Angles made by angular momentum vector with magnetic field

In summary, for an orbital quantum number of l=3, the possible angles that the orbital angular momentum vector (L) can make with the z axis are 30, 54.7, and 73.2 degrees. The book's given answers of 60, 35.3, and 16.8 degrees may be incorrect as they can only be obtained by using sine instead of cosine and may be defining the angle in a different location.
  • #1
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Homework Statement



1) For l=3, find the possible angles that the orbital angular momentum vector(L) makes with the z axis? Here the magnetic field acts along the z axis. l is the orbital quantum number.


Homework Equations





The Attempt at a Solution



I solved it in the following way:
Let m(l) represent the magnetic quantum number. Phi is the angle between L and z axis. For l=3, m(l)=-3,-2,-1,0,1,2,3
m(l)x(h/2(pi)) = L[cos(phi)]{[lx(l+1)]^(1/2)}(h/2(pi))
i.e. cos(phi) = m(l)/{[lx(l+1)]^(1/2)}
For m(l) = 3,
Cos(phi) = 3/(12^(1/2))
phi = 30 degrees
For m(l) = 2,
Cos(phi) = 2/(12^(1/2))
phi = 54.7 degrees

For m(l) = 1,
Cos(phi) = 1/(12^(1/2))
phi = 73.2 degrees
But the answer given in my book is 60, 35.3 and 16.8 degrees. I would get this answer if I take sine instead of cosine.
 

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  • #2
I think your answers are correct. If I draw it out to scale, I get your values for theta. Either they made a mistake by using sin, or are for some reason defining the angle in a different place (not between L and Lz). So personally I think the book is wrong.
 
  • #3
Thanx buddy.
 

1. What is the relationship between an angular momentum vector and a magnetic field?

The angular momentum vector and magnetic field are closely related in that the direction of the angular momentum vector is always perpendicular to the direction of the magnetic field. This means that the angular momentum vector will precess around the magnetic field at a constant rate.

2. How does the strength of the magnetic field affect the angle made by the angular momentum vector?

The strength of the magnetic field directly affects the angle made by the angular momentum vector. As the magnetic field increases in strength, the angle between the two vectors will also increase, causing the angular momentum vector to precess at a faster rate.

3. Can the angle between the angular momentum vector and magnetic field change over time?

Yes, the angle between the angular momentum vector and magnetic field can change over time. This is because the angular momentum vector is constantly precessing around the magnetic field, causing the angle to change. However, the rate of precession will remain constant as long as the strength of the magnetic field does not change.

4. How does the direction of the angular momentum vector affect its angle with the magnetic field?

The direction of the angular momentum vector does not affect its angle with the magnetic field. As stated earlier, the angle between the two vectors will always be perpendicular, regardless of the direction of the angular momentum vector.

5. What are some real-world applications of the relationship between angular momentum vector and magnetic field?

The relationship between the angular momentum vector and magnetic field is used in many technical applications, such as particle accelerators, nuclear magnetic resonance imaging (MRI), and magnetic resonance spectroscopy. It is also important in understanding the behavior of celestial bodies, such as planets and stars, which have their own angular momentum and interact with magnetic fields in space.

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