What is the role of spherical harmonics in quantum mechanics?

In summary, the conversation discusses the use of spherical harmonics as a basis in quantum mechanics, specifically in relation to their completeness and orthonormality. The conversation also touches on the use of spherical coordinates and their usefulness in studying rotational symmetries and central forces. It is confirmed that |θφ> can be treated as a basis in Hilbert space, similar to the use of |x> in potential wells.
  • #1
KostasV
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Hello people !
I have been studying Zettili's book of quantum mechanics and found that spherical harmonics are written <θφ|L,M>.
Does this mean that |θφ> is a basis? What is more, is it complete and orthonormal basis in Hilbert?
More evidence that it is a basis, in the photo i uploaded , in (5.163) it seems that he "opens" a complete basis to orthonormalization condition ... Furthermore, why does it have integrals of φ and θ ?
I am confussed ... I have never heared about |θφ> basis ... (In expression |θφ> is it a product between θ and φ or should be a decimal point between them like in |L,M> ? )
I am asking because in potential wells we write the wave function like this <x|y> in position representation. So we use the |x> basis there.
I want to know if i can treat |θφ> Like i treat |x> as a base.
Thank you!
 
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  • #2
Hello Kostas, welcome to PF :smile: !

Seems you have seem so much detail that you have lost a little bit of the oversight.
Yes, ##\theta## and ##\phi## are coordinates. Spherical coordinates, very useful when studying situations with rotational symmetries such as with central forces. Spherical symmetry allows separation of variables and this section looks at angular dependences only.

And yes, you can treat ## | r, \theta, \phi > ## and its subspace ## | \theta, \phi > ## as bases in Hilbert space, in the same way as ## | x, y, z > ##
 
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Likes KostasV
  • #3
Thank u very much for the welcome and the answer ! :)
 

Related to What is the role of spherical harmonics in quantum mechanics?

1. What are spherical harmonics?

Spherical harmonics are a set of mathematical functions that describe the behavior of waves in a spherical environment. They are commonly used in physics and engineering to describe phenomena such as electromagnetic radiation, sound waves, and quantum mechanical systems.

2. How are spherical harmonics used in science?

Spherical harmonics are used in a variety of scientific fields, including physics, astronomy, and geophysics. They are particularly useful for describing the behavior of waves in spherical systems, which are common in nature. They are also used in image processing and computer graphics to represent 3D objects and lighting effects.

3. What is a basis in relation to spherical harmonics?

A basis is a set of functions that can be combined to represent other functions. In the case of spherical harmonics, they form a complete set of functions that can be used to describe any wave in a spherical environment. This means that any spherical harmonic can be expressed as a linear combination of other spherical harmonics, making them a powerful tool for analyzing and modeling complex systems.

4. How are spherical harmonics different from regular harmonics?

Spherical harmonics are similar to regular harmonics in that they both describe the behavior of waves. However, spherical harmonics are specifically designed to describe waves in a spherical environment, while regular harmonics are more general and can describe waves in any shape or form. Spherical harmonics also have unique properties, such as orthogonality and symmetry, that make them well-suited for certain applications.

5. What are some real-world applications of spherical harmonics?

Spherical harmonics have numerous applications in science and engineering. They are commonly used in quantum mechanics to describe the behavior of electrons in atoms and molecules. They are also used in geophysics to model the Earth's magnetic field and in atmospheric science to study weather patterns. In addition, spherical harmonics are used in computer graphics and virtual reality to create realistic lighting and shading effects.

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