- #1
Domnu
- 178
- 0
Problem
Show for the one-dimensional simple harmonic oscillator
[tex]\langle 0 | e^{ikx} | 0 \rangle = \exp{[-k^2 \langle 0 | x^2 | 0 \rangle / 2]} [/tex]
where [tex]x[/tex] is the position operator (here, [tex]k[/tex] is a number, not an operator, with dimensions 1/length).
My Solution
Well, I already know how to do this problem, but my solution isn't as clean. I was searching for a more elegant solution. Here's the outline to my solution:
1. We know that [tex]e^{ikx}|p'\rangle = |p' + \hbar k\rangle[/tex] (pretty simple to prove).
2. We can show that
3. We then just need to find the harmonic oscillator's ground state in the momentum representation, which is just a Fourier transform of the ground state in the position representation.
4. Substitute into the thing got in #2 and compute away, leading to the equality. (and this works, after expanding everything)
As we can see, computing the Fourier transform in step 3 and computing everything in step 4 takes some time, and isn't exactly the "winning" solution. So would someone help me find or post a more elegant solution?
Show for the one-dimensional simple harmonic oscillator
[tex]\langle 0 | e^{ikx} | 0 \rangle = \exp{[-k^2 \langle 0 | x^2 | 0 \rangle / 2]} [/tex]
where [tex]x[/tex] is the position operator (here, [tex]k[/tex] is a number, not an operator, with dimensions 1/length).
My Solution
Well, I already know how to do this problem, but my solution isn't as clean. I was searching for a more elegant solution. Here's the outline to my solution:
1. We know that [tex]e^{ikx}|p'\rangle = |p' + \hbar k\rangle[/tex] (pretty simple to prove).
2. We can show that
[tex]\langle 0 |e^{ikx}| 0 \rangle = \int dp' \langle 0|p' \rangle \langle p'-\hbar k | 0 \rangle[/tex]
by putting everything in the momentum basis.3. We then just need to find the harmonic oscillator's ground state in the momentum representation, which is just a Fourier transform of the ground state in the position representation.
4. Substitute into the thing got in #2 and compute away, leading to the equality. (and this works, after expanding everything)
As we can see, computing the Fourier transform in step 3 and computing everything in step 4 takes some time, and isn't exactly the "winning" solution. So would someone help me find or post a more elegant solution?