Raising and lowering operators

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In summary, raising and lowering operators are mathematical operators used to increase or decrease the energy level or angular momentum of a function or quantum state. They have the main purpose of simplifying calculations and are widely used in quantum mechanics and the theory of angular momentum. These operators work by acting on a quantum state and are defined as linear combinations of position and momentum operators. They are related to each other as adjoints, with the raising operator being the Hermitian conjugate of the lowering operator. Some examples of raising and lowering operators include ladder operators for the harmonic oscillator, spin operators, and creation and annihilation operators for photons in quantum optics.
  • #1
intervoxel
195
1
I'm confused about these two forms of the raising/lowering operators for the harmonic oscillator.
When each one is used?

[tex]
a_+\psi_n=i\sqrt{(n+1)\hbar\omega} \psi_{n+1}
[/tex]
[tex]
a_-\psi_n=-i\sqrt{n\hbar\omega} \psi_{n-1}
[/tex]

[tex]
a_+|\psi_n\rangle=\sqrt{n+1} |\psi_{n+1}\rangle
[/tex]
[tex]
a_-|\psi_n\rangle=\sqrt{n} |\psi_{n-1}\rangle
[/tex]
 
Last edited:
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  • #2
what?
 
  • #3
ansgar said:
what?

Oops! I had pushed the submit button before ending the post. Now it is corrected.
 
  • #4
it depends if you use bra-ket or "function" notation for your physics.
 

Related to Raising and lowering operators

What are raising and lowering operators?

Raising and lowering operators are mathematical operators that act on a function or a quantum state to increase or decrease its energy level or angular momentum.

What is the purpose of raising and lowering operators?

The main purpose of raising and lowering operators is to simplify mathematical calculations and make them more efficient. They also have important applications in quantum mechanics and the theory of angular momentum.

How do raising and lowering operators work?

Raising and lowering operators are defined as linear combinations of the position and momentum operators. In quantum mechanics, they act on a quantum state to increase or decrease its energy level or angular momentum by a set amount.

What is the relationship between raising and lowering operators?

Raising and lowering operators are related to each other in that they are each other's adjoint. This means that the raising operator is the Hermitian conjugate of the lowering operator, and vice versa.

What are some examples of raising and lowering operators?

Some examples of raising and lowering operators include the ladder operators for the harmonic oscillator, the spin raising and lowering operators, and the creation and annihilation operators for photons in quantum optics.

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