- #1
soul
- 61
- 0
I tried to use the eigenvalue of the operators but I couldn't get the result.
Can anyone help me to understand this relationship?
Thank you.
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SUMMARY
The discussion focuses on the eigenvalues of the raising (L+) and lowering (L-) momentum operators in quantum mechanics. It is established that applying the raising operator L+ to the state |l,m2> yields the eigenvalue multiplied by the state |l,m2+1>. The orthogonality condition indicates that the inner product
- Understanding of quantum mechanics principles
- Familiarity with eigenvalues and eigenvectors
- Knowledge of Dirac notation
- Basic concepts of angular momentum operators
- Study the mathematical properties of raising and lowering operators in quantum mechanics
- Learn about the implications of orthogonality in quantum states
- Explore the role of Dirac delta functions in quantum mechanics
- Investigate the application of angular momentum in quantum systems
Students and professionals in quantum mechanics, physicists working with angular momentum, and anyone seeking to deepen their understanding of operator theory in quantum systems.
I tried to use the eigenvalue of the operators but I couldn't get the result.
Can anyone help me to understand this relationship?
Thank you.
- #2
Sisplat
- 9
- 0
if you multiply out the brackets inside the square root, you will find that they are in fact the eigenvalues of the L+ and L- operators.
Remember that L+|l,m2> = Eigenvalue*|l,m2+1>
Once you have operated with L+ on the left hand side you can move the eigenvalue out to the front as it is just a number. You are left with:
<l,m1|l,m2+1>, which, by orthogonality, is 0 unless m1 = m2+1. This is precisely what the dirac delta functions on the right hand side represent.
Remember that L+|l,m2> = Eigenvalue*|l,m2+1>
Once you have operated with L+ on the left hand side you can move the eigenvalue out to the front as it is just a number. You are left with:
<l,m1|l,m2+1>, which, by orthogonality, is 0 unless m1 = m2+1. This is precisely what the dirac delta functions on the right hand side represent.
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