Operator acting on ket state n

  • Context: Undergrad 
  • Thread starter Thread starter Jean-Mathys du bois
  • Start date Start date
  • Tags Tags
    Operator State
Click For Summary

Discussion Overview

The discussion revolves around the action of operators on ket states, specifically focusing on the number operator and its implications in the context of quantum harmonic oscillators. Participants explore the mathematical properties and interpretations of these operators.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant seeks assistance with the number operator's eigenvalue equation, indicating difficulty in understanding its application.
  • Another participant states that if $$\hat{N}\left|n\right\rangle =n\left|n\right\rangle$$, then a function of the operator can be expressed as $$f(\hat{N})\left|n\right\rangle =f(n)\left|n\right\rangle$$.
  • A participant summarizes the inquiry about how the operator $$(N+1)^{-1/2} \alpha$$ acts on a ket state $$|n\rangle$$, noting that $$N$$ is the number operator and $$\alpha$$ is the annihilation operator.
  • Another participant suggests interpreting non-polynomial functions of operators as Taylor series, providing an example expansion for $$(1 + N)^{-1/2}$$.
  • One participant refers to the operator $$(N+1)^{-1/2} \alpha$$ as being associated with Susskind.

Areas of Agreement / Disagreement

The discussion contains multiple viewpoints and interpretations regarding the action of operators on ket states, with no consensus reached on the specific implications or applications of these operators.

Contextual Notes

Participants express uncertainty about the mathematical steps involved in applying the operators and the interpretations of non-polynomial functions of operators. The discussion does not resolve these uncertainties.

Jean-Mathys du bois
Messages
2
Reaction score
0
TL;DR
how does the operator (N+1)^-1/2 α act on a |n> state of harmonic osciliator? N is the number operator N|n>=n|n> and α anihilation operator
I tried playing with the number's operator eigenvalues equation but couldn't get anywhere, can s/b help me out?
 
Physics news on Phys.org
If $$\hat{N}\left|n\right\rangle =n\left|n\right\rangle $$
then, by the very definition of function of operators, we have that
$$f(\hat{N})\left|n\right\rangle =f(n)\left|n\right\rangle $$
 
Reply
  • Like
Likes   Reactions: DrClaude and vanhees71
Jean-Mathys du bois said:
Summary:: how does the operator (N+1)^-1/2 α act on a |n> state of harmonic osciliator? N is the number operator N|n>=n|n> and α anihilation operator

I tried playing with the number's operator eigenvalues equation but couldn't get anywhere, can s/b help me out?
To make sense of a non-polynomial function of an operator, you can interpret it as a Taylor series:
$$(1 + N)^{-1/2} = 1 - \frac 1 2 N + \frac 3 8 N^2 \dots $$
 
The operator (N+1)^-1/2 α , i think is called Susskind
 

Similar threads

Undergrad Is the result of an operator acting on a ket vector always a ket？
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad Motivation behind the Operator-Formalism in QM
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Density matrix representation of a density operator
  • · Replies 8 ·
Replies
8
Views
2K
Graduate What happens when an operator maps a vector out of the Hilbert space?
  • · Replies 59 ·
2
Replies
59
Views
5K
Undergrad Vector operator acting on a ket gives a ket out of the state space
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Action of an Hermitian unitary operator member of SU(2)
  • · Replies 15 ·
Replies
15
Views
982
Graduate Can anyone give me the details of creating a cat state in Circuit QED?
  • · Replies 16 ·
Replies
16
Views
4K
Undergrad What is the Role of the Radial Position Operator in Quantum Mechanics?
  • · Replies 21 ·
Replies
21
Views
3K
Undergrad How to Show \(\psi_{i}\ket{\xi} = \xi_{i}\ket{\xi}\)?
  • · Replies 4 ·
Replies
4
Views
1K
Graduate Eigenvalues of Hyperfine Hamiltonian
  • · Replies 1 ·
Replies
1
Views
2K