Solving for <p>, <x> and <x^2> using raising and lowering operators

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Homework Help Overview

The discussion revolves around evaluating expectation values for position and momentum

using raising and lowering operators in the context of quantum mechanics, specifically for a harmonic oscillator. The original poster is tasked with showing that =

= 0 and finding an expression for .

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to set up equations for and

    but expresses uncertainty about using the operators correctly. They mention trying to set up an integral involving the raising and lowering operators and the wave function for the harmonic oscillator.

  • Some participants question the completeness of the problem statement, noting the absence of a specified state for which to compute the expectation values.
  • Others suggest that the problem may be misleading without clarification on the state, pointing out that the assumption of energy eigenstates might be necessary.

Discussion Status

The discussion is ongoing, with participants exploring the implications of the missing state information. Some guidance has been offered regarding the assumption of energy eigenstates, but no consensus has been reached on how to proceed with the calculations.

Contextual Notes

The problem lacks explicit information about the state for which the expectation values should be computed, leading to confusion among participants. There is an acknowledgment that the results may vary depending on the chosen state.

njdevils45

Homework Statement


A) Show that <x>=<p>=0
hint: use orthogonality
B) Use the raising and lowering operators to evaluate an expression for < x2 >

Homework Equations


d60319a2d0031cbc5dcae0218b0668ad.png

Also A- and A+ will definitely come in handy

The Attempt at a Solution


I tried setting up the equations for <x> and <p> but I don't know how the operators are meant to be used in this scenario. I think that the integral is meant to be set up as ∫eq1*xop*the general equation for ψn for a harmonic oscillator, however whatever I do I can't get the math to come out to 0 in the end for either.
 
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Is this the entire problem exactly as stated? Clearly it is possible to find states such that ##\langle x\rangle \neq 0## and the same for ##\langle p\rangle##.
 
Yes that's the entire problem
 
Thereis no mention of what state you should compute the expectation values for?
 
None at all. That's why I'm confused, I might just ask the professor for help on the setup to be honest
 
If the state is not mentioned the problem statement is misleading at best. Now, it is true for the energy eigenstates so this is presumably the missing assumption. It is generally not true for linear combinations of energy eigenstates that contain adjacent energy states.
 

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