Harmonic Oscillator violating Heisenberg's Uncertainity

  • #1

Homework Statement


Does the n = 2 state of a quantum harmonic oscillator violate the Heisenberg Uncertainty Principle?

Homework Equations



$$\sigma_x\sigma_p = \frac{\hbar}{2}$$

The Attempt at a Solution


[/B]
I worked out the solution for the second state of the harmonic oscillator.
$$\frac{1}{\sqrt{2}}(\frac{mw}{\pi\hbar})^{1/4}(\frac{1}{\hbar})(e^{\frac{-mwx^2}{2\hbar}}(2mwx^2 - \hbar)$$

Should I be solving for standard deviation of position and momentum? Or is there another way to do this problem. I'm not sure what the next step is. Thank you for the guidance.
 

Answers and Replies

  • #2
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,992
6,787
Should I be solving for standard deviation of position and momentum?
Yes.

Or is there another way to do this problem.
If you work in terms of creation and annihilation operators you never need to compute the wave function at all.
 
  • #3
Yes.

If you work in terms of creation and annihilation operators you never need to compute the wave function at all.

You mean by using the following and the momentum operator in terms of creation/annihilation operators?
$$x = \sqrt{\frac{\hbar}{2mw}}(a_+ + a_{-})$$
 
  • #4
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,992
6,787
Along with the properly normalised definitions of the energy eigenstates ##|n\rangle##, yes.
 
  • #5
Along with the properly normalised definitions of the energy eigenstates ##|n\rangle##, yes.

Edit: I figured it out I got standard deviation of x * standard deviation of p = $$2\hbar$$. Which works out.
 
  • #6
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,992
6,787
I just worked through it and after evaluating the limit over all space I getting that the standard deviation of the expectation value of position is 0. Does this make sense?
No, it does not. However, it is impossible to see where you went wrong without seeing your maths.
 
  • Like
Likes Safder Aree
  • #7
No, it does not. However, it is impossible to see where you went wrong without seeing your maths.

I realized I messed up the <x^2> calculation. Thank you for your help!
 
  • #8
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,992
6,787
So the challenge is now to generalise the problem to the state ##|n\rangle## and then to an arbitrary linear combination of the ##|n\rangle##.
 
  • #9
So the challenge is now to generalise the problem to the state ##|n\rangle## and then to an arbitrary linear combination of the ##|n\rangle##.

I was actually working through Griffiths problems just now and I've seen the nth generalization, very cool stuff.
 

Related Threads on Harmonic Oscillator violating Heisenberg's Uncertainity

Replies
10
Views
3K
  • Last Post
Replies
0
Views
982
  • Last Post
Replies
10
Views
9K
Replies
6
Views
627
Replies
0
Views
3K
  • Last Post
Replies
4
Views
566
Replies
3
Views
3K
  • Last Post
Replies
2
Views
4K
Replies
12
Views
853
Replies
8
Views
2K
Top