L sub z below the origin. The vector L makes an angle of theta three with the z axis. The radius of the orbit is the component of L perpendicular to the z 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 = −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.


The vectors L and L z (in the z -direction) form a right triangle, where L is the hypotenuse and L z is the adjacent side. The ratio of L z to | L | is the cosine of the angle of interest. The magnitudes L = | L | and L z are given by

L = l ( l + 1 ) and L z = m .


We are given l = 1 , so ml can be + 1 , 0 , or 1 . Thus, L has the value given by

L = l ( l + 1 ) = 2 .

The quantity L z can have three values, given by L z = m l .

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

As you can see in [link] , cos θ = L z / L , so for m = + 1 , we have

cos θ 1 = L Z L = 2 = 1 2 = 0.707 .


θ 1 = cos −1 0.707 = 45.0 ° .

Similarly, for m = 0 , we find cos θ 2 = 0 ; this gives

θ 2 = cos −1 0 = 90.0 ° .

Then for m l = −1 :

cos θ 3 = L Z L = 2 = 1 2 = −0.707 ,

so that

θ 3 = cos −1 ( −0.707 ) = 135.0 ° .


The angles are consistent with the figure. Only the angle relative to the z -axis is quantized. L can point in any direction as long as it makes the proper angle with the z -axis. Thus, the angular momentum vectors lie on cones, as illustrated. 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 , namely m l = l . For that smallest angle,

cos θ = L z L = l l ( l + 1 ) ,

which approaches 1 as l becomes very large. If cos θ = 1 , then θ = 0 º . Furthermore, for large l , there are many values of m l , so that all angles become possible as l gets very large.

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Check Your Understanding Can the magnitude of L z ever be equal to L ?

No. The quantum number m = l , l + 1 ,… , 0 ,… , l 1 , l . Thus, the magnitude of L z is always less than L because < l ( l + 1 )

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Using the wave function to make predictions

As we saw earlier, we can use quantum mechanics to make predictions about physical events by the use of probability statements. It is therefore proper to state, “An electron is located within this volume with this probability at this time,” but not, “An electron is located at the position ( x , y , z ) at this time.” To determine the probability of finding an electron in a hydrogen atom in a particular region of space, it is necessary to integrate the probability density | ψ n l m | 2 over that region:

Probability = volume | ψ n l m | 2 d V ,

where dV is an infinitesimal volume element. If this integral is computed for all space, the result is 1, because the probability of the particle to be located somewhere is 100% (the normalization condition). In a more advanced course on modern physics, you will find that | ψ n l m | 2 = ψ n l m * ψ n l m , where ψ n l m * is the complex conjugate. This eliminates the occurrences of i = −1 in the above calculation.

Consider an electron in a state of zero angular momentum ( l = 0 ). In this case, the electron’s wave function depends only on the radial coordinate r . (Refer to the states ψ 100 and ψ 200 in [link] .) The infinitesimal volume element corresponds to a spherical shell of radius r and infinitesimal thickness dr , written as

Questions & Answers

why we have physics
Anil Reply
because is the study of mater and natural world
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
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Physics Reply
explain l-s coupling
Depk Reply
how can we say dirac equation is also called a relativistic equation in one word
preeti Reply
what is the electronic configration of Al
usman Reply
what's the signeficance of dirac equetion.?
Sibghat Reply
what is the effect of heat on refractive index
Nepal Reply
As refractive index depend on other factors also but if we supply heat on any system or media its refractive index decrease. i.e. it is inversely proportional to the heat.
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if we heated the ice then the refractive index be change from natural water
can someone explain normalization condition
Priyojit Reply
please tell
1 millimeter is How many metres
Darling Reply
1millimeter =0.001metre
The photoelectric effect is the emission of electrons when light shines on a material. 
Chris Reply
What is photoelectric effect
Amit Reply
it gives practical evidence of particke nature of light.
particle nature
photoelectric effect is the phenomenon of emission of electrons from a material(i.e Metal) when it is exposed to sunlight. Emitted electrons are called as photo electrons.
what are the applications of quantum mechanics to medicine?
application of quantum mechanics in medicine: 1) improved disease screening and treatment ; using a relatively new method known as BIO- BARCODE ASSAY we can detect disease-specific clues in our blood using gold nanoparticles. 2) in Genomic medicine 3) in protein folding 4) in radio theraphy(MRI)
Quantam physics ki basic concepts?
Laxmikanta Reply
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electrons have negative
Proton and meltdown has greater mass than electron. So it naturally electron will move around nucleus such as gases surrounded earth
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electron is the smallest negetive charge...An anaion i.e., negetive ion contains extra electrons. How ever an atom is neutral so it must contains proton and electron
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quantum theory tells us that both light and matter consists of tiny particles which have wave like propertise associated with them.
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plot a graph of MP against tan ( Angle/2) and determine the slope of the graph and find the error in it.
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Practice Key Terms 5

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