Question Regarding Harmonic Oscillator Eigenkets

  • #1
Hi everyone!

Given that a harmonic oscillator has eigenkstates |n> where n = 1,2,3,..., how can we calculate <X>, <P>, <X^2>, etc. Is there a need to define a wavefunction in the |n> basis?

Thanks!
 

Answers and Replies

  • #2
dextercioby
Science Advisor
Homework Helper
Insights Author
13,024
579
Essentially no, because |n>'s are eigenkets of the number operator/Hamiltonian and X and P, though unbounded & with purely continuous spectrum, can be expressed as linear combinations of the raising & lowering ladder operators whose action on the eigenket's space becomes known once you establish that |n>'s are eigenkets of H and N.
 
  • #3
So that means just express the |n> kets as linear combinations of the ladder operators, and then use them as ψ in the formula <X> = <ψ|X|ψ>? But how would you deal with the infinite dimensionality? Will the answer be finite in that case?

Thank you by the way for the idea!
 
  • #4
649
3
Consider that
[tex]
\hat{a}=\frac{1}{\sqrt{2}}(\hat{x}+i\hat{p})
[/tex]
and
[tex]
\hat{a}^{\dagger}=\frac{1}{\sqrt{2}}(\hat{x}-i\hat{p})
[/tex]
You can use these to write [itex]\hat{x}[/itex] and [itex]\hat{p}[/itex] in terms of [itex]\hat{a}[/itex] and [itex]\hat{a}^{\dagger}[/itex]. Then you know
[tex]
\hat{a}|n\rangle = \sqrt{n}|n-1\rangle
[/tex]
and
[tex]
\hat{a}^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle
[/tex]
You have the tools to take the expectation value.
 
  • #5
Matterwave
Science Advisor
Gold Member
3,965
326
So that means just express the |n> kets as linear combinations of the ladder operators, and then use them as ψ in the formula <X> = <ψ|X|ψ>? But how would you deal with the infinite dimensionality? Will the answer be finite in that case?

Thank you by the way for the idea!
You don't express the kets as ladder operators acting on the vacuum, you express x and p as ladder operators.
 

Related Threads on Question Regarding Harmonic Oscillator Eigenkets

  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
9
Views
760
  • Last Post
Replies
1
Views
3K
Top