Creation Operator of Harmonic Oscillator

In summary, the creation operator of harmonic oscillator is a mathematical operator used in quantum mechanics to describe the creation of a particle in a harmonic oscillator potential. It acts on a quantum state to increase the number of particles in that state by one and is also used to calculate the expectation value of the energy of the system. Its commutation relation with the annihilation operator indicates that it is a fundamental concept in quantum mechanics. It is related to the harmonic oscillator wave function through Hermite polynomials and plays a crucial role in the quantization of the harmonic oscillator and in quantum field theory.
  • #1
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For creation operator of hamonic oscillator, we have

[tex]a^\dagger |n> = \sqrt{n+1}|n+1>[/tex]

if I consider the creation operator operate on the bar vector, should I also get the same thing? namely

[tex]<n|a^\dagger = \sqrt{n+1}<n+1|[/tex]
 
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  • #2
No. Take the hermitian conjugate of your first equation; what do you get?
 
  • #3
You have to consider the dual correspondence:

[tex] X|\alpha > [/tex] corresponds to [tex] < \alpha | X^{\dagger} [/tex]

Also you can consider the number operator: [tex] N = a^{\dagger}a [/tex], with [tex] N|n> = n |n> [/tex]

Sandwhich it between: [tex] <n|N|n> [/tex], try it!
 

Related to Creation Operator of Harmonic Oscillator

What is the Creation Operator of Harmonic Oscillator?

The creation operator of harmonic oscillator is a mathematical operator used in quantum mechanics to describe the creation of a particle in a harmonic oscillator potential. It is represented by the symbol a+ and it acts on the energy eigenstates of the system.

How does the Creation Operator of Harmonic Oscillator work?

The creation operator acts on a quantum state to increase the number of particles in that state by one. For example, if the state has n particles, the application of the creation operator will result in a state with n+1 particles. It is also used to calculate the expectation value of the energy of the system.

What is the commutation relation of the Creation Operator of Harmonic Oscillator?

The creation operator of harmonic oscillator has a commutation relation with the annihilation operator, represented by a, given by [a, a+] = 1. This indicates that the creation and annihilation operators do not commute, which is a characteristic of quantum systems.

What is the significance of the Creation Operator of Harmonic Oscillator in quantum mechanics?

The creation operator of harmonic oscillator is a fundamental concept in quantum mechanics, used to describe the dynamics of a system with a harmonic potential. It plays a crucial role in the quantization of the harmonic oscillator and is also used in the creation and annihilation of particles in quantum field theory.

How is the Creation Operator of Harmonic Oscillator related to the Harmonic Oscillator wave function?

The creation operator of harmonic oscillator is related to the harmonic oscillator wave function through the Hermite polynomials. The wave function is given by the product of the Hermite polynomial and the Gaussian function, and the creation operator acts on this wave function to produce higher energy states.

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