- #1
soul
- 62
- 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.
In summary, The eigenvalues of the L+ and L- operators can be found by multiplying out the brackets inside the square root. Remember that L+|l,m2> = Eigenvalue*|l,m2+1> and by operating with L+ on the left hand side, the eigenvalue can be moved to the front. This results in <l,m1|l,m2+1>, which, by orthogonality, is 0 unless m1 = m2+1. The dirac delta functions on the right hand side represent this relationship.
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.
FAQ: Raising and Lowering momentum operators
What is the difference between raising and lowering momentum operators?
Raising and lowering momentum operators are mathematical operators used in quantum mechanics to describe the behavior of particles. The main difference between them is that the raising operator increases the momentum of a particle by a fixed amount, while the lowering operator decreases the momentum by the same amount.
What are the properties of raising and lowering momentum operators?
The raising and lowering momentum operators have several important properties. These include being Hermitian conjugates of each other, having eigenvalues that are complex conjugates of each other, and satisfying the commutation relation [a, a†] = 1, where a is the lowering operator and a† is the raising operator.
How do raising and lowering momentum operators affect the wave function of a particle?
Raising and lowering momentum operators act on the wave function of a particle to change its momentum state. The raising operator increases the momentum while the lowering operator decreases it. This results in a change in the shape and position of the wave function.
What is the physical significance of raising and lowering momentum operators?
Raising and lowering momentum operators have physical significance in quantum mechanics as they represent the creation and annihilation of particles with a specific momentum. These operators are used to describe the behavior of particles in quantum systems and are essential in calculating the probability of different momentum states.
How are raising and lowering momentum operators related to the uncertainty principle?
The uncertainty principle states that there is a fundamental limit to how accurately we can measure certain physical quantities, such as position and momentum, simultaneously. The raising and lowering momentum operators are related to this principle as they are non-commuting operators, meaning that their measurements cannot be known with complete certainty at the same time.