Change for position to energy basis

  • Thread starter lrf
  • Start date
  • #1
lrf
4
0

Homework Statement



Give expressions for computing the matrix elements Xmn of the matrix X representing the position operator X in the energy basis (using eigenvectors of the Harmiltonian operator)

Also told to consider the example of the harmonic oscillator where energy eigenvalues are En=(1/2+n)hω

Homework Equations



Xmn=<em|X|en>

H|en>=En|en>

The Attempt at a Solution



I'm thrown off a bit by how Xmn is defined here - if it is originally in the |x> basis, why is Xmn defined using |em> and |en>. Shouldn't these be inserted using the completeness relation to convert the matrix into the energy basis representation?

Here goes...
Xmn=<em|X|en>
Xmn=ƩƩ<em|x><x|X|x'><x'|en>
Xmn=ƩƩem(x)X(x,x')en(x')
 

Answers and Replies

  • #2
dextercioby
Science Advisor
Homework Helper
Insights Author
13,138
689
You can use the algebraic formulation of the harmonic oscilator and write X in terms of a and a^dagger. The action of a and a^dagger on the standard basis (eigenvectors of N and H) is already known, so...
 
  • #3
lrf
4
0
You can use the algebraic formulation of the harmonic oscilator and write X in terms of a and a^dagger.

Sorry, still very confused!

So use -h2/2m d2ψ/dx2+1/2mω2x2ψ=Eψ how?
 
  • #4
lrf
4
0
or use 1/2P2+1/2m2X2=H ?
 
  • #5
dextercioby
Science Advisor
Homework Helper
Insights Author
13,138
689
No, X and P need to be replaced by the raising and the lowering ladder operators, a and [itex] a^{\dagger} [/itex]. You should be familiar with them, I hope...
 
  • #6
lrf
4
0
yes, got it now, thank you for the push in the right direction!
 

Related Threads on Change for position to energy basis

  • Last Post
Replies
2
Views
2K
Replies
1
Views
2K
  • Last Post
Replies
3
Views
4K
Replies
1
Views
716
Replies
5
Views
2K
Replies
1
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
1K
Replies
4
Views
3K
Replies
3
Views
3K
Top