Quantum Theory: Operator Exponentiation

AI Thread Summary
The discussion focuses on demonstrating that U^n|x⟩ is an eigenstate of position x and calculating the corresponding eigenvalue for U = e^{ip}. It highlights that (e^{ip})^{n} simplifies to e^{inp}, and emphasizes the role of momentum as the generator of translations. The translation operator T(x') is noted for its property of transforming |x⟩ to |x+x'⟩, confirming it remains an eigenstate of x. Participants clarify the use of the commutator [U^n, x] in their calculations, leading to a successful resolution of the problem. The conversation concludes with a sense of accomplishment regarding the solution.
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Homework Statement



Let \left|x\right\rangle and \left|p\right\rangle denote position and momentum eigenstates, respectively. Show that U^n\left|x\right\rangle is an eigenstate for x and compute the eigenvalue, for U = e^{ip}. Show that V^n\left|p\right\rangle is an eigenstate for p and compute the eigenvalue, for V = e^{ix}.

The Attempt at a Solution



I know that (e^{ip})^{n} = e^{inp}, since p obviously commutes with itself; I also know that momentum is defined as the generator of translations, which leads to a translation operator T(x') = e^{(\frac{ipx'}{\hbar})} with the property that T(x') \left| x \right\rangle = \left|x+x'\right\rangle, which is also an eigenstate of x. Is the solution as simple as identifying U^n with the translation operator, with n in units of length/action?
 
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WisheDeom said:

Homework Statement



Let \left|x\right\rangle and \left|p\right\rangle denote position and momentum eigenstates, respectively. Show that U^n\left|x\right\rangle is an eigenstate for x and compute the eigenvalue, for U = e^{ip}. Show that V^n\left|p\right\rangle is an eigenstate for p and compute the eigenvalue, for V = e^{ix}.

The Attempt at a Solution



I know that (e^{ip})^{n} = e^{inp}, since p obviously commutes with itself; I also know that momentum is defined as the generator of translations, which leads to a translation operator T(x') = e^{(\frac{ipx'}{\hbar})} with the property that T(x') \left| x \right\rangle = \left|x+x'\right\rangle, which is also an eigenstate of x. Is the solution as simple as identifying U^n with the translation operator, with n in units of length/action?

Hint: Consider the action of the commutator [U^n, p] on the ket |x\rangle
 
I think you meant [U^n,x], and if so, I got it! Thanks a lot.
 
WisheDeom said:
I think you meant [U^n,x], and if so, I got it! Thanks a lot.

I did, and you're welcome! :smile:
 
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